rocSOLVER LAPACK-like Functions

Other Lapack-like routines provided by rocSOLVER. These are divided into the following subcategories:

• Triangular factorizations. Based on Gaussian elimination.

• Linear-systems solvers. Based on triangular factorizations.

• Symmetric eigensolvers. Eigenproblems for symmetric matrices.

• Singular value decomposition. Singular values and related problems for general matrices.

Note

Throughout the APIs’ descriptions, we use the following notations:

• i, j, and k are used as general purpose indices. In some legacy LAPACK APIs, k could be a parameter indicating some problem/matrix dimension.

• Depending on the context, when it is necessary to index rows, columns and blocks or submatrices, i is assigned to rows, j to columns and k to blocks. \(l\) is always used to index matrices/problems in a batch.

• x[i] stands for the i-th element of vector x, while A[i,j] represents the element in the i-th row and j-th column of matrix A. Indices are 1-based, i.e. x[1] is the first element of x.

• To identify a block in a matrix or a matrix in the batch, k and \(l\) are used as sub-indices

• x_i \(=x_i\); we sometimes use both notations, \(x_i\) when displaying mathematical equations, and x_i in the text describing the function parameters.

• If X is a real vector or matrix, \(X^T\) indicates its transpose; if X is complex, then \(X^H\) represents its conjugate transpose. When X could be real or complex, we use X’ to indicate X transposed or X conjugate transposed, accordingly.

• When a matrix A is formed as the product of several matrices, the following notation is used: A=M(1)M(2)…M(t).

Triangular factorizations

rocsolver_<type>getf2_npvt()

rocblas_status rocsolver_zgetf2_npvt_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_double_complex *A, const int64_t lda, int64_t *info)

rocblas_status rocsolver_cgetf2_npvt_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_float_complex *A, const int64_t lda, int64_t *info)

rocblas_status rocsolver_dgetf2_npvt_64(rocblas_handle handle, const int64_t m, const int64_t n, double *A, const int64_t lda, int64_t *info)

rocblas_status rocsolver_sgetf2_npvt_64(rocblas_handle handle, const int64_t m, const int64_t n, float *A, const int64_t lda, int64_t *info)

rocblas_status rocsolver_zgetf2_npvt(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, rocblas_int *info)

rocblas_status rocsolver_cgetf2_npvt(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, rocblas_int *info)

rocblas_status rocsolver_dgetf2_npvt(rocblas_handle handle, const rocblas_int m, const rocblas_int n, double *A, const rocblas_int lda, rocblas_int *info)

rocblas_status rocsolver_sgetf2_npvt(rocblas_handle handle, const rocblas_int m, const rocblas_int n, float *A, const rocblas_int lda, rocblas_int *info)

GETF2_NPVT computes the LU factorization of a general m-by-n matrix A without partial pivoting.

(This is the unblocked Level-2-BLAS version of the algorithm. An optimized internal implementation without rocBLAS calls could be executed with small and mid-size matrices if optimizations are enabled (default option). For more details, see the “Tuning rocSOLVER performance” section of the Library Design Guide).

The factorization has the form

\[ A = LU \]

where L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

Note

Although this routine can offer better performance, Gaussian elimination without pivoting is not backward stable. If numerical accuracy is compromised, use the legacy-LAPACK API GETF2 routines instead.

Parameters:

• handle – [in] rocblas_handle.

• m – [in] rocblas_int. m >= 0. The number of rows of the matrix A.

• n – [in] rocblas_int. n >= 0. The number of columns of the matrix A.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the m-by-n matrix A to be factored. On exit, the factors L and U from the factorization. The unit diagonal elements of L are not stored.

• lda – [in] rocblas_int. lda >= m. Specifies the leading dimension of A.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i > 0, U is singular. U[i,i] is the first zero element in the diagonal. The factorization from this point might be incomplete.

rocsolver_<type>getf2_npvt_batched()

rocblas_status rocsolver_zgetf2_npvt_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_double_complex *const A[], const int64_t lda, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_cgetf2_npvt_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_float_complex *const A[], const int64_t lda, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_dgetf2_npvt_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, double *const A[], const int64_t lda, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_sgetf2_npvt_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, float *const A[], const int64_t lda, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_zgetf2_npvt_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetf2_npvt_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetf2_npvt_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, double *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetf2_npvt_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, float *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

GETF2_NPVT_BATCHED computes the LU factorization of a batch of general m-by-n matrices without partial pivoting.

(This is the unblocked Level-2-BLAS version of the algorithm. An optimized internal implementation without rocBLAS calls could be executed with small and mid-size matrices if optimizations are enabled (default option). For more details, see the “Tuning rocSOLVER performance” section of the Library Design Guide).

The factorization of matrix \(A_l\) in the batch has the form

\[ A_l = L_lU_l \]

where \(L_l\) is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and \(U_l\) is upper triangular (upper trapezoidal if m < n).

Note

Although this routine can offer better performance, Gaussian elimination without pivoting is not backward stable. If numerical accuracy is compromised, use the legacy-LAPACK API GETF2_BATCHED routines instead.

Parameters:

• handle – [in] rocblas_handle.

• m – [in] rocblas_int. m >= 0. The number of rows of all matrices A_l in the batch.

• n – [in] rocblas_int. n >= 0. The number of columns of all matrices A_l in the batch.

• A – [inout] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the m-by-n matrices A_l to be factored. On exit, the factors L_l and U_l from the factorizations. The unit diagonal elements of L_l are not stored.

• lda – [in] rocblas_int. lda >= m. Specifies the leading dimension of matrices A_l.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for factorization of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero element in the diagonal. The factorization from this point might be incomplete.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>getf2_npvt_strided_batched()

rocblas_status rocsolver_zgetf2_npvt_strided_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_double_complex *A, const int64_t lda, const rocblas_stride strideA, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_cgetf2_npvt_strided_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_float_complex *A, const int64_t lda, const rocblas_stride strideA, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_dgetf2_npvt_strided_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, double *A, const int64_t lda, const rocblas_stride strideA, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_sgetf2_npvt_strided_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, float *A, const int64_t lda, const rocblas_stride strideA, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_zgetf2_npvt_strided_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetf2_npvt_strided_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetf2_npvt_strided_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetf2_npvt_strided_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

GETF2_NPVT_STRIDED_BATCHED computes the LU factorization of a batch of general m-by-n matrices without partial pivoting.

(This is the unblocked Level-2-BLAS version of the algorithm. An optimized internal implementation without rocBLAS calls could be executed with small and mid-size matrices if optimizations are enabled (default option). For more details, see the “Tuning rocSOLVER performance” section of the Library Design Guide).

The factorization of matrix \(A_l\) in the batch has the form

\[ A_l = L_lU_l \]

where \(L_l\) is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and \(U_l\) is upper triangular (upper trapezoidal if m < n).

Note

Although this routine can offer better performance, Gaussian elimination without pivoting is not backward stable. If numerical accuracy is compromised, use the legacy-LAPACK-like API GETF2_STRIDED_BATCHED routines instead.

Parameters:

• handle – [in] rocblas_handle.

• m – [in] rocblas_int. m >= 0. The number of rows of all matrices A_l in the batch.

• n – [in] rocblas_int. n >= 0. The number of columns of all matrices A_l in the batch.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the m-by-n matrices A_l to be factored. On exit, the factors L_l and U_l from the factorization. The unit diagonal elements of L_l are not stored.

• lda – [in] rocblas_int. lda >= m. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for factorization of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero element in the diagonal. The factorization from this point might be incomplete.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>getrf_npvt()

rocblas_status rocsolver_zgetrf_npvt_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_double_complex *A, const int64_t lda, int64_t *info)

rocblas_status rocsolver_cgetrf_npvt_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_float_complex *A, const int64_t lda, int64_t *info)

rocblas_status rocsolver_dgetrf_npvt_64(rocblas_handle handle, const int64_t m, const int64_t n, double *A, const int64_t lda, int64_t *info)

rocblas_status rocsolver_sgetrf_npvt_64(rocblas_handle handle, const int64_t m, const int64_t n, float *A, const int64_t lda, int64_t *info)

rocblas_status rocsolver_zgetrf_npvt(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, rocblas_int *info)

rocblas_status rocsolver_cgetrf_npvt(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, rocblas_int *info)

rocblas_status rocsolver_dgetrf_npvt(rocblas_handle handle, const rocblas_int m, const rocblas_int n, double *A, const rocblas_int lda, rocblas_int *info)

rocblas_status rocsolver_sgetrf_npvt(rocblas_handle handle, const rocblas_int m, const rocblas_int n, float *A, const rocblas_int lda, rocblas_int *info)

GETRF_NPVT computes the LU factorization of a general m-by-n matrix A without partial pivoting.

(This is the blocked Level-3-BLAS version of the algorithm. An optimized internal implementation without rocBLAS calls could be executed with mid-size matrices if optimizations are enabled (default option). For more details, see the “Tuning rocSOLVER performance” section of the Library Design Guide).

The factorization has the form

\[ A = LU \]

where L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

Note: Although this routine can offer better performance, Gaussian elimination without pivoting is not backward stable. If numerical accuracy is compromised, use the legacy-LAPACK-like API GETRF routines instead.

Parameters:

• handle – [in] rocblas_handle.

• m – [in] rocblas_int. m >= 0. The number of rows of the matrix A.

• n – [in] rocblas_int. n >= 0. The number of columns of the matrix A.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the m-by-n matrix A to be factored. On exit, the factors L and U from the factorization. The unit diagonal elements of L are not stored.

• lda – [in] rocblas_int. lda >= m. Specifies the leading dimension of A.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i > 0, U is singular. U[i,i] is the first zero element in the diagonal. The factorization from this point might be incomplete.

rocsolver_<type>getrf_npvt_batched()

rocblas_status rocsolver_zgetrf_npvt_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_double_complex *const A[], const int64_t lda, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_cgetrf_npvt_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_float_complex *const A[], const int64_t lda, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_dgetrf_npvt_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, double *const A[], const int64_t lda, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_sgetrf_npvt_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, float *const A[], const int64_t lda, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_zgetrf_npvt_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetrf_npvt_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetrf_npvt_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, double *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetrf_npvt_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, float *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

GETRF_NPVT_BATCHED computes the LU factorization of a batch of general m-by-n matrices without partial pivoting.

(This is the blocked Level-3-BLAS version of the algorithm. An optimized internal implementation without rocBLAS calls could be executed with mid-size matrices if optimizations are enabled (default option). For more details, see the “Tuning rocSOLVER performance” section of the Library Design Guide).

The factorization of matrix \(A_l\) in the batch has the form

\[ A_l = L_lU_l \]

where \(L_l\) is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and \(U_l\) is upper triangular (upper trapezoidal if m < n).

Note: Although this routine can offer better performance, Gaussian elimination without pivoting is not backward stable. If numerical accuracy is compromised, use the legacy-LAPACK-like API GETRF_BATCHED routines instead.

Parameters:

• handle – [in] rocblas_handle.

• m – [in] rocblas_int. m >= 0. The number of rows of all matrices A_l in the batch.

• n – [in] rocblas_int. n >= 0. The number of columns of all matrices A_l in the batch.

• A – [inout] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the m-by-n matrices A_l to be factored. On exit, the factors L_l and U_l from the factorizations. The unit diagonal elements of L_l are not stored.

• lda – [in] rocblas_int. lda >= m. Specifies the leading dimension of matrices A_l.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for factorization of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero element in the diagonal. The factorization from this point might be incomplete.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>getrf_npvt_strided_batched()

rocblas_status rocsolver_zgetrf_npvt_strided_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_double_complex *A, const int64_t lda, const rocblas_stride strideA, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_cgetrf_npvt_strided_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, rocblas_float_complex *A, const int64_t lda, const rocblas_stride strideA, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_dgetrf_npvt_strided_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, double *A, const int64_t lda, const rocblas_stride strideA, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_sgetrf_npvt_strided_batched_64(rocblas_handle handle, const int64_t m, const int64_t n, float *A, const int64_t lda, const rocblas_stride strideA, int64_t *info, const int64_t batch_count)

rocblas_status rocsolver_zgetrf_npvt_strided_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetrf_npvt_strided_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetrf_npvt_strided_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetrf_npvt_strided_batched(rocblas_handle handle, const rocblas_int m, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

GETRF_NPVT_STRIDED_BATCHED computes the LU factorization of a batch of general m-by-n matrices without partial pivoting.

(This is the blocked Level-3-BLAS version of the algorithm. An optimized internal implementation without rocBLAS calls could be executed with mid-size matrices if optimizations are enabled (default option). For more details, see the “Tuning rocSOLVER performance” section of the Library Design Guide).

The factorization of matrix \(A_l\) in the batch has the form

\[ A_l = L_lU_l \]

where \(L_l\) is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and \(U_l\) is upper triangular (upper trapezoidal if m < n).

Note: Although this routine can offer better performance, Gaussian elimination without pivoting is not backward stable. If numerical accuracy is compromised, use the legacy-LAPACK-like API GETRF_STRIDED_BATCHED routines instead.

Parameters:

• handle – [in] rocblas_handle.

• m – [in] rocblas_int. m >= 0. The number of rows of all matrices A_l in the batch.

• n – [in] rocblas_int. n >= 0. The number of columns of all matrices A_l in the batch.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the m-by-n matrices A_l to be factored. On exit, the factors L_l and U_l from the factorization. The unit diagonal elements of L_l are not stored.

• lda – [in] rocblas_int. lda >= m. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for factorization of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero element in the diagonal. The factorization from this point might be incomplete.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>geblttrf_npvt()

rocblas_status rocsolver_zgeblttrf_npvt(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, rocblas_double_complex *A, const rocblas_int lda, rocblas_double_complex *B, const rocblas_int ldb, rocblas_double_complex *C, const rocblas_int ldc, rocblas_int *info)

rocblas_status rocsolver_cgeblttrf_npvt(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, rocblas_float_complex *A, const rocblas_int lda, rocblas_float_complex *B, const rocblas_int ldb, rocblas_float_complex *C, const rocblas_int ldc, rocblas_int *info)

rocblas_status rocsolver_dgeblttrf_npvt(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, double *A, const rocblas_int lda, double *B, const rocblas_int ldb, double *C, const rocblas_int ldc, rocblas_int *info)

rocblas_status rocsolver_sgeblttrf_npvt(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, float *A, const rocblas_int lda, float *B, const rocblas_int ldb, float *C, const rocblas_int ldc, rocblas_int *info)

GEBLTTRF_NPVT computes the LU factorization of a block tridiagonal matrix without partial pivoting.

The LU factorization of a block tridiagonal matrix

\[\begin{split} M = \left[\begin{array}{ccccc} B_1 & C_1\\ A_1 & B_2 & C_2\\ & \ddots & \ddots & \ddots \\ & & A_{n-2} & B_{n-1} & C_{n-1}\\ & & & A_{n-1} & B_n \end{array}\right] \end{split}\]

with \(n = \mathrm{nblocks}\) diagonal blocks of size nb, can be represented as

\[\begin{split} M = \left[\begin{array}{cccc} L_1 \\ A_1 & L_2\\ & \ddots & \ddots \\ & & A_{n-1} & L_n \end{array}\right] \left[\begin{array}{cccc} I & U_1 \\ & \ddots & \ddots \\ & & I & U_{n-1}\\ & & & I \end{array}\right] = LU \end{split}\]

where the blocks \(L_i\) and \(U_i\) are also general blocks of size nb.

Parameters:

• handle – [in] rocblas_handle.

• nb – [in] rocblas_int. nb >= 0. The number of rows and columns of each block.

• nblocks – [in] rocblas_int. nblocks >= 0. The number of blocks along the diagonal of the matrix.

• A – [in] pointer to type. Array on the GPU of dimension lda*nb*(nblocks-1). Contains the blocks A_i arranged one after the other.

• lda – [in] rocblas_int. lda >= nb. Specifies the leading dimension of blocks A_i.

• B – [inout] pointer to type. Array on the GPU of dimension ldb*nb*nblocks. On entry, contains the blocks B_i arranged one after the other. On exit it is overwritten by blocks L_i in factorized form as returned by GETRF_NPVT

• ldb – [in] rocblas_int. ldb >= nb. Specifies the leading dimension of blocks B_i.

• C – [inout] pointer to type. Array on the GPU of dimension ldc*nb*(nblocks-1). On entry, contains the blocks C_i arranged one after the other. On exit it is overwritten by blocks U_i.

• ldc – [in] rocblas_int. ldc >= nb. Specifies the leading dimension of blocks C_i.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i > 0, the matrix is singular.

rocsolver_<type>geblttrf_npvt_batched()

rocblas_status rocsolver_zgeblttrf_npvt_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_double_complex *const B[], const rocblas_int ldb, rocblas_double_complex *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgeblttrf_npvt_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_float_complex *const B[], const rocblas_int ldb, rocblas_float_complex *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgeblttrf_npvt_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, double *const A[], const rocblas_int lda, double *const B[], const rocblas_int ldb, double *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgeblttrf_npvt_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, float *const A[], const rocblas_int lda, float *const B[], const rocblas_int ldb, float *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

GEBLTTRF_NPVT_BATCHED computes the LU factorization of a batch of block tridiagonal matrices without partial pivoting.

The LU factorization of a block tridiagonal matrix \(M_l\) in the batch

\[\begin{split} M_l = \left[\begin{array}{ccccc} B_{l1} & C_{l1}\\ A_{l1} & B_{l2} & C_{l2}\\ & \ddots & \ddots & \ddots \\ & & A_{l(n-2)} & B_{l(n-1)} & C_{l(l-1)}\\ & & & A_{l(n-1)} & B_{ln} \end{array}\right] \end{split}\]

with \(n = \mathrm{nblocks}\) diagonal blocks of size nb, can be represented as

\[\begin{split} M_l = \left[\begin{array}{cccc} L_{l1} \\ A_{l1} & L_{l2}\\ & \ddots & \ddots \\ & & A_{l(n-1)} & L_{ln} \end{array}\right] \left[\begin{array}{cccc} I & U_{l1} \\ & \ddots & \ddots \\ & & I & U_{l(n-1)}\\ & & & I \end{array}\right] = L_lU_l \end{split}\]

where the blocks \(L_{li}\) and \(U_{li}\) are also general blocks of size nb.

