rocsolver_slatrd Interface Reference

rocsolver_slatrd Interface Reference#

HIPFORT API Reference: hipfort_rocsolver::rocsolver_slatrd Interface Reference

LATRD computes the tridiagonal form of k rows and columns of a symmetric/hermitian matrix A, as well as the matrix W needed to update the remaining part of A. More...

Public Member Functions
integer(kind(rocblas_status_success)) function	rocsolver_slatrd_ (handle, uplo, n, k, a, lda, e, tau, w, ldw)

integer(kind(rocblas_status_success)) function	rocsolver_slatrd_rank_0 (handle, uplo, n, k, a, lda, e, tau, w, ldw)

integer(kind(rocblas_status_success)) function	rocsolver_slatrd_rank_1 (handle, uplo, n, k, a, lda, e, tau, w, ldw)

integer(kind(rocblas_status_success)) function	rocsolver_slatrd_full_rank (handle, uplo, n, k, a, lda, e, tau, w, ldw)

Detailed Description

LATRD computes the tridiagonal form of k rows and columns of a symmetric/hermitian matrix A, as well as the matrix W needed to update the remaining part of A.

The reduced form is given by:

\[ T = Q^H A Q \]

If uplo is lower, the first k rows and columns of T form the tridiagonal block. If uplo is upper, then the last k rows and columns of T form the tridiagonal block. Q is an orthogonal/unitary matrix represented as the product of Householder matrices

\[ \begin{array}{cl} Q = H(1)H(2)\cdots H(k) & \text{if uplo indicates lower, or}\\% Q = H(n)H(n-1)\cdots H(n-k+1) & \text{if uplo is upper}. \end{array} \]

Each Householder matrix \(H(i)\) is given by

\[ H(i) = I - \text{tau}[i]\cdot v_i^{}v_i^H \]

where tau[ \(i\)] is the corresponding Householder scalar. When uplo indicates lower, the first \(i\) elements of the Householder vector \(v_i\) are zero, and \(v_i[i+1] = 1\). If uplo is upper, the last n- \(i\) elements of the Householder vector \(v_i\) are zero, and \(v_i[i] = 1\).

The unreduced part of the matrix A can be updated using a rank update of the form:

\[ A = A - VW^H - WV^H \]

where V is the n-by-k matrix formed by the vectors \(v_i\).

Parameters

[in]	handle	- rocblas_handle.
[in]	uplo	- rocblas_fill. Specifies whether the upper or lower part of the matrix A is stored. If uplo indicates lower (or upper), then the upper (or lower) part of A is not used.
[in]	n	- rocblas_int. n >= 0. The number of rows and columns of the matrix A.
[in]	k	- rocblas_int. 0 <= k <= n. The number of rows and columns of the matrix A to be reduced.
[in,out]	A	- pointer to type. Array on the GPU of dimension lda*n. On entry, the n-by-n matrix to be reduced. On exit, if uplo is lower, the first k columns have been reduced to tridiagonal form (given in the diagonal elements of A and the array E), the elements below the diagonal contain the possibly non-zero entries of the Householder vectors associated with Q, stored as columns. If uplo is upper, the last k columns have been reduced to tridiagonal form (given in the diagonal elements of A and the array E), the elements above the diagonal contain the possibly non-zero entries of the Householder vectors associated with Q, stored as columns.
[in]	lda	- rocblas_int. lda >= n. The leading dimension of A.
[out]	E	- pointer to real type. Array on the GPU of dimension n-1. If upper (lower), the last (first) k elements of E are the off-diagonal elements of the computed tridiagonal block.
[out]	tau	- pointer to type. Array on the GPU of dimension n-1. If upper (lower), the last (first) k elements of tau are the Householder scalars related to Q.
[out]	W	- pointer to type. Array on the GPU of dimension ldw*k. The n-by-k matrix needed to update the unreduced part of A.
[in]	ldw	- rocblas_int. ldw >= n. The leading dimension of W.

Member Function/Subroutine Documentation