Parameters:

• handle – [in] rocblas_handle.

• nb – [in] rocblas_int. nb >= 0. The number of rows and columns of each block.

• nblocks – [in] rocblas_int. nblocks >= 0. The number of blocks along the diagonal of each matrix in the batch.

• A – [in] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*nb*(nblocks-1). Contains the blocks A_{li} arranged one after the other.

• lda – [in] rocblas_int. lda >= nb. Specifies the leading dimension of blocks A_{li}.

• B – [inout] array of pointers to type. Each pointer points to an array on the GPU of dimension ldb*nb*nblocks. On entry, contains the blocks B_{li} arranged one after the other. On exit it is overwritten by blocks L_{li} in factorized form as returned by GETRF_NPVT

• ldb – [in] rocblas_int. ldb >= nb. Specifies the leading dimension of blocks B_{li}.

• C – [inout] array of pointers to type. Each pointer points to an array on the GPU of dimension ldc*nb*(nblocks-1). On entry, contains the blocks C_{li} arranged one after the other. On exit it is overwritten by blocks U_{li}.

• ldc – [in] rocblas_int. ldc >= nb. Specifies the leading dimension of blocks C_{li}.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for factorization of l-th batch instance. If info[l] = i > 0, the l-th batch instance is singular.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>geblttrf_npvt_strided_batched()

rocblas_status rocsolver_zgeblttrf_npvt_strided_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_double_complex *B, const rocblas_int ldb, const rocblas_stride strideB, rocblas_double_complex *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgeblttrf_npvt_strided_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_float_complex *B, const rocblas_int ldb, const rocblas_stride strideB, rocblas_float_complex *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgeblttrf_npvt_strided_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, double *A, const rocblas_int lda, const rocblas_stride strideA, double *B, const rocblas_int ldb, const rocblas_stride strideB, double *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgeblttrf_npvt_strided_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, float *A, const rocblas_int lda, const rocblas_stride strideA, float *B, const rocblas_int ldb, const rocblas_stride strideB, float *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

GEBLTTRF_NPVT_STRIDED_BATCHED computes the LU factorization of a batch of block tridiagonal matrices without partial pivoting.

The LU factorization of a block tridiagonal matrix \(M_l\) in the batch

\[\begin{split} M_l = \left[\begin{array}{ccccc} B_{l1} & C_{l1}\\ A_{l1} & B_{l2} & C_{l2}\\ & \ddots & \ddots & \ddots \\ & & A_{l(n-2)} & B_{l(n-1)} & C_{l(n-1)}\\ & & & A_{l(n-1)} & B_{ln} \end{array}\right] \end{split}\]

with \(n = \mathrm{nblocks}\) diagonal blocks of size nb, can be represented as

\[\begin{split} M_l = \left[\begin{array}{cccc} L_{l1} \\ A_{l1} & L_{l2}\\ & \ddots & \ddots \\ & & A_{l(n-1)} & L_{ln} \end{array}\right] \left[\begin{array}{cccc} I & U_{l1} \\ & \ddots & \ddots \\ & & I & U_{l(n-1)}\\ & & & I \end{array}\right] = L_lU_l \end{split}\]

where the blocks \(L_{li}\) and \(U_{li}\) are also general blocks of size nb.

Parameters:

• handle – [in] rocblas_handle.

• nb – [in] rocblas_int. nb >= 0. The number of rows and columns of each block.

• nblocks – [in] rocblas_int. nblocks >= 0. The number of blocks along the diagonal of each matrix in the batch.

• A – [in] pointer to type. Array on the GPU (the size depends on the value of strideA). Contains the blocks A_{li} arranged one after the other.

• lda – [in] rocblas_int. lda >= nb. Specifies the leading dimension of blocks A_{li}.

• strideA – [in] rocblas_stride. Stride from the start of one block A_{li} to the same block in the next batch instance A_{(l+1)i}. There is no restriction for the value of strideA. Normal use case is strideA >= lda*nb*nblocks.

• B – [inout] pointer to type. Array on the GPU (the size depends on the value of strideB). On entry, contains the blocks B_{li} arranged one after the other. On exit it is overwritten by blocks L_{li} in factorized form as returned by GETRF_NPVT

• ldb – [in] rocblas_int. ldb >= nb. Specifies the leading dimension of matrix blocks B_{li}.

• strideB – [in] rocblas_stride. Stride from the start of one block B_{li} to the same block in the next batch instance B_{(l+1)i}. There is no restriction for the value of strideB. Normal use case is strideB >= ldb*nb*nblocks.

• C – [inout] pointer to type. Array on the GPU (the size depends on the value of strideC). On entry, contains the blocks C_{li} arranged one after the other. On exit it is overwritten by blocks U_{li}.

• ldc – [in] rocblas_int. ldc >= nb. Specifies the leading dimension of matrix blocks C_{li}.

• strideC – [in] rocblas_stride. Stride from the start of one block B_{li} to the same block in the next batch instance B_{(l+1)i}. There is no restriction for the value of strideC. Normal use case is strideC >= ldc*nb*nblocks.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for factorization of l-th batch instance. If info[l] = i > 0, the l-th batch instance is singular.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>geblttrf_npvt_interleaved_batched()

rocblas_status rocsolver_zgeblttrf_npvt_interleaved_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, rocblas_double_complex *A, const rocblas_int inca, const rocblas_int lda, const rocblas_stride strideA, rocblas_double_complex *B, const rocblas_int incb, const rocblas_int ldb, const rocblas_stride strideB, rocblas_double_complex *C, const rocblas_int incc, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgeblttrf_npvt_interleaved_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, rocblas_float_complex *A, const rocblas_int inca, const rocblas_int lda, const rocblas_stride strideA, rocblas_float_complex *B, const rocblas_int incb, const rocblas_int ldb, const rocblas_stride strideB, rocblas_float_complex *C, const rocblas_int incc, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgeblttrf_npvt_interleaved_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, double *A, const rocblas_int inca, const rocblas_int lda, const rocblas_stride strideA, double *B, const rocblas_int incb, const rocblas_int ldb, const rocblas_stride strideB, double *C, const rocblas_int incc, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgeblttrf_npvt_interleaved_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, float *A, const rocblas_int inca, const rocblas_int lda, const rocblas_stride strideA, float *B, const rocblas_int incb, const rocblas_int ldb, const rocblas_stride strideB, float *C, const rocblas_int incc, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

GEBLTTRF_NPVT_INTERLEAVED_BATCHED computes the LU factorization of a batch of block tridiagonal matrices without partial pivoting.

The LU factorization of a block tridiagonal matrix \(M_l\) in the batch

\[\begin{split} M_l = \left[\begin{array}{ccccc} B_{l1} & C_{l1}\\ A_{l1} & B_{l2} & C_{l2}\\ & \ddots & \ddots & \ddots \\ & & A_{l(n-2)} & B_{l(n-1)} & C_{l(n-1)}\\ & & & A_{l(n-1)} & B_{ln} \end{array}\right] \end{split}\]

with \(n = \mathrm{nblocks}\) diagonal blocks of size nb, can be represented as

\[\begin{split} M_l = \left[\begin{array}{cccc} L_{l1} \\ A_{l1} & L_{l2}\\ & \ddots & \ddots \\ & & A_{l(n-1)} & L_{ln} \end{array}\right] \left[\begin{array}{cccc} I & U_{l1} \\ & \ddots & \ddots \\ & & I & U_{l(n-1)}\\ & & & I \end{array}\right] = L_lU_l \end{split}\]

where the blocks \(L_{li}\) and \(U_{li}\) are also general blocks of size nb.

Parameters:

• handle – [in] rocblas_handle.

• nb – [in] rocblas_int. nb >= 0. The number of rows and columns of each block.

• nblocks – [in] rocblas_int. nblocks >= 0. The number of blocks along the diagonal of each matrix in the batch.

• A – [in] pointer to type. Array on the GPU (the size depends on the value of strideA). Contains the blocks A_{li} arranged one after the other.

• inca – [in] rocblas_int. inca > 0. Stride from the start of one row of A_{li} to the next. Normal use cases are inca = 1 (strided batched case) or inca = batch_count (interleaved batched case).

• lda – [in] rocblas_int. lda >= inca * nb. Specifies the leading dimension of blocks A_{li}, i.e. the stride from the start of one column of A_{li} to the next.

• strideA – [in] rocblas_stride. Stride from the start of one block A_{li} to the same block in the next batch instance A_{(l+1)i}. There is no restriction for the value of strideA. Normal use cases are strideA >= lda*nb*nblocks (strided batched case) or strideA = 1 (interleaved batched case).

• B – [inout] pointer to type. Array on the GPU (the size depends on the value of strideB). On entry, contains the blocks B_{li} arranged one after the other. On exit it is overwritten by blocks L_{li} in factorized form as returned by GETRF_NPVT

• incb – [in] rocblas_int. incb > 0. Stride from the start of one row of B_{li} to the next. Normal use cases are incb = 1 (strided batched case) or incb = batch_count (interleaved batched case).

• ldb – [in] rocblas_int. ldb >= incb * nb. Specifies the leading dimension of blocks B_{li}, i.e. the stride from the start of one column of B_{li} to the next.

• strideB – [in] rocblas_stride. Stride from the start of one block B_{li} to the same block in the next batch instance B_{(l+1)i}. There is no restriction for the value of strideB. Normal use cases are strideB >= ldb*nb*nblocks (strided batched case) or strideB = 1 (interleaved batched case).

• C – [inout] pointer to type. Array on the GPU (the size depends on the value of strideC). On entry, contains the blocks C_{li} arranged one after the other. On exit it is overwritten by blocks U_{li}.

• incc – [in] rocblas_int. incc > 0. Stride from the start of one row of C_{li} to the next. Normal use cases are incc = 1 (strided batched case) or incc = batch_count (interleaved batched case).

• ldc – [in] rocblas_int. ldc >= incc * nb. Specifies the leading dimension of blocks C_{li}, i.e. the stride from the start of one column of C_{li} to the next.

• strideC – [in] rocblas_stride. Stride from the start of one block B_{li} to the same block in the next batch instance B_{(l+1)i}. There is no restriction for the value of strideC. Normal use cases are strideC >= ldc*nb*nblocks (strided batched case) or strideC = 1 (interleaved batched case).

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for factorization of l-th batch instance. If info[l] = i > 0, the l-th batch instance is singular.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

Linear-systems solvers

rocsolver_<type>getri_npvt()

rocblas_status rocsolver_zgetri_npvt(rocblas_handle handle, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, rocblas_int *info)

rocblas_status rocsolver_cgetri_npvt(rocblas_handle handle, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, rocblas_int *info)

rocblas_status rocsolver_dgetri_npvt(rocblas_handle handle, const rocblas_int n, double *A, const rocblas_int lda, rocblas_int *info)

rocblas_status rocsolver_sgetri_npvt(rocblas_handle handle, const rocblas_int n, float *A, const rocblas_int lda, rocblas_int *info)

GETRI_NPVT inverts a general n-by-n matrix A using the LU factorization computed by GETRF_NPVT.

The inverse is computed by solving the linear system

\[ A^{-1}L = U^{-1} \]

where L is the lower triangular factor of A with unit diagonal elements, and U is the upper triangular factor.

Parameters:

• handle – [in] rocblas_handle.

• n – [in] rocblas_int. n >= 0. The number of rows and columns of the matrix A.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the factors L and U of the factorization A = L*U returned by GETRF_NPVT. On exit, the inverse of A if info = 0; otherwise undefined.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of A.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i > 0, U is singular. U[i,i] is the first zero pivot.

rocsolver_<type>getri_npvt_batched()

rocblas_status rocsolver_zgetri_npvt_batched(rocblas_handle handle, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetri_npvt_batched(rocblas_handle handle, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetri_npvt_batched(rocblas_handle handle, const rocblas_int n, double *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetri_npvt_batched(rocblas_handle handle, const rocblas_int n, float *const A[], const rocblas_int lda, rocblas_int *info, const rocblas_int batch_count)

GETRI_NPVT_BATCHED inverts a batch of general n-by-n matrices using the LU factorization computed by GETRF_NPVT_BATCHED.

The inverse of matrix \(A_l\) in the batch is computed by solving the linear system

\[ A_l^{-1} L_l^{} = U_l^{-1} \]

where \(L_l\) is the lower triangular factor of \(A_l\) with unit diagonal elements, and \(U_l\) is the upper triangular factor.

Parameters:

• handle – [in] rocblas_handle.

• n – [in] rocblas_int. n >= 0. The number of rows and columns of all matrices A_l in the batch.

• A – [inout] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the factors L_l and U_l of the factorization A_l = L_l*U_l returned by GETRF_NPVT_BATCHED. On exit, the inverses of A_l if info[l] = 0; otherwise undefined.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for inversion of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero pivot.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>getri_npvt_strided_batched()

rocblas_status rocsolver_zgetri_npvt_strided_batched(rocblas_handle handle, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetri_npvt_strided_batched(rocblas_handle handle, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetri_npvt_strided_batched(rocblas_handle handle, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetri_npvt_strided_batched(rocblas_handle handle, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *info, const rocblas_int batch_count)

GETRI_NPVT_STRIDED_BATCHED inverts a batch of general n-by-n matrices using the LU factorization computed by GETRF_NPVT_STRIDED_BATCHED.

The inverse of matrix \(A_l\) in the batch is computed by solving the linear system

\[ A_l^{-1} L_l^{} = U_l^{-1} \]

where \(L_l\) is the lower triangular factor of \(A_l\) with unit diagonal elements, and \(U_l\) is the upper triangular factor.

Parameters:

• handle – [in] rocblas_handle.

• n – [in] rocblas_int. n >= 0. The number of rows and columns of all matrices A_l in the batch.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the factors L_l and U_l of the factorization A_l = L_l*U_l returned by GETRF_NPVT_STRIDED_BATCHED. On exit, the inverses of A_l if info[l] = 0; otherwise undefined.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for inversion of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero pivot.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>getri_outofplace()

rocblas_status rocsolver_zgetri_outofplace(rocblas_handle handle, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, rocblas_int *ipiv, rocblas_double_complex *C, const rocblas_int ldc, rocblas_int *info)

rocblas_status rocsolver_cgetri_outofplace(rocblas_handle handle, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, rocblas_int *ipiv, rocblas_float_complex *C, const rocblas_int ldc, rocblas_int *info)

rocblas_status rocsolver_dgetri_outofplace(rocblas_handle handle, const rocblas_int n, double *A, const rocblas_int lda, rocblas_int *ipiv, double *C, const rocblas_int ldc, rocblas_int *info)

rocblas_status rocsolver_sgetri_outofplace(rocblas_handle handle, const rocblas_int n, float *A, const rocblas_int lda, rocblas_int *ipiv, float *C, const rocblas_int ldc, rocblas_int *info)

GETRI_OUTOFPLACE computes the inverse \(C = A^{-1}\) of a general n-by-n matrix A.

The inverse is computed by solving the linear system

\[ AC = I \]

where I is the identity matrix, and A is factorized as \(A = PLU\) as given by GETRF.

Parameters:

• handle – [in] rocblas_handle.

• n – [in] rocblas_int. n >= 0. The number of rows and columns of the matrix A.

• A – [in] pointer to type. Array on the GPU of dimension lda*n. The factors L and U of the factorization A = P*L*U returned by GETRF.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of A.

• ipiv – [in] pointer to rocblas_int. Array on the GPU of dimension n. The pivot indices returned by GETRF.

• C – [out] pointer to type. Array on the GPU of dimension ldc*n. If info = 0, the inverse of A. Otherwise, undefined.

• ldc – [in] rocblas_int. ldc >= n. Specifies the leading dimension of C.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i > 0, U is singular. U[i,i] is the first zero pivot.

rocsolver_<type>getri_outofplace_batched()

rocblas_status rocsolver_zgetri_outofplace_batched(rocblas_handle handle, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_int *ipiv, const rocblas_stride strideP, rocblas_double_complex *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetri_outofplace_batched(rocblas_handle handle, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_int *ipiv, const rocblas_stride strideP, rocblas_float_complex *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetri_outofplace_batched(rocblas_handle handle, const rocblas_int n, double *const A[], const rocblas_int lda, rocblas_int *ipiv, const rocblas_stride strideP, double *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetri_outofplace_batched(rocblas_handle handle, const rocblas_int n, float *const A[], const rocblas_int lda, rocblas_int *ipiv, const rocblas_stride strideP, float *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

GETRI_OUTOFPLACE_BATCHED computes the inverse \(C_l = A_l^{-1}\) of a batch of general n-by-n matrices \(A_l\).

The inverse is computed by solving the linear system

\[ A_l C_l = I \]

where I is the identity matrix, and \(A_l\) is factorized as \(A_l = P_l L_l U_l\) as given by GETRF_BATCHED.

Parameters:

• handle – [in] rocblas_handle.

• n – [in] rocblas_int. n >= 0. The number of rows and columns of all matrices A_l in the batch.

• A – [in] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. The factors L_l and U_l of the factorization A_l = P_l*L_l*U_l returned by GETRF_BATCHED.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• ipiv – [in] pointer to rocblas_int. Array on the GPU (the size depends on the value of strideP). The pivot indices returned by GETRF_BATCHED.

• strideP – [in] rocblas_stride. Stride from the start of one vector ipiv_l to the next one ipiv_(l+1). There is no restriction for the value of strideP. Normal use case is strideP >= n.

• C – [out] array of pointers to type. Each pointer points to an array on the GPU of dimension ldc*n. If info[l] = 0, the inverse of matrices A_l. Otherwise, undefined.

• ldc – [in] rocblas_int. ldc >= n. Specifies the leading dimension of C_l.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for inversion of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero pivot.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>getri_outofplace_strided_batched()

rocblas_status rocsolver_zgetri_outofplace_strided_batched(rocblas_handle handle, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *ipiv, const rocblas_stride strideP, rocblas_double_complex *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetri_outofplace_strided_batched(rocblas_handle handle, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *ipiv, const rocblas_stride strideP, rocblas_float_complex *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetri_outofplace_strided_batched(rocblas_handle handle, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *ipiv, const rocblas_stride strideP, double *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetri_outofplace_strided_batched(rocblas_handle handle, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_int *ipiv, const rocblas_stride strideP, float *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

GETRI_OUTOFPLACE_STRIDED_BATCHED computes the inverse \(C_l = A_l^{-1}\) of a batch of general n-by-n matrices \(A_l\).

The inverse is computed by solving the linear system

\[ A_l C_l = I \]

where I is the identity matrix, and \(A_l\) is factorized as \(A_l = P_l L_l U_l\) as given by GETRF_STRIDED_BATCHED.

Parameters:

• handle – [in] rocblas_handle.

• n – [in] rocblas_int. n >= 0. The number of rows and columns of all matrices A_l in the batch.

• A – [in] pointer to type. Array on the GPU (the size depends on the value of strideA). The factors L_l and U_l of the factorization A_l = P_l*L_l*U_l returned by GETRF_STRIDED_BATCHED.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n

• ipiv – [in] pointer to rocblas_int. Array on the GPU (the size depends on the value of strideP). The pivot indices returned by GETRF_STRIDED_BATCHED.

• strideP – [in] rocblas_stride. Stride from the start of one vector ipiv_l to the next one ipiv_(l+1). There is no restriction for the value of strideP. Normal use case is strideP >= n.

• C – [out] pointer to type. Array on the GPU (the size depends on the value of strideC). If info[l] = 0, the inverse of matrices A_l. Otherwise, undefined.

• ldc – [in] rocblas_int. ldc >= n. Specifies the leading dimension of C_l.

• strideC – [in] rocblas_stride. Stride from the start of one matrix C_l to the next one C_(l+1). There is no restriction for the value of strideC. Normal use case is strideC >= ldc*n

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for inversion of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero pivot.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>getri_npvt_outofplace()

rocblas_status rocsolver_zgetri_npvt_outofplace(rocblas_handle handle, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, rocblas_double_complex *C, const rocblas_int ldc, rocblas_int *info)

rocblas_status rocsolver_cgetri_npvt_outofplace(rocblas_handle handle, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, rocblas_float_complex *C, const rocblas_int ldc, rocblas_int *info)

rocblas_status rocsolver_dgetri_npvt_outofplace(rocblas_handle handle, const rocblas_int n, double *A, const rocblas_int lda, double *C, const rocblas_int ldc, rocblas_int *info)

rocblas_status rocsolver_sgetri_npvt_outofplace(rocblas_handle handle, const rocblas_int n, float *A, const rocblas_int lda, float *C, const rocblas_int ldc, rocblas_int *info)

GETRI_NPVT_OUTOFPLACE computes the inverse \(C = A^{-1}\) of a general n-by-n matrix A without partial pivoting.

The inverse is computed by solving the linear system

\[ AC = I \]

where I is the identity matrix, and A is factorized as \(A = LU\) as given by GETRF_NPVT.

Parameters:

• handle – [in] rocblas_handle.

• n – [in] rocblas_int. n >= 0. The number of rows and columns of the matrix A.

• A – [in] pointer to type. Array on the GPU of dimension lda*n. The factors L and U of the factorization A = L*U returned by GETRF_NPVT.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of A.

• C – [out] pointer to type. Array on the GPU of dimension ldc*n. If info = 0, the inverse of A. Otherwise, undefined.

• ldc – [in] rocblas_int. ldc >= n. Specifies the leading dimension of C.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i > 0, U is singular. U[i,i] is the first zero pivot.

rocsolver_<type>getri_npvt_outofplace_batched()

rocblas_status rocsolver_zgetri_npvt_outofplace_batched(rocblas_handle handle, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_double_complex *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetri_npvt_outofplace_batched(rocblas_handle handle, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_float_complex *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetri_npvt_outofplace_batched(rocblas_handle handle, const rocblas_int n, double *const A[], const rocblas_int lda, double *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetri_npvt_outofplace_batched(rocblas_handle handle, const rocblas_int n, float *const A[], const rocblas_int lda, float *const C[], const rocblas_int ldc, rocblas_int *info, const rocblas_int batch_count)

GETRI_NPVT_OUTOFPLACE_BATCHED computes the inverse \(C_l^{} = A_l^{-1}\) of a batch of general n-by-n matrices \(A_l\) without partial pivoting.

The inverse is computed by solving the linear system

\[ A_l C_l = I \]

where I is the identity matrix, and \(A_l\) is factorized as \(A_l = L_l U_l\) as given by GETRF_NPVT_BATCHED.

Parameters:

• handle – [in] rocblas_handle.

• n – [in] rocblas_int. n >= 0. The number of rows and columns of all matrices A_l in the batch.

• A – [in] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. The factors L_l and U_l of the factorization A_l = L_l*U_l returned by GETRF_NPVT_BATCHED.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• C – [out] array of pointers to type. Each pointer points to an array on the GPU of dimension ldc*n. If info[l] = 0, the inverse of matrices A_l. Otherwise, undefined.

• ldc – [in] rocblas_int. ldc >= n. Specifies the leading dimension of C_l.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for inversion of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero pivot.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>getri_npvt_outofplace_strided_batched()

rocblas_status rocsolver_zgetri_npvt_outofplace_strided_batched(rocblas_handle handle, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_double_complex *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgetri_npvt_outofplace_strided_batched(rocblas_handle handle, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_float_complex *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgetri_npvt_outofplace_strided_batched(rocblas_handle handle, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, double *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgetri_npvt_outofplace_strided_batched(rocblas_handle handle, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, float *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_int *info, const rocblas_int batch_count)

GETRI_NPVT_OUTOFPLACE_STRIDED_BATCHED computes the inverse \(C_l^{} = A_l^{-1}\) of a batch of general n-by-n matrices \(A_l\) without partial pivoting.

The inverse is computed by solving the linear system

\[ A_l C_l = I \]

where I is the identity matrix, and \(A_l\) is factorized as \(A_l = L_l U_l\) as given by GETRF_NPVT_STRIDED_BATCHED.

Parameters:

• handle – [in] rocblas_handle.

• n – [in] rocblas_int. n >= 0. The number of rows and columns of all matrices A_l in the batch.

• A – [in] pointer to type. Array on the GPU (the size depends on the value of strideA). The factors L_l and U_l of the factorization A_l = L_l*U_l returned by GETRF_NPVT_STRIDED_BATCHED.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n

• C – [out] pointer to type. Array on the GPU (the size depends on the value of strideC). If info[l] = 0, the inverse of matrices A_l. Otherwise, undefined.

• ldc – [in] rocblas_int. ldc >= n. Specifies the leading dimension of C_l.

• strideC – [in] rocblas_stride. Stride from the start of one matrix C_l to the next one C_(l+1). There is no restriction for the value of strideC. Normal use case is strideC >= ldc*n

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for inversion of A_l. If info[l] = i > 0, U_l is singular. U_l[i,i] is the first zero pivot.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>geblttrs_npvt()

rocblas_status rocsolver_zgeblttrs_npvt(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, rocblas_double_complex *A, const rocblas_int lda, rocblas_double_complex *B, const rocblas_int ldb, rocblas_double_complex *C, const rocblas_int ldc, rocblas_double_complex *X, const rocblas_int ldx)

rocblas_status rocsolver_cgeblttrs_npvt(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, rocblas_float_complex *A, const rocblas_int lda, rocblas_float_complex *B, const rocblas_int ldb, rocblas_float_complex *C, const rocblas_int ldc, rocblas_float_complex *X, const rocblas_int ldx)

rocblas_status rocsolver_dgeblttrs_npvt(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, double *A, const rocblas_int lda, double *B, const rocblas_int ldb, double *C, const rocblas_int ldc, double *X, const rocblas_int ldx)

rocblas_status rocsolver_sgeblttrs_npvt(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, float *A, const rocblas_int lda, float *B, const rocblas_int ldb, float *C, const rocblas_int ldc, float *X, const rocblas_int ldx)

GEBLTTRS_NPVT solves a system of linear equations given by a block tridiagonal matrix in its factorized form (without partial pivoting).

The linear system has the form

\[\begin{split} MX = \left[\begin{array}{ccccc} B_1 & C_1\\ A_1 & B_2 & C_2\\ & \ddots & \ddots & \ddots \\ & & A_{n-2} & B_{n-1} & C_{n-1}\\ & & & A_{n-1} & B_n \end{array}\right]\left[\begin{array}{c} X_1\\ X_2\\ X_3\\ \vdots\\ X_n \end{array}\right]=\left[\begin{array}{c} R_1\\ R_2\\ R_3\\ \vdots\\ R_n \end{array}\right]=R \end{split}\]

where matrix M has \(n = \mathrm{nblocks}\) diagonal blocks of size nb, and the right-hand-side blocks \(R_i\) are general blocks of size nb-by-nrhs. The blocks of matrix M should be in the factorized form as returned by GEBLTTRF_NPVT.

Parameters:

• handle – [in] rocblas_handle.

• nb – [in] rocblas_int. nb >= 0. The number of rows and columns of each block.

• nblocks – [in] rocblas_int. nblocks >= 0. The number of blocks along the diagonal of the matrix.

• nrhs – [in] rocblas_int. nrhs >= 0. The number of right hand sides, i.e., the number of columns of blocks R_i.

• A – [in] pointer to type. Array on the GPU of dimension lda*nb*(nblocks-1). Contains the blocks A_i as returned by GEBLTTRF_NPVT.

• lda – [in] rocblas_int. lda >= nb. Specifies the leading dimension of blocks A_i.

• B – [in] pointer to type. Array on the GPU of dimension ldb*nb*nblocks. Contains the blocks B_i as returned by GEBLTTRF_NPVT.

• ldb – [in] rocblas_int. ldb >= nb. Specifies the leading dimension of blocks B_i.

• C – [in] pointer to type. Array on the GPU of dimension ldc*nb*(nblocks-1). Contains the blocks C_i as returned by GEBLTTRF_NPVT.

• ldc – [in] rocblas_int. ldc >= nb. Specifies the leading dimension of blocks C_i.

• X – [inout] pointer to type. Array on the GPU of dimension ldx*nblocks*nrhs. On entry, X contains the right-hand-side blocks R_i. It is overwritten by solution vectors X_i on exit.

• ldx – [in] rocblas_int. ldx >= nb. Specifies the leading dimension of blocks X_i.

rocsolver_<type>geblttrs_npvt_batched()

rocblas_status rocsolver_zgeblttrs_npvt_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_double_complex *const B[], const rocblas_int ldb, rocblas_double_complex *const C[], const rocblas_int ldc, rocblas_double_complex *const X[], const rocblas_int ldx, const rocblas_int batch_count)

rocblas_status rocsolver_cgeblttrs_npvt_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_float_complex *const B[], const rocblas_int ldb, rocblas_float_complex *const C[], const rocblas_int ldc, rocblas_float_complex *const X[], const rocblas_int ldx, const rocblas_int batch_count)

rocblas_status rocsolver_dgeblttrs_npvt_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, double *const A[], const rocblas_int lda, double *const B[], const rocblas_int ldb, double *const C[], const rocblas_int ldc, double *const X[], const rocblas_int ldx, const rocblas_int batch_count)

rocblas_status rocsolver_sgeblttrs_npvt_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, float *const A[], const rocblas_int lda, float *const B[], const rocblas_int ldb, float *const C[], const rocblas_int ldc, float *const X[], const rocblas_int ldx, const rocblas_int batch_count)

GEBLTTRS_NPVT_BATCHED solves a batch of system of linear equations given by block tridiagonal matrices in its factorized form (without partial pivoting).

Each linear system has the form

\[\begin{split} M_lX_l = \left[\begin{array}{ccccc} B_{l1} & C_{l1}\\ A_{l1} & B_{l2} & C_{l2}\\ & \ddots & \ddots & \ddots \\ & & A_{l(n-2)} & B_{l(n-1)} & C_{l(n-1)}\\ & & & A_{l(n-1)} & B_{ln} \end{array}\right]\left[\begin{array}{c} X_{l1}\\ X_{l2}\\ X_{l3}\\ \vdots\\ X_{ln} \end{array}\right]=\left[\begin{array}{c} R_{l1}\\ R_{l2}\\ R_{l3}\\ \vdots\\ R_{ln} \end{array}\right]=R_l \end{split}\]

where matrix \(M_l\) has \(n = \mathrm{nblocks}\) diagonal blocks of size nb, and the right-hand-side blocks \(R_{li}\) are general blocks of size nb-by-nrhs. The blocks of matrix \(M_l\) should be in the factorized form as returned by GEBLTTRF_NPVT_BATCHED.

Parameters:

• handle – [in] rocblas_handle.

• nb – [in] rocblas_int. nb >= 0. The number of rows and columns of each block.

• nblocks – [in] rocblas_int. nblocks >= 0. The number of blocks along the diagonal of each matrix in the batch.

• nrhs – [in] rocblas_int. nrhs >= 0. The number of right hand sides, i.e., the number of columns of blocks R_{li}.

• A – [in] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*nb*(nblocks-1). Contains the blocks A_{li} as returned by GEBLTTRF_NPVT_BATCHED.

• lda – [in] rocblas_int. lda >= nb. Specifies the leading dimension of blocks A_{li}.

• B – [in] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*nb*nblocks. Contains the blocks B_{li} as returned by GEBLTTRF_NPVT_BATCHED.

• ldb – [in] rocblas_int. ldb >= nb. Specifies the leading dimension of blocks B_{li}.

• C – [in] array of pointers to type. Each pointer points to an array on the GPU of dimension ldc*nb*(nblocks-1). Contains the blocks C_{li} as returned by GEBLTTRF_NPVT_BATCHED.

• ldc – [in] rocblas_int. ldc >= nb. Specifies the leading dimension of blocks C_{li}.

• X – [inout] array of pointers to type. Each pointer points to an array on the GPU of dimension ldx*nblocks*nrhs. On entry, X contains the right-hand-side blocks R_{li}. It is overwritten by solution vectors X_{li} on exit.

• ldx – [in] rocblas_int. ldx >= nb. Specifies the leading dimension of blocks X_{li}.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>geblttrs_npvt_strided_batched()

rocblas_status rocsolver_zgeblttrs_npvt_strided_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_double_complex *B, const rocblas_int ldb, const rocblas_stride strideB, rocblas_double_complex *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_double_complex *X, const rocblas_int ldx, const rocblas_stride strideX, const rocblas_int batch_count)

rocblas_status rocsolver_cgeblttrs_npvt_strided_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_float_complex *B, const rocblas_int ldb, const rocblas_stride strideB, rocblas_float_complex *C, const rocblas_int ldc, const rocblas_stride strideC, rocblas_float_complex *X, const rocblas_int ldx, const rocblas_stride strideX, const rocblas_int batch_count)

rocblas_status rocsolver_dgeblttrs_npvt_strided_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, double *A, const rocblas_int lda, const rocblas_stride strideA, double *B, const rocblas_int ldb, const rocblas_stride strideB, double *C, const rocblas_int ldc, const rocblas_stride strideC, double *X, const rocblas_int ldx, const rocblas_stride strideX, const rocblas_int batch_count)

rocblas_status rocsolver_sgeblttrs_npvt_strided_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, float *A, const rocblas_int lda, const rocblas_stride strideA, float *B, const rocblas_int ldb, const rocblas_stride strideB, float *C, const rocblas_int ldc, const rocblas_stride strideC, float *X, const rocblas_int ldx, const rocblas_stride strideX, const rocblas_int batch_count)

GEBLTTRS_NPVT_STRIDED_BATCHED solves a batch of system of linear equations given by block tridiagonal matrices in its factorized form (without partial pivoting).

Each linear system has the form

\[\begin{split} M_lX_l = \left[\begin{array}{ccccc} B_{l1} & C_{l1}\\ A_{l1} & B_{l2} & C_{l2}\\ & \ddots & \ddots & \ddots \\ & & A_{l(n-2)} & B_{l(n-1)} & C_{l(n-1)}\\ & & & A_{l(n-1)} & B_{ln} \end{array}\right]\left[\begin{array}{c} X_{l1}\\ X_{l2}\\ X_{l3}\\ \vdots\\ X_{ln} \end{array}\right]=\left[\begin{array}{c} R_{l1}\\ R_{l2}\\ R_{l3}\\ \vdots\\ R_{ln} \end{array}\right]=R_l \end{split}\]

where matrix \(M_l\) has \(n = \mathrm{nblocks}\) diagonal blocks of size nb, and the right-hand-side blocks \(R_{li}\) are general blocks of size nb-by-nrhs. The blocks of matrix \(M_l\) should be in the factorized form as returned by GEBLTTRF_NPVT_STRIDED_BATCHED.

Parameters:

• handle – [in] rocblas_handle.

• nb – [in] rocblas_int. nb >= 0. The number of rows and columns of each block.

• nblocks – [in] rocblas_int. nblocks >= 0. The number of blocks along the diagonal of each matrix in the batch.

• nrhs – [in] rocblas_int. nrhs >= 0. The number of right hand sides, i.e., the number of columns of blocks R_{li}.

• A – [in] pointer to type. Array on the GPU (the size depends on the value of strideA). Contains the blocks A_{li} as returned by GEBLTTRF_NPVT_STRIDED_BATCHED.

• lda – [in] rocblas_int. lda >= nb. Specifies the leading dimension of blocks A_{li}.

• strideA – [in] rocblas_stride. Stride from the start of one block A_{li} to the same block in the next batch instance A_{(l+1)i}. There is no restriction for the value of strideA. Normal use case is strideA >= lda*nb*nblocks

• B – [in] pointer to type. Array on the GPU (the size depends on the value of strideB). Contains the blocks B_{li} as returned by GEBLTTRF_NPVT_STRIDED_BATCHED.

• ldb – [in] rocblas_int. ldb >= nb. Specifies the leading dimension of blocks B_{li}.

• strideB – [in] rocblas_stride. Stride from the start of one block B_{li} to the same block in the next batch instance B_{(l+1)i}. There is no restriction for the value of strideB. Normal use case is strideB >= ldb*nb*nblocks

• C – [in] pointer to type. Array on the GPU (the size depends on the value of strideC). Contains the blocks C_{li} as returned by GEBLTTRF_NPVT_STRIDED_BATCHED.

• ldc – [in] rocblas_int. ldc >= nb. Specifies the leading dimension of blocks C_{li}.

• strideC – [in] rocblas_stride. Stride from the start of one block C_{li} to the same block in the next batch instance C_{(l+1)i}. There is no restriction for the value of strideC. Normal use case is strideC >= ldc*nb*nblocks

• X – [inout] pointer to type. Array on the GPU (the size depends on the value of strideX). On entry, X contains the right-hand-side blocks R_{li}. It is overwritten by solution vectors X_{li} on exit.

• ldx – [in] rocblas_int. ldx >= nb. Specifies the leading dimension of blocks X_{li}.

• strideX – [in] rocblas_stride. Stride from the start of one block X_{li} to the same block in the next batch instance X_{(l+1)i}. There is no restriction for the value of strideX. Normal use case is strideX >= ldx*nblocks*nrhs

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>geblttrs_npvt_interleaved_batched()

rocblas_status rocsolver_zgeblttrs_npvt_interleaved_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, rocblas_double_complex *A, const rocblas_int inca, const rocblas_int lda, const rocblas_stride strideA, rocblas_double_complex *B, const rocblas_int incb, const rocblas_int ldb, const rocblas_stride strideB, rocblas_double_complex *C, const rocblas_int incc, const rocblas_int ldc, const rocblas_stride strideC, rocblas_double_complex *X, const rocblas_int incx, const rocblas_int ldx, const rocblas_stride strideX, const rocblas_int batch_count)

rocblas_status rocsolver_cgeblttrs_npvt_interleaved_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, rocblas_float_complex *A, const rocblas_int inca, const rocblas_int lda, const rocblas_stride strideA, rocblas_float_complex *B, const rocblas_int incb, const rocblas_int ldb, const rocblas_stride strideB, rocblas_float_complex *C, const rocblas_int incc, const rocblas_int ldc, const rocblas_stride strideC, rocblas_float_complex *X, const rocblas_int incx, const rocblas_int ldx, const rocblas_stride strideX, const rocblas_int batch_count)

rocblas_status rocsolver_dgeblttrs_npvt_interleaved_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, double *A, const rocblas_int inca, const rocblas_int lda, const rocblas_stride strideA, double *B, const rocblas_int incb, const rocblas_int ldb, const rocblas_stride strideB, double *C, const rocblas_int incc, const rocblas_int ldc, const rocblas_stride strideC, double *X, const rocblas_int incx, const rocblas_int ldx, const rocblas_stride strideX, const rocblas_int batch_count)

rocblas_status rocsolver_sgeblttrs_npvt_interleaved_batched(rocblas_handle handle, const rocblas_int nb, const rocblas_int nblocks, const rocblas_int nrhs, float *A, const rocblas_int inca, const rocblas_int lda, const rocblas_stride strideA, float *B, const rocblas_int incb, const rocblas_int ldb, const rocblas_stride strideB, float *C, const rocblas_int incc, const rocblas_int ldc, const rocblas_stride strideC, float *X, const rocblas_int incx, const rocblas_int ldx, const rocblas_stride strideX, const rocblas_int batch_count)

GEBLTTRS_NPVT_INTERLEAVED_BATCHED solves a batch of system of linear equations given by block tridiagonal matrices in its factorized form (without partial pivoting).

Each linear system has the form

\[\begin{split} M_lX_l = \left[\begin{array}{ccccc} B_{l1} & C_{ll}\\ A_{l1} & B_{ll} & C_{ll}\\ & \ddots & \ddots & \ddots \\ & & A_{l(n-2)} & B_{l(n-1)} & C_{l(n-1)}\\ & & & A_{l(n-1)} & B_{ln} \end{array}\right]\left[\begin{array}{c} X_{l1}\\ X_{l2}\\ X_{l3}\\ \vdots\\ X_{ln} \end{array}\right]=\left[\begin{array}{c} R_{l1}\\ R_{l2}\\ R_{l3}\\ \vdots\\ R_{ln} \end{array}\right]=R_l \end{split}\]

where matrix \(M_l\) has \(n = \mathrm{nblocks}\) diagonal blocks of size nb, and the right-hand-side blocks \(R_{li}\) are general blocks of size nb-by-nrhs. The blocks of matrix \(M_l\) should be in the factorized form as returned by GEBLTTRF_NPVT_INTERLEAVED_BATCHED.

Parameters:

• handle – [in] rocblas_handle.

• nb – [in] rocblas_int. nb >= 0. The number of rows and columns of each block.

• nblocks – [in] rocblas_int. nblocks >= 0. The number of blocks along the diagonal of each matrix in the batch.

• nrhs – [in] rocblas_int. nrhs >= 0. The number of right hand sides, i.e., the number of columns of blocks R_{li}.

• A – [in] pointer to type. Array on the GPU (the size depends on the value of strideA). Contains the blocks A_{li} as returned by GEBLTTRF_NPVT_INTERLEAVED_BATCHED.

• inca – [in] rocblas_int. inca > 0. Stride from the start of one row of A_{li} to the next. Normal use cases are inca = 1 (strided batched case) or inca = batch_count (interleaved batched case).

• lda – [in] rocblas_int. lda >= inca * nb. Specifies the leading dimension of blocks A_{li}, i.e. the stride from the start of one column of A_{li} to the next.

• strideA – [in] rocblas_stride. Stride from the start of one block A_{li} to the same block in the next batch instance A_{(l+1)i}. There is no restriction for the value of strideA. Normal use cases are strideA >= lda*nb*nblocks (strided batched case) or strideA = 1 (interleaved batched case).

• B – [in] pointer to type. Array on the GPU (the size depends on the value of strideB). Contains the blocks B_{li} as returned by GEBLTTRF_NPVT_INTERLEAVED_BATCHED.

• incb – [in] rocblas_int. incb > 0. Stride from the start of one row of B_{li} to the next. Normal use cases are incb = 1 (strided batched case) or incb = batch_count (interleaved batched case).

• ldb – [in] rocblas_int. ldb >= incb * nb. Specifies the leading dimension of blocks B_{li}, i.e. the stride from the start of one column of B_{li} to the next.

• strideB – [in] rocblas_stride. Stride from the start of one block B_{li} to the same block in the next batch instance B_{(l+1)i}. There is no restriction for the value of strideB. Normal use cases are strideB >= ldb*nb*nblocks (strided batched case) or strideB = 1 (interleaved batched case).

• C – [in] pointer to type. Array on the GPU (the size depends on the value of strideC). Contains the blocks C_{li} as returned by GEBLTTRF_NPVT_INTERLEAVED_BATCHED.

• incc – [in] rocblas_int. incc > 0. Stride from the start of one row of C_{li} to the next. Normal use cases are incc = 1 (strided batched case) or incc = batch_count (interleaved batched case).

• ldc – [in] rocblas_int. ldc >= incc * nb. Specifies the leading dimension of blocks C_{li}, i.e. the stride from the start of one column of C_{li} to the next.

• strideC – [in] rocblas_stride. Stride from the start of one block C_{li} to the same block in the next batch instance C_{(l+1)i}. There is no restriction for the value of strideC. Normal use cases are strideC >= ldc*nb*nblocks (strided batched case) or strideC = 1 (interleaved batched case).

• X – [inout] pointer to type. Array on the GPU (the size depends on the value of strideX). On entry, X contains the right-hand-side blocks R_{li}. It is overwritten by solution vectors X_{li} on exit.

• incx – [in] rocblas_int. incx > 0. Stride from the start of one row of X_{li} to the next. Normal use cases are incx = 1 (strided batched case) or incx = batch_count (interleaved batched case).

• ldx – [in] rocblas_int. ldx >= incx * nb. Specifies the leading dimension of blocks X_{li}, i.e. the stride from the start of one column of X_{li} to the next.

• strideX – [in] rocblas_stride. Stride from the start of one block X_{li} to the same block in the next batch instance X_{(l+1)i}. There is no restriction for the value of strideX. Normal use cases are strideX >= ldx*nrhs*nblocks (strided batched case) or strideX = 1 (interleaved batched case).

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

Symmetric eigensolvers

rocsolver_<type>syevj()

rocblas_status rocsolver_dsyevj(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, rocblas_int *info)

rocblas_status rocsolver_ssyevj(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, rocblas_int *info)

SYEVJ computes the eigenvalues and optionally the eigenvectors of a real symmetric matrix A.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in an order depending on the value of esort. The eigenvectors are computed depending on the value of evect. The computed eigenvectors are orthonormal.

At the \(k\)-th iteration (or “sweep”), \(A\) is transformed by a product of Jacobi rotations \(V\) as

\[ A^{(k)} = V' A^{(k-1)} V \]

such that \(off(A^{(k)}) < off(A^{(k-1)})\), where \(A^{(0)} = A\) and \(off(A^{(k)})\) is the Frobenius norm of the off-diagonal elements of \(A^{(k)}\). As \(off(A^{(k)}) \rightarrow 0\), the diagonal elements of \(A^{(k)}\) increasingly resemble the eigenvalues of \(A\).

Parameters:

• handle – [in] rocblas_handle.

• esort – [in] rocblas_esort. Specifies the order of the returned eigenvalues. If esort is rocblas_esort_ascending, then the eigenvalues are sorted and returned in ascending order. If esort is rocblas_esort_none, then the order of the returned eigenvalues is unspecified.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the symmetric matrix A is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrix A.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the matrix A. On exit, the eigenvectors of A if they were computed and the algorithm converged; otherwise the contents of A are unchanged.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrix A.

• abstol – [in] type. The absolute tolerance. The algorithm is considered to have converged once off(A) is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to type on the GPU. The Frobenius norm of the off-diagonal elements of A (i.e. off(A)) at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to a rocblas_int on the GPU. The actual number of sweeps (iterations) used by the algorithm.

• W – [out] pointer to type. Array on the GPU of dimension n. The eigenvalues of A in increasing order.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = 1, the algorithm did not converge.

rocsolver_<type>syevj_batched()

rocblas_status rocsolver_dsyevj_batched(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *const A[], const rocblas_int lda, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssyevj_batched(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *const A[], const rocblas_int lda, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

SYEVJ_BATCHED computes the eigenvalues and optionally the eigenvectors of a batch of real symmetric matrices A_l.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in an order depending on the value of esort. The eigenvectors are computed depending on the value of evect. The computed eigenvectors are orthonormal.

At the \(k\)-th iteration (or “sweep”), \(A_l\) is transformed by a product of Jacobi rotations \(V_l\) as

\[ A_l^{(k)} = V_l' A_l^{(k-1)} V_l^{} \]

such that \(off(A_l^{(k)}) < off(A_l^{(k-1)})\), where \(A_l^{(0)} = A_l\) and \(off(A_l^{(k)})\) is the Frobenius norm of the off-diagonal elements of \(A_l^{(k)}\). As \(off(A_l^{(k)}) \rightarrow 0\), the diagonal elements of \(A_l^{(k)}\) increasingly resemble the eigenvalues of \(A_l\).

Parameters:

• handle – [in] rocblas_handle.

• esort – [in] rocblas_esort. Specifies the order of the returned eigenvalues. If esort is rocblas_esort_ascending, then the eigenvalues are sorted and returned in ascending order. If esort is rocblas_esort_none, then the order of the returned eigenvalues is unspecified.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the symmetric matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrices A_l.

• A – [inout] Array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the matrices A_l. On exit, the eigenvectors of A_l if they were computed and the algorithm converged; otherwise the contents of A_l are unchanged.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• abstol – [in] type. The absolute tolerance. The algorithm is considered to have converged once off(A_l) is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to type. Array of batch_count scalars on the GPU. The Frobenius norm of the off-diagonal elements of A_l (i.e. off(A_l)) at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• W – [out] pointer to type. Array on the GPU (the size depends on the value of strideW). The eigenvalues of A_l in increasing order.

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for matrix A_l. If info[l] = 1, the algorithm did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>syevj_strided_batched()

rocblas_status rocsolver_dsyevj_strided_batched(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssyevj_strided_batched(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

SYEVJ_STRIDED_BATCHED computes the eigenvalues and optionally the eigenvectors of a batch of real symmetric matrices A_l.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in an order depending on the value of esort. The eigenvectors are computed depending on the value of evect. The computed eigenvectors are orthonormal.

At the \(k\)-th iteration (or “sweep”), \(A_l\) is transformed by a product of Jacobi rotations \(V_l\) as

\[ A_l^{(k)} = V_l' A_l^{(k-1)} V_l^{} \]

such that \(off(A_l^{(k)}) < off(A_l^{(k-1)})\), where \(A_l^{(0)} = A_l\) and \(off(A_l^{(k)})\) is the Frobenius norm of the off-diagonal elements of \(A_l^{(k)}\). As \(off(A_l^{(k)}) \rightarrow 0\), the diagonal elements of \(A_l^{(k)}\) increasingly resemble the eigenvalues of \(A_l\).

Parameters:

• handle – [in] rocblas_handle.

• esort – [in] rocblas_esort. Specifies the order of the returned eigenvalues. If esort is rocblas_esort_ascending, then the eigenvalues are sorted and returned in ascending order. If esort is rocblas_esort_none, then the order of the returned eigenvalues is unspecified.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the symmetric matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrices A_l.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the matrices A_l. On exit, the eigenvectors of A_l if they were computed and the algorithm converged; otherwise the contents of A_l are unchanged.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n.

• abstol – [in] type. The absolute tolerance. The algorithm is considered to have converged once off(A_l) is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to type. Array of batch_count scalars on the GPU. The Frobenius norm of the off-diagonal elements of A_l (i.e. off(A_l)) at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• W – [out] pointer to type. Array on the GPU (the size depends on the value of strideW). The eigenvalues of A_l in increasing order.

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for matrix A_l. If info[l] = 1, the algorithm did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>heevj()

rocblas_status rocsolver_zheevj(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, rocblas_int *info)

rocblas_status rocsolver_cheevj(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, rocblas_int *info)

HEEVJ computes the eigenvalues and optionally the eigenvectors of a complex Hermitian matrix A.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in an order depending on the value of esort. The eigenvectors are computed depending on the value of evect. The computed eigenvectors are orthonormal.

At the \(k\)-th iteration (or “sweep”), \(A\) is transformed by a product of Jacobi rotations \(V\) as

\[ A^{(k)} = V' A^{(k-1)} V \]

such that \(off(A^{(k)}) < off(A^{(k-1)})\), where \(A^{(0)} = A\) and \(off(A^{(k)})\) is the Frobenius norm of the off-diagonal elements of \(A^{(k)}\). As \(off(A^{(k)}) \rightarrow 0\), the diagonal elements of \(A^{(k)}\) increasingly resemble the eigenvalues of \(A\).

Parameters:

• handle – [in] rocblas_handle.

• esort – [in] rocblas_esort. Specifies the order of the returned eigenvalues. If esort is rocblas_esort_ascending, then the eigenvalues are sorted and returned in ascending order. If esort is rocblas_esort_none, then the order of the returned eigenvalues is unspecified.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the Hermitian matrix A is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrix A.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the matrix A. On exit, the eigenvectors of A if they were computed and the algorithm converged; otherwise the contents of A are unchanged.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrix A.

• abstol – [in] real type. The absolute tolerance. The algorithm is considered to have converged once off(A) is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to real type on the GPU. The Frobenius norm of the off-diagonal elements of A (i.e. off(A)) at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to a rocblas_int on the GPU. The actual number of sweeps (iterations) used by the algorithm.

• W – [out] pointer to real type. Array on the GPU of dimension n. The eigenvalues of A in increasing order.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = 1, the algorithm did not converge.

rocsolver_<type>heevj_batched()

rocblas_status rocsolver_zheevj_batched(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cheevj_batched(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

HEEVJ_BATCHED computes the eigenvalues and optionally the eigenvectors of a batch of complex Hermitian matrices A_l.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in an order depending on the value of esort. The eigenvectors are computed depending on the value of evect. The computed eigenvectors are orthonormal.

At the \(k\)-th iteration (or “sweep”), \(A_l\) is transformed by a product of Jacobi rotations \(V_l\) as

\[ A_l^{(k)} = V_l' A_l^{(k-1)} V_l^{} \]

such that \(off(A_l^{(k)}) < off(A_l^{(k-1)})\), where \(A_l^{(0)} = A_l\) and \(off(A_l^{(k)})\) is the Frobenius norm of the off-diagonal elements of \(A_l^{(k)}\). As \(off(A_l^{(k)}) \rightarrow 0\), the diagonal elements of \(A_l^{(k)}\) increasingly resemble the eigenvalues of \(A_l\).

Parameters:

• handle – [in] rocblas_handle.

• esort – [in] rocblas_esort. Specifies the order of the returned eigenvalues. If esort is rocblas_esort_ascending, then the eigenvalues are sorted and returned in ascending order. If esort is rocblas_esort_none, then the order of the returned eigenvalues is unspecified.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the Hermitian matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrices A_l.

• A – [inout] Array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the matrices A_l. On exit, the eigenvectors of A_l if they were computed and the algorithm converged; otherwise the contents of A_l are unchanged.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• abstol – [in] real type. The absolute tolerance. The algorithm is considered to have converged once off(A_l) is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to real type. Array of batch_count scalars on the GPU. The Frobenius norm of the off-diagonal elements of A_l (i.e. off(A_l)) at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• W – [out] pointer to real type. Array on the GPU (the size depends on the value of strideW). The eigenvalues of A_l in increasing order.

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for matrix A_l. If info[l] = 1, the algorithm did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>heevj_strided_batched()

rocblas_status rocsolver_zheevj_strided_batched(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cheevj_strided_batched(rocblas_handle handle, const rocblas_esort esort, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

HEEVJ_STRIDED_BATCHED computes the eigenvalues and optionally the eigenvectors of a batch of complex Hermitian matrices A_l.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in an order depending on the value of esort. The eigenvectors are computed depending on the value of evect. The computed eigenvectors are orthonormal.

At the \(k\)-th iteration (or “sweep”), \(A_l\) is transformed by a product of Jacobi rotations \(V_l\) as

\[ A_l^{(k)} = V_l' A_l^{(k-1)} V_l^{} \]

such that \(off(A_l^{(k)}) < off(A_l^{(k-1)})\), where \(A_l^{(0)} = A_l\) and \(off(A_l^{(k)})\) is the Frobenius norm of the off-diagonal elements of \(A_l^{(k)}\). As \(off(A_l^{(k)}) \rightarrow 0\), the diagonal elements of \(A_l^{(k)}\) increasingly resemble the eigenvalues of \(A_l\).

Parameters:

• handle – [in] rocblas_handle.

• esort – [in] rocblas_esort. Specifies the order of the returned eigenvalues. If esort is rocblas_esort_ascending, then the eigenvalues are sorted and returned in ascending order. If esort is rocblas_esort_none, then the order of the returned eigenvalues is unspecified.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the Hermitian matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrices A_l.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the matrices A_l. On exit, the eigenvectors of A_l if they were computed and the algorithm converged; otherwise the contents of A_l are unchanged.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n.

• abstol – [in] real type. The absolute tolerance. The algorithm is considered to have converged once off(A_l) is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to real type. Array of batch_count scalars on the GPU. The Frobenius norm of the off-diagonal elements of A_l (i.e. off(A_l)) at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• W – [out] pointer to real type. Array on the GPU (the size depends on the value of strideW). The eigenvalues of A_l in increasing order.

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for matrix A_l. If info[l] = 1, the algorithm did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>sygvj()

rocblas_status rocsolver_dsygvj(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, double *B, const rocblas_int ldb, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, rocblas_int *info)

rocblas_status rocsolver_ssygvj(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, float *B, const rocblas_int ldb, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, rocblas_int *info)

SYGVJ computes the eigenvalues and (optionally) eigenvectors of a real generalized symmetric-definite eigenproblem.

The problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A X = \lambda B X & \: \text{1st form,}\\ A B X = \lambda X & \: \text{2nd form, or}\\ B A X = \lambda X & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed depending on the value of evect.

When computed, the matrix Z of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^T B Z=I & \: \text{if 1st or 2nd form, or}\\ Z^T B^{-1} Z=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblem.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A and B are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A and B are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the symmetric matrix A. On exit, if evect is original, the normalized matrix Z of eigenvectors. If evect is none, then the upper or lower triangular part of the matrix A (including the diagonal) is destroyed, depending on the value of uplo.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of A.

• B – [out] pointer to type. Array on the GPU of dimension ldb*n. On entry, the symmetric positive definite matrix B. On exit, the triangular factor of B as returned by POTRF.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B.

• abstol – [in] type. The absolute tolerance. The algorithm is considered to have converged once the residual is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to type on the GPU. The Frobenius norm of the off-diagonal elements at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to a rocblas_int on the GPU. The actual number of sweeps (iterations) used by the algorithm.

• W – [out] pointer to type. Array on the GPU of dimension n. On exit, the eigenvalues in increasing order.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = 1, the algorithm did not converge. If info = n + i, the leading minor of order i of B is not positive definite.

rocsolver_<type>sygvj_batched()

rocblas_status rocsolver_dsygvj_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *const A[], const rocblas_int lda, double *const B[], const rocblas_int ldb, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssygvj_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *const A[], const rocblas_int lda, float *const B[], const rocblas_int ldb, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

SYGVJ_BATCHED computes the eigenvalues and (optionally) eigenvectors of a batch of real generalized symmetric-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed depending on the value of evect.

When computed, the matrix \(Z_l\) of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z_l^T B_l^{} Z_l^{}=I & \: \text{if 1st or 2nd form, or}\\ Z_l^T B_l^{-1} Z_l^{}=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the symmetric matrices A_l. On exit, if evect is original, the normalized matrix Z_l of eigenvectors. If evect is none, then the upper or lower triangular part of the matrices A_l (including the diagonal) are destroyed, depending on the value of uplo.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of A_l.

• B – [out] array of pointers to type. Each pointer points to an array on the GPU of dimension ldb*n. On entry, the symmetric positive definite matrices B_l. On exit, the triangular factor of B_l as returned by POTRF_BATCHED.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B_l.

• abstol – [in] type. The absolute tolerance. The algorithm is considered to have converged once the residual is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to type. Array of batch_count scalars on the GPU. The Frobenius norm of the off-diagonal elements at the final iteration for each batch instance.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• W – [out] pointer to type. Array on the GPU (the size depends on the value of strideW). On exit, the eigenvalues in increasing order.

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use is strideW >= n.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit of batch instance l. If info[l] = 1, the algorithm did not converge. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>sygvj_strided_batched()

rocblas_status rocsolver_dsygvj_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, double *B, const rocblas_int ldb, const rocblas_stride strideB, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssygvj_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, float *B, const rocblas_int ldb, const rocblas_stride strideB, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

SYGVJ_STRIDED_BATCHED computes the eigenvalues and (optionally) eigenvectors of a batch of real generalized symmetric-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed depending on the value of evect.

When computed, the matrix \(Z_l\) of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z_l^T B_l^{} Z_l^{}=I & \: \text{if 1st or 2nd form, or}\\ Z_l^T B_l^{-1} Z_l^{}=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the symmetric matrices A_l. On exit, if evect is original, the normalized matrix Z_l of eigenvectors. If evect is none, then the upper or lower triangular part of the matrices A_l (including the diagonal) are destroyed, depending on the value of uplo.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use is strideA >= lda*n.

• B – [out] pointer to type. Array on the GPU (the size depends on the value of strideB). On entry, the symmetric positive definite matrices B_l. On exit, the triangular factor of B_l as returned by POTRF_STRIDED_BATCHED.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B_l.

• strideB – [in] rocblas_stride. Stride from the start of one matrix B_l to the next one B_(l+1). There is no restriction for the value of strideB. Normal use is strideB >= ldb*n.

• abstol – [in] type. The absolute tolerance. The algorithm is considered to have converged once the residual is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to type. Array of batch_count scalars on the GPU. The Frobenius norm of the off-diagonal elements at the final iteration for each batch instance.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• W – [out] pointer to type. Array on the GPU (the size depends on the value of strideW). On exit, the eigenvalues in increasing order.

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use is strideW >= n.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit of batch l. If info[l] = 1, the algorithm did not converge. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>hegvj()

rocblas_status rocsolver_zhegvj(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, rocblas_double_complex *B, const rocblas_int ldb, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, rocblas_int *info)

rocblas_status rocsolver_chegvj(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, rocblas_float_complex *B, const rocblas_int ldb, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, rocblas_int *info)

HEGVJ computes the eigenvalues and (optionally) eigenvectors of a complex generalized hermitian-definite eigenproblem.

The problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A X = \lambda B X & \: \text{1st form,}\\ A B X = \lambda X & \: \text{2nd form, or}\\ B A X = \lambda X & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed depending on the value of evect.

When computed, the matrix Z of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^H B Z=I & \: \text{if 1st or 2nd form, or}\\ Z^H B^{-1} Z=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblem.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A and B are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A and B are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the hermitian matrix A. On exit, if evect is original, the normalized matrix Z of eigenvectors. If evect is none, then the upper or lower triangular part of the matrix A (including the diagonal) is destroyed, depending on the value of uplo.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of A.

• B – [out] pointer to type. Array on the GPU of dimension ldb*n. On entry, the hermitian positive definite matrix B. On exit, the triangular factor of B as returned by POTRF.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B.

• abstol – [in] real type. The absolute tolerance. The algorithm is considered to have converged once the residual is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to real type on the GPU. The Frobenius norm of the off-diagonal elements at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to a rocblas_int on the GPU. The actual number of sweeps (iterations) used by the algorithm.

• W – [out] pointer to real type. Array on the GPU of dimension n. On exit, the eigenvalues in increasing order.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = 1, the algorithm did not converge. If info = n + i, the leading minor of order i of B is not positive definite.

rocsolver_<type>hegvj_batched()

rocblas_status rocsolver_zhegvj_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_double_complex *const B[], const rocblas_int ldb, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_chegvj_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_float_complex *const B[], const rocblas_int ldb, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

HEGVJ_BATCHED computes the eigenvalues and (optionally) eigenvectors of a batch of complex generalized hermitian-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed depending on the value of evect.

When computed, the matrix \(Z_l\) of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z_l^H B_l^{} Z_l^{}=I & \: \text{if 1st or 2nd form, or}\\ Z_l^H B_l^{-1} Z_l^{}=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the hermitian matrices A_l. On exit, if evect is original, the normalized matrix Z_l of eigenvectors. If evect is none, then the upper or lower triangular part of the matrices A_l (including the diagonal) are destroyed, depending on the value of uplo.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of A_l.

• B – [out] array of pointers to type. Each pointer points to an array on the GPU of dimension ldb*n. On entry, the hermitian positive definite matrices B_l. On exit, the triangular factor of B_l as returned by POTRF_BATCHED.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B_l.

• abstol – [in] real type. The absolute tolerance. The algorithm is considered to have converged once the residual is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to real type. Array of batch_count scalars on the GPU. The Frobenius norm of the off-diagonal elements at the final iteration for each batch instance.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• W – [out] pointer to real type. Array on the GPU (the size depends on the value of strideW). On exit, the eigenvalues in increasing order.

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use is strideW >= n.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit of batch l. If info[l] = 1, the algorithm did not converge. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>hegvj_strided_batched()

rocblas_status rocsolver_zhegvj_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_double_complex *B, const rocblas_int ldb, const rocblas_stride strideB, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_chegvj_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_float_complex *B, const rocblas_int ldb, const rocblas_stride strideB, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *W, const rocblas_stride strideW, rocblas_int *info, const rocblas_int batch_count)

HEGVJ_STRIDED_BATCHED computes the eigenvalues and (optionally) eigenvectors of a batch of complex generalized hermitian-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed depending on the value of evect.

When computed, the matrix \(Z_l\) of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z_l^H B_l^{} Z_l^{}=I & \: \text{if 1st or 2nd form, or}\\ Z_l^H B_l^{-1} Z_l^{}=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the hermitian matrices A_l. On exit, if evect is original, the normalized matrix Z_l of eigenvectors. If evect is none, then the upper or lower triangular part of the matrices A_l (including the diagonal) are destroyed, depending on the value of uplo.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use is strideA >= lda*n.

• B – [out] pointer to type. Array on the GPU (the size depends on the value of strideB). On entry, the hermitian positive definite matrices B_l. On exit, the triangular factor of B_l as returned by POTRF_STRIDED_BATCHED.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B_l.

• strideB – [in] rocblas_stride. Stride from the start of one matrix B_l to the next one B_(l+1). There is no restriction for the value of strideB. Normal use is strideB >= ldb*n.

• abstol – [in] real type. The absolute tolerance. The algorithm is considered to have converged once the residual is <= abstol. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to real type. Array of batch_count scalars on the GPU. The Frobenius norm of the off-diagonal elements at the final iteration for each batch instance.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• W – [out] pointer to real type. Array on the GPU (the size depends on the value of strideW). On exit, the eigenvalues in increasing order.

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use is strideW >= n.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit of batch l. If info[l] = 1, the algorithm did not converge. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>syevdj()

rocblas_status rocsolver_dsyevdj(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, double *D, rocblas_int *info)

rocblas_status rocsolver_ssyevdj(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, float *D, rocblas_int *info)

SYEVDJ computes the eigenvalues and optionally the eigenvectors of a real symmetric matrix A.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer approach depending on the value of evect. The computed eigenvectors are orthonormal.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower part of the symmetric matrix A is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A is not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrix A.

• A – [inout]

  pointer to type. Array on the GPU of dimension lda*n.

  On entry, the matrix A. On exit, the eigenvectors of A if they were computed and the algorithm converged; otherwise the contents of A are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrix A.

• D – [out]

  pointer to type. Array on the GPU of dimension n.

  The eigenvalues of A in increasing order.

• info – [out]

  pointer to a rocblas_int on the GPU.

  If info = 0, successful exit. If info = 1, the algorithm did not converge.

rocsolver_<type>syevdj_batched()

rocblas_status rocsolver_dsyevdj_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *const A[], const rocblas_int lda, double *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssyevdj_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *const A[], const rocblas_int lda, float *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

SYEVDJ_BATCHED computes the eigenvalues and optionally the eigenvectors of a batch of real symmetric matrices A_l.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer approach depending on the value of evect. The computed eigenvectors are orthonormal.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower part of the symmetric matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrices A_l.

• A – [inout]

  Array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n.

  On entry, the matrices A_l. On exit, the eigenvectors of A_l if they were computed and the algorithm converged; otherwise the contents of A_l are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrices A_l.

• D – [out]

  pointer to type. Array on the GPU (the size depends on the value of strideD).

  The eigenvalues of A_l in increasing order.

• strideD – [in]

  rocblas_stride.

  Stride from the start of one vector D_l to the next one D_(l+1). There is no restriction for the value of strideD. Normal use case is strideD >= n.

• info – [out]

  pointer to rocblas_int. Array of batch_count integers on the GPU.

  If info[l] = 0, successful exit for A_l. If info[l] = 1, the algorithm did not converge for A_l.

• batch_count – [in]

  rocblas_int. batch_count >= 0.

  Number of matrices in the batch.

rocsolver_<type>syevdj_strided_batched()

rocblas_status rocsolver_dsyevdj_strided_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, double *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssyevdj_strided_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, float *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

SYEVDJ_STRIDED_BATCHED computes the eigenvalues and optionally the eigenvectors of a batch of real symmetric matrices A_l.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer approach depending on the value of evect. The computed eigenvectors are orthonormal.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower part of the symmetric matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrices A_l.

• A – [inout]

  Pointer to type. Array on the GPU (the size depends on the value of strideA).

  On entry, the matrices A_l. On exit, the eigenvectors of A_l if they were computed and the algorithm converged; otherwise the contents of A_l are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrices A_l.

• strideA – [in]

  rocblas_stride.

  Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n.

• D – [out]

  pointer to type. Array on the GPU (the size depends on the value of strideD).

  The eigenvalues of A_l in increasing order.

• strideD – [in]

  rocblas_stride.

  Stride from the start of one vector D_l to the next one D_(l+1). There is no restriction for the value of strideD. Normal use case is strideD >= n.

• info – [out]

  pointer to rocblas_int. Array of batch_count integers on the GPU.

  If info[l] = 0, successful exit for A_l. If info[l] = 1, the algorithm did not converge for A_l.

• batch_count – [in]

  rocblas_int. batch_count >= 0.

  Number of matrices in the batch.

rocsolver_<type>heevdj()

rocblas_status rocsolver_zheevdj(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, double *D, rocblas_int *info)

rocblas_status rocsolver_cheevdj(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, float *D, rocblas_int *info)

HEEVDJ computes the eigenvalues and optionally the eigenvectors of a complex Hermitian matrix A.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer approach depending on the value of evect. The computed eigenvectors are orthonormal.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower part of the Hermitian matrix A is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A is not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrix A.

• A – [inout]

  pointer to type. Array on the GPU of dimension lda*n.

  On entry, the matrix A. On exit, the eigenvectors of A if they were computed and the algorithm converged; otherwise the contents of A are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrix A.

• D – [out]

  pointer to real type. Array on the GPU of dimension n.

  The eigenvalues of A in increasing order.

• info – [out]

  pointer to a rocblas_int on the GPU.

  If info = 0, successful exit. If info = 1, the algorithm did not converge.

rocsolver_<type>heevdj_batched()

rocblas_status rocsolver_zheevdj_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, double *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cheevdj_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, float *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

HEEVDJ_BATCHED computes the eigenvalues and optionally the eigenvectors of a batch of complex Hermitian matrices A_l.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer approach depending on the value of evect. The computed eigenvectors are orthonormal.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower part of the Hermitian matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrices A_l.

• A – [inout]

  Array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n.

  On entry, the matrices A_l. On exit, the eigenvectors of A_l if they were computed and the algorithm converged; otherwise the contents of A_l are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrices A_l.

• D – [out]

  pointer to real type. Array on the GPU (the size depends on the value of strideD).

  The eigenvalues of A_l in increasing order.

• strideD – [in]

  rocblas_stride.

  Stride from the start of one vector D_l to the next one D_(l+1). There is no restriction for the value of strideD. Normal use case is strideD >= n.

• info – [out]

  pointer to rocblas_int. Array of batch_count integers on the GPU.

  If info[l] = 0, successful exit for A_l. If info[l] = 1, the algorithm did not converge for A_l.

• batch_count – [in]

  rocblas_int. batch_count >= 0.

  Number of matrices in the batch.

rocsolver_<type>heevdj_strided_batched()

rocblas_status rocsolver_zheevdj_strided_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, double *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cheevdj_strided_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, float *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

HEEVDJ_STRIDED_BATCHED computes the eigenvalues and optionally the eigenvectors of a batch of complex Hermitian matrices A_l.

The eigenvalues are found using the iterative Jacobi algorithm and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer approach depending on the value of evect. The computed eigenvectors are orthonormal.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower part of the Hermitian matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrices A_l.

• A – [inout]

  Pointer to type. Array on the GPU (the size depends on the value of strideA).

  On entry, the matrices A_l. On exit, the eigenvectors of A_l if they were computed and the algorithm converged; otherwise the contents of A_l are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrices A_l.

• strideA – [in]

  rocblas_stride.

  Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n.

• D – [out]

  pointer to real type. Array on the GPU (the size depends on the value of strideD).

  The eigenvalues of A_l in increasing order.

• strideD – [in]

  rocblas_stride.

  Stride from the start of one vector D_l to the next one D_(l+1). There is no restriction for the value of strideD. Normal use case is strideD >= n.

• info – [out]

  pointer to rocblas_int. Array of batch_count integers on the GPU.

  If info[l] = 0, successful exit for A_l. If info[l] = 1, the algorithm did not converge for A_l.

• batch_count – [in]

  rocblas_int. batch_count >= 0.

  Number of matrices in the batch.

rocsolver_<type>sygvdj()

rocblas_status rocsolver_dsygvdj(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, double *B, const rocblas_int ldb, double *D, rocblas_int *info)

rocblas_status rocsolver_ssygvdj(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, float *B, const rocblas_int ldb, float *D, rocblas_int *info)

SYGVDJ computes the eigenvalues and (optionally) eigenvectors of a real generalized symmetric-definite eigenproblem.

The problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A X = \lambda B X & \: \text{1st form,}\\ A B X = \lambda X & \: \text{2nd form, or}\\ B A X = \lambda X & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer algorithm, depending on the value of evect.

When computed, the matrix Z of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^T B Z=I & \: \text{if 1st or 2nd form, or}\\ Z^T B^{-1} Z=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform

  .

  Specifies the form of the generalized eigenproblem.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower parts of the matrices A and B are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A and B are not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrix A.

• A – [inout]

  pointer to type. Array on the GPU of dimension lda*n.

  On entry, the matrix A. On exit, the normalized matrix Z of eigenvectors if they were computed and the algorithm converged; otherwise the contents of A are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrix A.

• B – [inout]

  pointer to type. Array on the GPU of dimension ldb*n.

  On entry, the symmetric positive definite matrix B. On exit, the triangular factor of B as returned by

  POTRF.

• ldb – [in]

  rocblas_int. ldb >= n.

  Specifies the leading dimension of matrix B.

• D – [out]

  pointer to type. Array on the GPU of dimension n.

  The eigenvalues in increasing order.

• info – [out]

  pointer to a rocblas_int on the GPU.

  If info = 0, successful exit. If info = 1, the algorithm did not converge. If info = n + i, the leading minor of order i of B is not positive definite.

rocsolver_<type>sygvdj_batched()

rocblas_status rocsolver_dsygvdj_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *const A[], const rocblas_int lda, double *const B[], const rocblas_int ldb, double *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssygvdj_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *const A[], const rocblas_int lda, float *const B[], const rocblas_int ldb, float *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

SYGVDJ_BATCHED computes the eigenvalues and (optionally) eigenvectors of batch of real generalized symmetric-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer algorithm, depending on the value of evect.

When computed, the matrix Z_l of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^T_l B_l Z_l=I & \: \text{if 1st or 2nd form, or}\\ Z^T_l B^{-1}_l Z_l=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform

  .

  Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrix A_l.

• A – [inout]

  array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n.

  On entry, the matrices A_l. On exit, the normalized matrices Z_l of eigenvectors if they were computed and the algorithm converged; otherwise the contents of A_l are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrices A_l.

• B – [inout]

  array of pointers to type. Each pointer points to an array on the GPU of dimension ldb*n.

  On entry, the symmetric positive definite matrices B_l. On exit, the triangular factor of B_l as returned by

  POTRF_BATCHED.

• ldb – [in]

  rocblas_int. ldb >= n.

  Specifies the leading dimension of matrices B_l.

• D – [out]

  pointer to type. Array on the GPU (the size depends on the value of strideD).

  The eigenvalues in increasing order.

• strideD – [in]

  rocblas_stride.

  Stride from the start of one vector D_l to the next one D_(l+1). There is no restriction for the value of strideD. Normal use is strideD >= n.

• info – [out]

  pointer to rocblas_int. Array of batch_count integers on the GPU.

  If info[l] = 0, successful exit. If info[l] = 1, the algorithm did not converge for matrix A_l. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in]

  rocblas_int. batch_count >= 0.

  Number of eigenproblems in the batch.

rocsolver_<type>sygvdj_strided_batched()

rocblas_status rocsolver_dsygvdj_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, double *B, const rocblas_int ldb, const rocblas_stride strideB, double *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssygvdj_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, float *B, const rocblas_int ldb, const rocblas_stride strideB, float *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

SYGVDJ_STRIDED_BATCHED computes the eigenvalues and (optionally) eigenvectors of batch of real generalized symmetric-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer algorithm, depending on the value of evect.

When computed, the matrix Z_l of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^T_l B_l Z_l=I & \: \text{if 1st or 2nd form, or}\\ Z^T_l B^{-1}_l Z_l=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform

  .

  Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrix A_l.

• A – [inout]

  pointer to type. Array on the GPU (the size depends on the value of strideA).

  On entry, the matrices A_l. On exit, the normalized matrices Z_l of eigenvectors if they were computed and the algorithm converged; otherwise the contents of A_l are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrices A_l.

• strideA – [in]

  rocblas_stride.

  Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use is strideA >= lda*n.

• B – [inout]

  pointer to type. Array on the GPU (the size depends on the value of strideB).

  On entry, the symmetric positive definite matrices B_l. On exit, the triangular factor of B_l as returned by

  POTRF_STRIDED_BATCHED.

• ldb – [in]

  rocblas_int. ldb >= n.

  Specifies the leading dimension of matrices B_l.

• strideB – [in]

  rocblas_stride.

  Stride from the start of one matrix B_l to the next one B_(l+1). There is no restriction for the value of strideB. Normal use is strideB >= ldb*n.

• D – [out]

  pointer to type. Array on the GPU (the size depends on the value of strideD).

  The eigenvalues in increasing order.

• strideD – [in]

  rocblas_stride.

  Stride from the start of one vector D_l to the next one D_(l+1). There is no restriction for the value of strideD. Normal use is strideD >= n.

• info – [out]

  pointer to rocblas_int. Array of batch_count integers on the GPU.

  If info[l] = 0, successful exit. If info[l] = 1, the algorithm did not converge for matrix A_l. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in]

  rocblas_int. batch_count >= 0.

  Number of eigenproblems in the batch.

rocsolver_<type>hegvdj()

rocblas_status rocsolver_zhegvdj(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, rocblas_double_complex *B, const rocblas_int ldb, double *D, rocblas_int *info)

rocblas_status rocsolver_chegvdj(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, rocblas_float_complex *B, const rocblas_int ldb, float *D, rocblas_int *info)

HEGVDJ computes the eigenvalues and (optionally) eigenvectors of a complex generalized Hermitian-definite eigenproblem.

The problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A X = \lambda B X & \: \text{1st form,}\\ A B X = \lambda X & \: \text{2nd form, or}\\ B A X = \lambda X & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer algorithm, depending on the value of evect.

When computed, the matrix Z of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^H B Z=I & \: \text{if 1st or 2nd form, or}\\ Z^H B^{-1} Z=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform

  .

  Specifies the form of the generalized eigenproblem.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower parts of the matrices A and B are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A and B are not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrix A.

• A – [inout]

  pointer to type. Array on the GPU of dimension lda*n.

  On entry, the matrix A. On exit, the normalized matrix Z of eigenvectors if they were computed and the algorithm converged; otherwise the contents of A are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrix A.

• B – [inout]

  pointer to type. Array on the GPU of dimension ldb*n.

  On entry, the Hermitian positive definite matrix B. On exit, the triangular factor of B as returned by

  POTRF.

• ldb – [in]

  rocblas_int. ldb >= n.

  Specifies the leading dimension of matrix B.

• D – [out]

  pointer to real type. Array on the GPU of dimension n.

  The eigenvalues in increasing order.

• info – [out]

  pointer to a rocblas_int on the GPU.

  If info = 0, successful exit. If info = 1, the algorithm did not converge. If info = n + i, the leading minor of order i of B is not positive definite.

rocsolver_<type>hegvdj_batched()

rocblas_status rocsolver_zhegvdj_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_double_complex *const B[], const rocblas_int ldb, double *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_chegvdj_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_float_complex *const B[], const rocblas_int ldb, float *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

HEGVDJ_BATCHED computes the eigenvalues and (optionally) eigenvectors of batch of complex generalized Hermitian-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer algorithm, depending on the value of evect.

When computed, the matrix Z_l of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^H_l B_l Z_l=I & \: \text{if 1st or 2nd form, or}\\ Z^H_l B^{-1}_l Z_l=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform

  .

  Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrix A_l.

• A – [inout]

  array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n.

  On entry, the matrices A_l. On exit, the normalized matrices Z_l of eigenvectors if they were computed and the algorithm converged; otherwise the contents of A_l are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrices A_l.

• B – [inout]

  array of pointers to type. Each pointer points to an array on the GPU of dimension ldb*n.

  On entry, the Hermitian positive definite matrices B_l. On exit, the triangular factor of B_l as returned by

  POTRF_BATCHED.

• ldb – [in]

  rocblas_int. ldb >= n.

  Specifies the leading dimension of matrices B_l.

• D – [out]

  pointer to real type. Array on the GPU (the size depends on the value of strideD).

  The eigenvalues in increasing order.

• strideD – [in]

  rocblas_stride.

  Stride from the start of one vector D_l to the next one D_(l+1). There is no restriction for the value of strideD. Normal use is strideD >= n.

• info – [out]

  pointer to rocblas_int. Array of batch_count integers on the GPU.

  If info[l] = 0, successful exit. If info[l] = 1, the algorithm did not converge for matrix A_l. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in]

  rocblas_int. batch_count >= 0.

  Number of eigenproblems in the batch.

rocsolver_<type>hegvdj_strided_batched()

rocblas_status rocsolver_zhegvdj_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_double_complex *B, const rocblas_int ldb, const rocblas_stride strideB, double *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_chegvdj_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_float_complex *B, const rocblas_int ldb, const rocblas_stride strideB, float *D, const rocblas_stride strideD, rocblas_int *info, const rocblas_int batch_count)

HEGVDJ_STRIDED_BATCHED computes the eigenvalues and (optionally) eigenvectors of batch of complex generalized Hermitian-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvalues are found using the iterative Jacobi algorithm, and are returned in ascending order. The eigenvectors are computed using a divide-and-conquer algorithm, depending on the value of evect.

When computed, the matrix Z_l of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^H_l B_l Z_l=I & \: \text{if 1st or 2nd form, or}\\ Z^H_l B^{-1}_l Z_l=I & \: \text{if 3rd form.} \end{array} \end{split}\]

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform

  .

  Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect

  .

  Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• uplo – [in]

  rocblas_fill.

  Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in]

  rocblas_int. n >= 0.

  Number of rows and columns of matrix A_l.

• A – [inout]

  pointer to type. Array on the GPU (the size depends on the value of strideA).

  On entry, the matrices A_l. On exit, the normalized matrices Z_l of eigenvectors if they were computed and the algorithm converged; otherwise the contents of A_l are destroyed.

• lda – [in]

  rocblas_int. lda >= n.

  Specifies the leading dimension of matrices A_l.

• strideA – [in]

  rocblas_stride.

  Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use is strideA >= lda*n.

• B – [inout]

  pointer to type. Array on the GPU (the size depends on the value of strideB).

  On entry, the Hermitian positive definite matrices B_l. On exit, the triangular factor of B_l as returned by

  POTRF_BATCHED.

• ldb – [in]

  rocblas_int. ldb >= n.

  Specifies the leading dimension of matrices B_l.

• strideB – [in]

  rocblas_stride.

  Stride from the start of one matrix B_l to the next one B_(l+1). There is no restriction for the value of strideB. Normal use is strideB >= ldb*n.

• D – [out]

  pointer to real type. Array on the GPU (the size depends on the value of strideD).

  The eigenvalues in increasing order.

• strideD – [in]

  rocblas_stride.

  Stride from the start of one vector D_l to the next one D_(l+1). There is no restriction for the value of strideD. Normal use is strideD >= n.

• info – [out]

  pointer to rocblas_int. Array of batch_count integers on the GPU.

  If info[l] = 0, successful exit. If info[l] = 1, the algorithm did not converge for matrix A_l. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in]

  rocblas_int. batch_count >= 0.

  Number of eigenproblems in the batch.

rocsolver_<type>syevdx()

rocblas_status rocsolver_dsyevdx(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, double *Z, const rocblas_int ldz, rocblas_int *info)

rocblas_status rocsolver_ssyevdx(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, float *Z, const rocblas_int ldz, rocblas_int *info)

SYEVDX computes a set of the eigenvalues and optionally the corresponding eigenvectors of a real symmetric matrix A.

This function computes all the eigenvalues of A, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the symmetric matrix A is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrix A.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the matrix A. On exit, the contents of A are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrix A.

• vl – [in] type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues within a set of indices.

• vu – [in] type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues in a half-open interval.

• nev – [out] pointer to a rocblas_int on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev = n. If erange is rocblas_erange_index, nev = iu - il + 1. Otherwise, 0 <= nev <= n.

• W – [out] pointer to type. Array on the GPU of dimension n. The first nev elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• Z – [out] pointer to type. Array on the GPU of dimension ldz*nev. On exit, if evect is not rocblas_evect_none and info = 0, the first nev columns contain the eigenvectors of A corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none. Note: If erange is rocblas_range_value, then the values of nev are not known in advance. The user should ensure that Z is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrix Z.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i > 0, the algorithm did not converge. i columns of Z did not converge.

rocsolver_<type>syevdx_batched()

rocblas_status rocsolver_dsyevdx_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, double *const A[], const rocblas_int lda, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, const rocblas_stride strideW, double *const Z[], const rocblas_int ldz, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssyevdx_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, float *const A[], const rocblas_int lda, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, const rocblas_stride strideW, float *const Z[], const rocblas_int ldz, rocblas_int *info, const rocblas_int batch_count)

SYEVDX_BATCHED computes a set of the eigenvalues and optionally the corresponding eigenvectors of a batch of real symmetric matrices A_l.

This function computes all the eigenvalues of A_l, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the symmetric matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrices A_l.

• A – [inout] Array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• vl – [in] type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• vu – [in] type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• nev – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev[l] = n. If erange is rocblas_erange_index, nev[l] = iu - il + 1. Otherwise, 0 <= nev[l] <= n.

• W – [out] pointer to type. Array on the GPU (the size depends on the value of strideW). The first nev[l] elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• Z – [out] Array of pointers to type. Each pointer points to an array on the GPU of dimension ldz*nev[l]. On exit, if evect is not rocblas_evect_none and info[l] = 0, the first nev[l] columns contain the eigenvectors of A_l corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none. Note: If erange is rocblas_range_value, then the values of nev[l] are not known in advance. The user should ensure that Z_l is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrices Z_l.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for matrix A_l. If info[l] = i > 0, the algorithm did not converge. i columns of Z_l did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>syevdx_strided_batched()

rocblas_status rocsolver_dsyevdx_strided_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, const rocblas_stride strideW, double *Z, const rocblas_int ldz, const rocblas_stride strideZ, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssyevdx_strided_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, const rocblas_stride strideW, float *Z, const rocblas_int ldz, const rocblas_stride strideZ, rocblas_int *info, const rocblas_int batch_count)

SYEVDX_STRIDED_BATCHED computes a set of the eigenvalues and optionally the corresponding eigenvectors of a batch of real symmetric matrices A_l.

This function computes all the eigenvalues of A_l, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the symmetric matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrices A_l.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n.

• vl – [in] type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• vu – [in] type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• nev – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev[l] = n. If erange is rocblas_erange_index, nev[l] = iu - il + 1. Otherwise, 0 <= nev[l] <= n.

• W – [out] pointer to type. Array on the GPU (the size depends on the value of strideW). The first nev[l] elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• Z – [out] pointer to type. Array on the GPU (the size depends on the value of strideZ). On exit, if evect is not rocblas_evect_none and info[l] = 0, the first nev[l] columns contain the eigenvectors of A_l corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrices Z_l.

• strideZ – [in] rocblas_stride. Stride from the start of one matrix Z_l to the next one Z_(l+1). There is no restriction for the value of strideZ. Normal use case is strideZ >= ldz*nev[l]. Note: If erange is rocblas_range_value, then the values of nev[l] are not known in advance. The user should ensure that Z_l is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for matrix A_l. If info[l] = i > 0, the algorithm did not converge. i columns of Z_l did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>heevdx()

Warning

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rocblas_status rocsolver_cheevdx(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, rocblas_float_complex *Z, const rocblas_int ldz, rocblas_int *info)

HEEVDX computes a set of the eigenvalues and optionally the corresponding eigenvectors of a Hermitian matrix A.

This function computes all the eigenvalues of A, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the symmetric matrix A is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrix A.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the matrix A. On exit, the contents of A are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrix A.

• vl – [in] real type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues within a set of indices.

• vu – [in] real type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues in a half-open interval.

• nev – [out] pointer to a rocblas_int on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev = n. If erange is rocblas_erange_index, nev = iu - il + 1. Otherwise, 0 <= nev <= n.

• W – [out] pointer to real type. Array on the GPU of dimension n. The first nev elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• Z – [out] pointer to type. Array on the GPU of dimension ldz*nev. On exit, if evect is not rocblas_evect_none and info = 0, the first nev columns contain the eigenvectors of A corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none. Note: If erange is rocblas_range_value, then the values of nev are not known in advance. The user should ensure that Z is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrix Z.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i > 0, the algorithm did not converge. i columns of Z did not converge.

rocsolver_<type>heevdx_batched()

rocblas_status rocsolver_zheevdx_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, const rocblas_stride strideW, rocblas_double_complex *const Z[], const rocblas_int ldz, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cheevdx_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, const rocblas_stride strideW, rocblas_float_complex *const Z[], const rocblas_int ldz, rocblas_int *info, const rocblas_int batch_count)

HEEVDX_BATCHED computes a set of the eigenvalues and optionally the corresponding eigenvectors of a batch of Hermitian matrices A_l.

This function computes all the eigenvalues of A_l, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the symmetric matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrices A_l.

• A – [inout] Array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• vl – [in] real type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• vu – [in] real type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• nev – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev[l] = n. If erange is rocblas_erange_index, nev[l] = iu - il + 1. Otherwise, 0 <= nev[l] <= n.

• W – [out] pointer to real type. Array on the GPU (the size depends on the value of strideW). The first nev[l] elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• Z – [out] Array of pointers to type. Each pointer points to an array on the GPU of dimension ldz*nev[l]. On exit, if evect is not rocblas_evect_none and info[l] = 0, the first nev[l] columns contain the eigenvectors of A_l corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none. Note: If erange is rocblas_range_value, then the values of nev[l] are not known in advance. The user should ensure that Z_l is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrices Z_l.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for matrix A_l. If info[l] = i > 0, the algorithm did not converge. i columns of Z_l did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>heevdx_strided_batched()

rocblas_status rocsolver_zheevdx_strided_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, const rocblas_stride strideW, rocblas_double_complex *Z, const rocblas_int ldz, const rocblas_stride strideZ, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cheevdx_strided_batched(rocblas_handle handle, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, const rocblas_stride strideW, rocblas_float_complex *Z, const rocblas_int ldz, const rocblas_stride strideZ, rocblas_int *info, const rocblas_int batch_count)

HEEVDX_STRIDED_BATCHED computes a set of the eigenvalues and optionally the corresponding eigenvectors of a batch of Hermitian matrices A_l.

This function computes all the eigenvalues of A_l, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower part of the symmetric matrices A_l is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_l is not used.

• n – [in] rocblas_int. n >= 0. Number of rows and columns of matrices A_l.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n.

• vl – [in] real type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• vu – [in] real type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• nev – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev[l] = n. If erange is rocblas_erange_index, nev[l] = iu - il + 1. Otherwise, 0 <= nev[l] <= n.

• W – [out] pointer to real type. Array on the GPU (the size depends on the value of strideW). The first nev[l] elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• Z – [out] pointer to type. Array on the GPU (the size depends on the value of strideZ). On exit, if evect is not rocblas_evect_none and info[l] = 0, the first nev[l] columns contain the eigenvectors of A_l corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrices Z_l.

• strideZ – [in] rocblas_stride. Stride from the start of one matrix Z_l to the next one Z_(l+1). There is no restriction for the value of strideZ. Normal use case is strideZ >= ldz*nev[l]. Note: If erange is rocblas_range_value, then the values of nev[l] are not known in advance. The user should ensure that Z_l is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit for matrix A_l. If info[l] = i > 0, the algorithm did not converge. i columns of Z_l did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>sygvdx()

rocblas_status rocsolver_dsygvdx(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, double *B, const rocblas_int ldb, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, double *Z, const rocblas_int ldz, rocblas_int *info)

rocblas_status rocsolver_ssygvdx(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, float *B, const rocblas_int ldb, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, float *Z, const rocblas_int ldz, rocblas_int *info)

SYGVDX computes a set of the eigenvalues and optionally the corresponding eigenvectors of a real generalized symmetric-definite eigenproblem.

The problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A X = \lambda B X & \: \text{1st form,}\\ A B X = \lambda X & \: \text{2nd form, or}\\ B A X = \lambda X & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvectors are computed depending on the value of evect.

When computed, the matrix Z of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^T B Z=I & \: \text{if 1st or 2nd form, or}\\ Z^T B^{-1} Z=I & \: \text{if 3rd form.} \end{array} \end{split}\]

This function computes all the eigenvalues, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblem.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A and B are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A and B are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the matrix A. On exit, the contents of A are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrix A.

• B – [out] pointer to type. Array on the GPU of dimension ldb*n. On entry, the symmetric positive definite matrix B. On exit, the triangular factor of B as returned by POTRF.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B.

• vl – [in] type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues within a set of indices.

• vu – [in] type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues in a half-open interval.

• nev – [out] pointer to a rocblas_int on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev = n. If erange is rocblas_erange_index, nev = iu - il + 1. Otherwise, 0 <= nev <= n.

• W – [out] pointer to type. Array on the GPU of dimension n. The first nev elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• Z – [out] pointer to type. Array on the GPU of dimension ldz*nev. On exit, if evect is not rocblas_evect_none and info = 0, the first nev columns contain the eigenvectors of A corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none. Note: If erange is rocblas_range_value, then the values of nev are not known in advance. The user should ensure that Z is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrix Z.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i <= n, i columns of Z did not converge. If info = n + i, the leading minor of order i of B is not positive definite.

rocsolver_<type>sygvdx_batched()

rocblas_status rocsolver_dsygvdx_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, double *const A[], const rocblas_int lda, double *const B[], const rocblas_int ldb, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, const rocblas_stride strideW, double *const Z[], const rocblas_int ldz, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssygvdx_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, float *const A[], const rocblas_int lda, float *const B[], const rocblas_int ldb, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, const rocblas_stride strideW, float *const Z[], const rocblas_int ldz, rocblas_int *info, const rocblas_int batch_count)

SYGVDX_BATCHED computes a set of the eigenvalues and optionally the corresponding eigenvectors of a batch of real generalized symmetric-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvectors are computed depending on the value of evect.

When computed, the matrix \(Z_l\) of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z_l^T B_l^{} Z_l^{}=I & \: \text{if 1st or 2nd form, or}\\ Z_l^T B_l^{-1} Z_l^{}=I & \: \text{if 3rd form.} \end{array} \end{split}\]

This function computes all the eigenvalues, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] Array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• B – [out] Array of pointers to type. Each pointer points to an array on the GPU of dimension ldb*n. On entry, the symmetric positive definite matrices B_l. On exit, the triangular factor of B_l as returned by POTRF_BATCHED.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B_l.

• vl – [in] type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• vu – [in] type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• nev – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev[l] = n. If erange is rocblas_erange_index, nev[l] = iu - il + 1. Otherwise, 0 <= nev[l] <= n.

• W – [out] pointer to type. Array on the GPU (the size depends on the value of strideW). The first nev[l] elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• Z – [out] Array of pointers to type. Each pointer points to an array on the GPU of dimension ldz*nev[l]. On exit, if evect is not rocblas_evect_none and info[l] = 0, the first nev[l] columns contain the eigenvectors of A_l corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none. Note: If erange is rocblas_range_value, then the values of nev[l] are not known in advance. The user should ensure that Z_l is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrices Z_l.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit of batch instance l. If info[l] = i <= n, i columns of Z_l did not converge. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>sygvdx_strided_batched()

rocblas_status rocsolver_dsygvdx_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, double *B, const rocblas_int ldb, const rocblas_stride strideB, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, const rocblas_stride strideW, double *Z, const rocblas_int ldz, const rocblas_stride strideZ, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_ssygvdx_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, float *B, const rocblas_int ldb, const rocblas_stride strideB, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, const rocblas_stride strideW, float *Z, const rocblas_int ldz, const rocblas_stride strideZ, rocblas_int *info, const rocblas_int batch_count)

SYGVDX_STRIDED_BATCHED computes a set of the eigenvalues and optionally the corresponding eigenvectors of a batch of real generalized symmetric-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvectors are computed depending on the value of evect.

When computed, the matrix \(Z_l\) of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z_l^T B_l^{} Z_l^{}=I & \: \text{if 1st or 2nd form, or}\\ Z_l^T B_l^{-1} Z_l^{}=I & \: \text{if 3rd form.} \end{array} \end{split}\]

This function computes all the eigenvalues, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n.

• B – [out] pointer to type. Array on the GPU (the size depends on the value of strideB). On entry, the symmetric positive definite matrices B_l. On exit, the triangular factor of B_l as returned by POTRF_STRIDED_BATCHED.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B_l.

• strideB – [in] rocblas_stride. Stride from the start of one matrix B_l to the next one B_(l+1). There is no restriction for the value of strideB. Normal use is strideB >= ldb*n.

• vl – [in] type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• vu – [in] type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• nev – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev[l] = n. If erange is rocblas_erange_index, nev[l] = iu - il + 1. Otherwise, 0 <= nev[l] <= n.

• W – [out] pointer to type. Array on the GPU (the size depends on the value of strideW). The first nev[l] elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• Z – [out] pointer to type. Array on the GPU (the size depends on the value of strideZ). On exit, if evect is not rocblas_evect_none and info[l] = 0, the first nev[l] columns contain the eigenvectors of A_l corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrices Z_l.

• strideZ – [in] rocblas_stride. Stride from the start of one matrix Z_l to the next one Z_(l+1). There is no restriction for the value of strideZ. Normal use case is strideZ >= ldz*nev[l]. Note: If erange is rocblas_range_value, then the values of nev[l] are not known in advance. The user should ensure that Z_l is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit of batch l. If info[l] = i <= n, i columns of Z_l did not converge. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>hegvdx()

rocblas_status rocsolver_zhegvdx(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, rocblas_double_complex *B, const rocblas_int ldb, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, rocblas_double_complex *Z, const rocblas_int ldz, rocblas_int *info)

rocblas_status rocsolver_chegvdx(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, rocblas_float_complex *B, const rocblas_int ldb, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, rocblas_float_complex *Z, const rocblas_int ldz, rocblas_int *info)

HEGVDX computes a set of the eigenvalues and optionally the corresponding eigenvectors of a complex generalized hermitian-definite eigenproblem.

The problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A X = \lambda B X & \: \text{1st form,}\\ A B X = \lambda X & \: \text{2nd form, or}\\ B A X = \lambda X & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvectors are computed depending on the value of evect.

When computed, the matrix Z of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z^H B Z=I & \: \text{if 1st or 2nd form, or}\\ Z^H B^{-1} Z=I & \: \text{if 3rd form.} \end{array} \end{split}\]

This function computes all the eigenvalues, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblem.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A and B are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A and B are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the matrix A. On exit, the contents of A are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrix A.

• B – [out] pointer to type. Array on the GPU of dimension ldb*n. On entry, the hermitian positive definite matrix B. On exit, the triangular factor of B as returned by POTRF.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B.

• vl – [in] real type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues within a set of indices.

• vu – [in] real type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A or the eigenvalues in a half-open interval.

• nev – [out] pointer to a rocblas_int on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev = n. If erange is rocblas_erange_index, nev = iu - il + 1. Otherwise, 0 <= nev <= n.

• W – [out] pointer to real type. Array on the GPU of dimension n. The first nev elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• Z – [out] pointer to type. Array on the GPU of dimension ldz*nev. On exit, if evect is not rocblas_evect_none and info = 0, the first nev columns contain the eigenvectors of A corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none. Note: If erange is rocblas_range_value, then the values of nev are not known in advance. The user should ensure that Z is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrix Z.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = i <= n, i columns of Z did not converge. If info = n + i, the leading minor of order i of B is not positive definite.

rocsolver_<type>hegvdx_batched()

rocblas_status rocsolver_zhegvdx_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, rocblas_double_complex *const B[], const rocblas_int ldb, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, const rocblas_stride strideW, rocblas_double_complex *const Z[], const rocblas_int ldz, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_chegvdx_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, rocblas_float_complex *const B[], const rocblas_int ldb, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, const rocblas_stride strideW, rocblas_float_complex *const Z[], const rocblas_int ldz, rocblas_int *info, const rocblas_int batch_count)

HEGVDX_BATCHED computes a set of the eigenvalues and optionally the corresponding eigenvectors of a batch of complex generalized hermitian-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvectors are computed depending on the value of evect.

When computed, the matrix \(Z_l\) of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z_l^H B_l^{} Z_l^{}=I & \: \text{if 1st or 2nd form, or}\\ Z_l^H B_l^{-1} Z_l^{}=I & \: \text{if 3rd form.} \end{array} \end{split}\]

This function computes all the eigenvalues, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] Array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• B – [out] Array of pointers to type. Each pointer points to an array on the GPU of dimension ldb*n. On entry, the hermitian positive definite matrices B_l. On exit, the triangular factor of B_l as returned by POTRF_BATCHED.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B_l.

• vl – [in] real type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• vu – [in] real type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• nev – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev[l] = n. If erange is rocblas_erange_index, nev[l] = iu - il + 1. Otherwise, 0 <= nev[l] <= n.

• W – [out] pointer to real type. Array on the GPU (the size depends on the value of strideW). The first nev[l] elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• Z – [out] Array of pointers to type. Each pointer points to an array on the GPU of dimension ldz*nev[l]. On exit, if evect is not rocblas_evect_none and info[l] = 0, the first nev[l] columns contain the eigenvectors of A_l corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none. Note: If erange is rocblas_range_value, then the values of nev[l] are not known in advance. The user should ensure that Z_l is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrices Z_l.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit of batch instance l. If info[l] = i <= n, i columns of Z_l did not converge. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>hegvdx_strided_batched()

rocblas_status rocsolver_zhegvdx_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_double_complex *B, const rocblas_int ldb, const rocblas_stride strideB, const double vl, const double vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, double *W, const rocblas_stride strideW, rocblas_double_complex *Z, const rocblas_int ldz, const rocblas_stride strideZ, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_chegvdx_strided_batched(rocblas_handle handle, const rocblas_eform itype, const rocblas_evect evect, const rocblas_erange erange, const rocblas_fill uplo, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, rocblas_float_complex *B, const rocblas_int ldb, const rocblas_stride strideB, const float vl, const float vu, const rocblas_int il, const rocblas_int iu, rocblas_int *nev, float *W, const rocblas_stride strideW, rocblas_float_complex *Z, const rocblas_int ldz, const rocblas_stride strideZ, rocblas_int *info, const rocblas_int batch_count)

HEGVDX_STRIDED_BATCHED computes a set of the eigenvalues and optionally the corresponding eigenvectors of a batch of complex generalized hermitian-definite eigenproblems.

For each instance in the batch, the problem solved by this function is either of the form

\[\begin{split} \begin{array}{cl} A_l X_l = \lambda B_l X_l & \: \text{1st form,}\\ A_l B_l X_l = \lambda X_l & \: \text{2nd form, or}\\ B_l A_l X_l = \lambda X_l & \: \text{3rd form,} \end{array} \end{split}\]

depending on the value of itype. The eigenvectors are computed depending on the value of evect.

When computed, the matrix \(Z_l\) of eigenvectors is normalized as follows:

\[\begin{split} \begin{array}{cl} Z_l^H B_l^{} Z_l^{}=I & \: \text{if 1st or 2nd form, or}\\ Z_l^H B_l^{-1} Z_l^{}=I & \: \text{if 3rd form.} \end{array} \end{split}\]

This function computes all the eigenvalues, all the eigenvalues in the half-open interval \((vl, vu]\), or the il-th through iu-th eigenvalues, depending on the value of erange. If evect is rocblas_evect_original, the eigenvectors for these eigenvalues will be computed as well. The eigenvectors are computed using a divide-and-conquer approach.

Parameters:

• handle – [in] rocblas_handle.

• itype – [in] rocblas_eform. Specifies the form of the generalized eigenproblems.

• evect – [in] rocblas_evect. Specifies whether the eigenvectors are to be computed. If evect is rocblas_evect_original, then the eigenvectors are computed. rocblas_evect_tridiagonal is not supported.

• erange – [in] rocblas_erange. Specifies the type of range or interval of the eigenvalues to be computed.

• uplo – [in] rocblas_fill. Specifies whether the upper or lower parts of the matrices A_l and B_l are stored. If uplo indicates lower (or upper), then the upper (or lower) parts of A_l and B_l are not used.

• n – [in] rocblas_int. n >= 0. The matrix dimensions.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= n. Specifies the leading dimension of matrices A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n.

• B – [out] pointer to type. Array on the GPU (the size depends on the value of strideB). On entry, the hermitian positive definite matrices B_l. On exit, the triangular factor of B_l as returned by POTRF_STRIDED_BATCHED.

• ldb – [in] rocblas_int. ldb >= n. Specifies the leading dimension of B_l.

• strideB – [in] rocblas_stride. Stride from the start of one matrix B_l to the next one B_(l+1). There is no restriction for the value of strideB. Normal use is strideB >= ldb*n.

• vl – [in] real type. vl < vu. The lower bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• vu – [in] real type. vl < vu. The upper bound of the search interval (vl, vu]. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues within a set of indices.

• il – [in] rocblas_int. il = 1 if n = 0; 1 <= il <= iu otherwise. The index of the smallest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• iu – [in] rocblas_int. iu = 0 if n = 0; 1 <= il <= iu otherwise.. The index of the largest eigenvalue to be computed. Ignored if range indicates to look for all the eigenvalues of A_l or the eigenvalues in a half-open interval.

• nev – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The total number of eigenvalues found. If erange is rocblas_erange_all, nev[l] = n. If erange is rocblas_erange_index, nev[l] = iu - il + 1. Otherwise, 0 <= nev[l] <= n.

• W – [out] pointer to real type. Array on the GPU (the size depends on the value of strideW). The first nev[l] elements contain the computed eigenvalues. (The remaining elements can be used as workspace for internal computations).

• strideW – [in] rocblas_stride. Stride from the start of one vector W_l to the next one W_(l+1). There is no restriction for the value of strideW. Normal use case is strideW >= n.

• Z – [out] pointer to type. Array on the GPU (the size depends on the value of strideZ). On exit, if evect is not rocblas_evect_none and info[l] = 0, the first nev[l] columns contain the eigenvectors of A_l corresponding to the output eigenvalues. Not referenced if evect is rocblas_evect_none.

• ldz – [in] rocblas_int. ldz >= n. Specifies the leading dimension of matrices Z_l.

• strideZ – [in] rocblas_stride. Stride from the start of one matrix Z_l to the next one Z_(l+1). There is no restriction for the value of strideZ. Normal use case is strideZ >= ldz*nev[l]. Note: If erange is rocblas_range_value, then the values of nev[l] are not known in advance. The user should ensure that Z_l is large enough to hold n columns, as all n columns can be used as workspace for internal computations.

• info – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. If info[l] = 0, successful exit of batch l. If info[l] = i <= n, i columns of Z_l did not converge. If info[l] = n + i, the leading minor of order i of B_l is not positive definite.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

Singular value decomposition

rocsolver_<type>gesvdj()

rocblas_status rocsolver_zgesvdj(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *S, rocblas_double_complex *U, const rocblas_int ldu, rocblas_double_complex *V, const rocblas_int ldv, rocblas_int *info)

rocblas_status rocsolver_cgesvdj(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *S, rocblas_float_complex *U, const rocblas_int ldu, rocblas_float_complex *V, const rocblas_int ldv, rocblas_int *info)

rocblas_status rocsolver_dgesvdj(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, double *A, const rocblas_int lda, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *S, double *U, const rocblas_int ldu, double *V, const rocblas_int ldv, rocblas_int *info)

rocblas_status rocsolver_sgesvdj(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, float *A, const rocblas_int lda, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *S, float *U, const rocblas_int ldu, float *V, const rocblas_int ldv, rocblas_int *info)

GESVDJ computes the singular values and optionally the singular vectors of a general m-by-n matrix A (Singular Value Decomposition).

The SVD of matrix A is given by:

\[ A = U S V' \]

where the m-by-n matrix S is zero except, possibly, for its min(m,n) diagonal elements, which are the singular values of A. U and V are orthogonal (unitary) matrices. The first min(m,n) columns of U and V are the left and right singular vectors of A, respectively.

The computation of the singular vectors is optional and it is controlled by the function arguments left_svect and right_svect as described below. When computed, this function returns the transpose (or transpose conjugate) of the right singular vectors, i.e. the rows of V’.

left_svect and right_svect are rocblas_svect enums that can take the following values:

• rocblas_svect_all: the entire matrix U (or V’) is computed,

• rocblas_svect_singular: the singular vectors (first min(m,n) columns of U or rows of V’) are computed, or

• rocblas_svect_none: no columns (or rows) of U (or V’) are computed, i.e. no singular vectors.

The singular values are computed by applying QR factorization to AV if m >= n (resp. LQ factorization to U’A if m < n), where V (resp. U) is found as the eigenvectors of A’A (resp. AA’) using the Jacobi eigenvalue algorithm.

Note

In order to carry out calculations, this method may synchronize the stream contained within the rocblas_handle.

Parameters:

• handle – [in] rocblas_handle.

• left_svect – [in] rocblas_svect. Specifies how the left singular vectors are computed. rocblas_svect_overwrite is not supported.

• right_svect – [in] rocblas_svect. Specifies how the right singular vectors are computed. rocblas_svect_overwrite is not supported.

• m – [in] rocblas_int. m >= 0. The number of rows of matrix A.

• n – [in] rocblas_int. n >= 0. The number of columns of matrix A.

• A – [inout] pointer to type. Array on the GPU of dimension lda*n. On entry, the matrix A. On exit, the contents of A are destroyed.

• lda – [in] rocblas_int. lda >= m. The leading dimension of A.

• abstol – [in] real type. The absolute tolerance. The algorithm is considered to have converged once off(A’A) is <= norm(A’A) * abstol [resp. off(AA’) <= norm(AA’) * abstol]. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to real type on the GPU. The Frobenius norm of the off-diagonal elements of A’A (resp. AA’) at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to a rocblas_int on the GPU. The actual number of sweeps (iterations) used by the algorithm.

• S – [out] pointer to real type. Array on the GPU of dimension min(m,n). The singular values of A in decreasing order.

• U – [out] pointer to type. Array on the GPU of dimension ldu*min(m,n) if left_svect is set to singular, or ldu*m when left_svect is equal to all. The matrix of left singular vectors stored as columns. Not referenced if left_svect is set to none.

• ldu – [in] rocblas_int. ldu >= m if left_svect is set to all or singular; ldu >= 1 otherwise. The leading dimension of U.

• V – [out] pointer to type. Array on the GPU of dimension ldv*n. The matrix of right singular vectors stored as rows (transposed / conjugate-transposed). Not referenced if right_svect is set to none.

• ldv – [in] rocblas_int. ldv >= n if right_svect is set to all; ldv >= min(m,n) if right_svect is set to singular; or ldv >= 1 otherwise. The leading dimension of V.

• info – [out] pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info = 1, the algorithm did not converge.

rocsolver_<type>gesvdj_batched()

rocblas_status rocsolver_zgesvdj_batched(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, rocblas_double_complex *const A[], const rocblas_int lda, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *S, const rocblas_stride strideS, rocblas_double_complex *U, const rocblas_int ldu, const rocblas_stride strideU, rocblas_double_complex *V, const rocblas_int ldv, const rocblas_stride strideV, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgesvdj_batched(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, rocblas_float_complex *const A[], const rocblas_int lda, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *S, const rocblas_stride strideS, rocblas_float_complex *U, const rocblas_int ldu, const rocblas_stride strideU, rocblas_float_complex *V, const rocblas_int ldv, const rocblas_stride strideV, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgesvdj_batched(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, double *const A[], const rocblas_int lda, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *S, const rocblas_stride strideS, double *U, const rocblas_int ldu, const rocblas_stride strideU, double *V, const rocblas_int ldv, const rocblas_stride strideV, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgesvdj_batched(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, float *const A[], const rocblas_int lda, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *S, const rocblas_stride strideS, float *U, const rocblas_int ldu, const rocblas_stride strideU, float *V, const rocblas_int ldv, const rocblas_stride strideV, rocblas_int *info, const rocblas_int batch_count)

GESVDJ_BATCHED computes the singular values and optionally the singular vectors of a batch of general m-by-n matrix A (Singular Value Decomposition).

The SVD of matrix A_l in the batch is given by:

\[ A_l = U_l S_l V_l' \]

where the m-by-n matrix \(S_l\) is zero except, possibly, for its min(m,n) diagonal elements, which are the singular values of \(A_l\). \(U_l\) and \(V_l\) are orthogonal (unitary) matrices. The first min(m,n) columns of \(U_l\) and \(V_l\) are the left and right singular vectors of \(A_l\), respectively.

The computation of the singular vectors is optional and it is controlled by the function arguments left_svect and right_svect as described below. When computed, this function returns the transpose (or transpose conjugate) of the right singular vectors, i.e. the rows of \(V_l'\).

left_svect and right_svect are rocblas_svect enums that can take the following values:

• rocblas_svect_all: the entire matrix \(U_l\) (or \(V_l'\)) is computed,

• rocblas_svect_singular: the singular vectors (first min(m,n) columns of \(U_l\) or rows of \(V_l'\)) are computed, or

• rocblas_svect_none: no columns (or rows) of \(U_l\) (or \(V_l'\)) are computed, i.e. no singular vectors.

The singular values are computed by applying QR factorization to \(A_lV_l\) if m >= n (resp. LQ factorization to \(U_l'A_l\) if m < n), where \(V_l\) (resp. \(U_l\)) is found as the eigenvectors of \(A_l'A_l\) (resp. \(A_lA_l'\)) using the Jacobi eigenvalue algorithm.

Note

In order to carry out calculations, this method may synchronize the stream contained within the rocblas_handle.

Parameters:

• handle – [in] rocblas_handle.

• left_svect – [in] rocblas_svect. Specifies how the left singular vectors are computed. rocblas_svect_overwrite is not supported.

• right_svect – [in] rocblas_svect. Specifies how the right singular vectors are computed. rocblas_svect_overwrite is not supported.

• m – [in] rocblas_int. m >= 0. The number of rows of all matrices A_l in the batch.

• n – [in] rocblas_int. n >= 0. The number of columns of all matrices A_l in the batch.

• A – [inout] Array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= m. The leading dimension of A_l.

• abstol – [in] real type. The absolute tolerance. The algorithm is considered to have converged once off(A_l’A_l) is <= norm(A_l’A_l) * abstol [resp. off(A_lA_l’) <= norm(A_lA_l’) * abstol]. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to real type on the GPU. The Frobenius norm of the off-diagonal elements of A_l’A_l (resp. A_lA_l’) at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• S – [out] pointer to real type. Array on the GPU (the size depends on the value of strideS). The singular values of A_l in decreasing order.

• strideS – [in] rocblas_stride. Stride from the start of one vector S_l to the next one S(l+1). There is no restriction for the value of strideS. Normal use case is strideS >= min(m,n).

• U – [out] pointer to type. Array on the GPU (the side depends on the value of strideU). The matrices U_l of left singular vectors stored as columns. Not referenced if left_svect is set to none.

• ldu – [in] rocblas_int. ldu >= m if left_svect is set to all or singular; ldu >= 1 otherwise. The leading dimension of U_l.

• strideU – [in] rocblas_stride. Stride from the start of one matrix U_l to the next one U(l+1). There is no restriction for the value of strideU. Normal use case is strideU >= ldu*min(m,n) if left_svect is set to singular, or strideU >= ldu*m when left_svect is equal to all.

• V – [out] pointer to type. Array on the GPU (the size depends on the value of strideV). The matrices V_l of right singular vectors stored as rows (transposed / conjugate-transposed). Not referenced if right_svect is set to none.

• ldv – [in] rocblas_int. ldv >= n if right_svect is set to all; ldv >= min(m,n) if right_svect is set to singular; or ldv >= 1 otherwise. The leading dimension of V.

• strideV – [in] rocblas_stride. Stride from the start of one matrix V_l to the next one V(l+1). There is no restriction for the value of strideV. Normal use case is strideV >= ldv*n.

• info – [out] pointer to a rocblas_int on the GPU. If info[l] = 0, successful exit. If info[l] = 1, the algorithm did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.

rocsolver_<type>gesvdj_strided_batched()

rocblas_status rocsolver_zgesvdj_strided_batched(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, rocblas_double_complex *A, const rocblas_int lda, const rocblas_stride strideA, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *S, const rocblas_stride strideS, rocblas_double_complex *U, const rocblas_int ldu, const rocblas_stride strideU, rocblas_double_complex *V, const rocblas_int ldv, const rocblas_stride strideV, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_cgesvdj_strided_batched(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, rocblas_float_complex *A, const rocblas_int lda, const rocblas_stride strideA, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *S, const rocblas_stride strideS, rocblas_float_complex *U, const rocblas_int ldu, const rocblas_stride strideU, rocblas_float_complex *V, const rocblas_int ldv, const rocblas_stride strideV, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_dgesvdj_strided_batched(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, double *A, const rocblas_int lda, const rocblas_stride strideA, const double abstol, double *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, double *S, const rocblas_stride strideS, double *U, const rocblas_int ldu, const rocblas_stride strideU, double *V, const rocblas_int ldv, const rocblas_stride strideV, rocblas_int *info, const rocblas_int batch_count)

rocblas_status rocsolver_sgesvdj_strided_batched(rocblas_handle handle, const rocblas_svect left_svect, const rocblas_svect right_svect, const rocblas_int m, const rocblas_int n, float *A, const rocblas_int lda, const rocblas_stride strideA, const float abstol, float *residual, const rocblas_int max_sweeps, rocblas_int *n_sweeps, float *S, const rocblas_stride strideS, float *U, const rocblas_int ldu, const rocblas_stride strideU, float *V, const rocblas_int ldv, const rocblas_stride strideV, rocblas_int *info, const rocblas_int batch_count)

GESVDJ_STRIDED_BATCHED computes the singular values and optionally the singular vectors of a batch of general m-by-n matrix A (Singular Value Decomposition).

The SVD of matrix A_l in the batch is given by:

\[ A_l = U_l S_l V_l' \]

where the m-by-n matrix \(S_l\) is zero except, possibly, for its min(m,n) diagonal elements, which are the singular values of \(A_l\). \(U_l\) and \(V_l\) are orthogonal (unitary) matrices. The first min(m,n) columns of \(U_l\) and \(V_l\) are the left and right singular vectors of \(A_l\), respectively.

The computation of the singular vectors is optional and it is controlled by the function arguments left_svect and right_svect as described below. When computed, this function returns the transpose (or transpose conjugate) of the right singular vectors, i.e. the rows of \(V_l'\).

left_svect and right_svect are rocblas_svect enums that can take the following values:

• rocblas_svect_all: the entire matrix \(U_l\) (or \(V_l'\)) is computed,

• rocblas_svect_singular: the singular vectors (first min(m,n) columns of \(U_l\) or rows of \(V_l'\)) are computed, or

• rocblas_svect_none: no columns (or rows) of \(U_l\) (or \(V_l'\)) are computed, i.e. no singular vectors.

The singular values are computed by applying QR factorization to \(A_lV_l\) if m >= n (resp. LQ factorization to \(U_l'A_l\) if m < n), where \(V_l\) (resp. \(U_l\)) is found as the eigenvectors of \(A_l'A_l\) (resp. \(A_lA_l'\)) using the Jacobi eigenvalue algorithm.

Note

In order to carry out calculations, this method may synchronize the stream contained within the rocblas_handle.

Parameters:

• handle – [in] rocblas_handle.

• left_svect – [in] rocblas_svect. Specifies how the left singular vectors are computed. rocblas_svect_overwrite is not supported.

• right_svect – [in] rocblas_svect. Specifies how the right singular vectors are computed. rocblas_svect_overwrite is not supported.

• m – [in] rocblas_int. m >= 0. The number of rows of all matrices A_l in the batch.

• n – [in] rocblas_int. n >= 0. The number of columns of all matrices A_l in the batch.

• A – [inout] pointer to type. Array on the GPU (the size depends on the value of strideA). On entry, the matrices A_l. On exit, the contents of A_l are destroyed.

• lda – [in] rocblas_int. lda >= m. The leading dimension of A_l.

• strideA – [in] rocblas_stride. Stride from the start of one matrix A_l to the next one A_(l+1). There is no restriction for the value of strideA. Normal use case is strideA >= lda*n.

• abstol – [in] real type. The absolute tolerance. The algorithm is considered to have converged once off(A_l’A_l) is <= norm(A_l’A_l) * abstol [resp. off(A_lA_l’) <= norm(A_lA_l’) * abstol]. If abstol <= 0, then the tolerance will be set to machine precision.

• residual – [out] pointer to real type on the GPU. The Frobenius norm of the off-diagonal elements of A_l’A_l (resp. A_lA_l’) at the final iteration.

• max_sweeps – [in] rocblas_int. max_sweeps > 0. Maximum number of sweeps (iterations) to be used by the algorithm.

• n_sweeps – [out] pointer to rocblas_int. Array of batch_count integers on the GPU. The actual number of sweeps (iterations) used by the algorithm for each batch instance.

• S – [out] pointer to real type. Array on the GPU (the size depends on the value of strideS). The singular values of A_l in decreasing order.

• strideS – [in] rocblas_stride. Stride from the start of one vector S_l to the next one S_(j+1). There is no restriction for the value of strideS. Normal use case is strideS >= min(m,n).

• U – [out] pointer to type. Array on the GPU (the side depends on the value of strideU). The matrices U_l of left singular vectors stored as columns. Not referenced if left_svect is set to none.

• ldu – [in] rocblas_int. ldu >= m if left_svect is set to all or singular; ldu >= 1 otherwise. The leading dimension of U_l.

• strideU – [in] rocblas_stride. Stride from the start of one matrix U_l to the next one U_(j+1). There is no restriction for the value of strideU. Normal use case is strideU >= ldu*min(m,n) if left_svect is set to singular, or strideU >= ldu*m when left_svect is equal to all.

• V – [out] pointer to type. Array on the GPU (the size depends on the value of strideV). The matrices V_l of right singular vectors stored as rows (transposed / conjugate-transposed). Not referenced if right_svect is set to none.

• ldv – [in] rocblas_int. ldv >= n if right_svect is set to all; ldv >= min(m,n) if right_svect is set to singular; or ldv >= 1 otherwise. The leading dimension of V.

• strideV – [in] rocblas_stride. Stride from the start of one matrix V_l to the next one V_(j+1). There is no restriction for the value of strideV. Normal use case is strideV >= ldv*n.

• info – [out] pointer to a rocblas_int on the GPU. If info[l] = 0, successful exit. If info[l] = 1, the algorithm did not converge.

• batch_count – [in] rocblas_int. batch_count >= 0. Number of matrices in the batch.