SYTF2_BATCHED computes the factorization of a batch of symmetric indefinite matrices using Bunch-Kaufman diagonal pivoting.

(This is the unblocked version of the algorithm).

The factorization has the form

$$\begin{array}{cl} A_i = U_i D_i U_i^T & \: \text{or}\newline A_i = L_i D_i L_i^T & \end{array}$$

where $U_i$ or $L_i$ is a product of permutation and unit upper/lower triangular matrices (depending on the value of uplo), and $D_i$ is a symmetric block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks $D_i(k)$.

Specifically, $U_i$ and $L_i$ are computed as

$$\begin{array}{cl} U_i = P_i(n) U_i(n) \cdots P_i(k) U_i(k) \cdots & \: \text{and}\newline L_i = P_i(1) L_i(1) \cdots P_i(k) L_i(k) \cdots & \end{array}$$

where $k$ decreases from $n$ to 1 (increases from 1 to $n$) in steps of 1 or 2, depending on the order of block $D_i(k)$, and $P_i(k)$ is a permutation matrix defined by $ipiv_i[k]$. If we let $s$ denote the order of block $D_i(k)$, then $U_i(k)$ and $L_i(k)$ are unit upper/lower triangular matrices defined as

$$U_i(k) = \left[ \begin{array}{ccc} I_{k-s} & v & 0 \newline 0 & I_s & 0 \newline 0 & 0 & I_{n-k} \end{array} \right]$$

and

$$L_i(k) = \left[ \begin{array}{ccc} I_{k-1} & 0 & 0 \newline 0 & I_s & 0 \newline 0 & v & I_{n-k-s+1} \end{array} \right].$$

If $s = 1$, then $D_i(k)$ is stored in $A_i[k,k]$ and $v$ is stored in the upper/lower part of column $k$ of $A_i$. If $s = 2$ and uplo is upper, then $D_i(k)$ is stored in $A_i[k-1,k-1]$, $A_i[k-1,k]$, and $A_i[k,k]$, and $v$ is stored in the upper parts of columns $k-1$ and $k$ of $A_i$. If $s = 2$ and uplo is lower, then $D_i(k)$ is stored in $A_i[k,k]$, $A_i[k+1,k]$, and $A_i[k+1,k+1]$, and $v$ is stored in the lower parts of columns $k$ and $k+1$ of $A_i$.

Parameters

[in]	handle	rocblas_handle.
[in]	uplo	rocblas_fill. Specifies whether the upper or lower part of the matrices A_i are stored. If uplo indicates lower (or upper), then the upper (or lower) part of A_i is not used.
[in]	n	rocblas_int. n >= 0. The number of rows and columns of all matrices A_i in the batch.
[in,out]	A	array of pointers to type. Each pointer points to an array on the GPU of dimension lda*n. On entry, the symmetric matrices A_i to be factored. On exit, the block diagonal matrices D_i and the multipliers needed to compute U_i or L_i.
[in]	lda	rocblas_int. lda >= n. Specifies the leading dimension of matrices A_i.
[out]	ipiv	pointer to rocblas_int. Array on the GPU of dimension n. The vector of pivot indices. Elements of ipiv are 1-based indices. For 1 <= k <= n, if ipiv_i[k] > 0 then rows and columns k and ipiv_i[k] were interchanged and D_i[k,k] is a 1-by-1 diagonal block. If, instead, ipiv_i[k] = ipiv_i[k-1] < 0 and uplo is upper (or ipiv_i[k] = ipiv_i[k+1] < 0 and uplo is lower), then rows and columns k-1 and -ipiv_i[k] (or rows and columns k+1 and -ipiv_i[k]) were interchanged and D_i[k-1,k-1] to D_i[k,k] (or D_i[k,k] to D_i[k+1,k+1]) is a 2-by-2 diagonal block.
[in]	strideP	rocblas_stride. Stride from the start of one vector ipiv_i to the next one ipiv_(i+1). There is no restriction for the value of strideP. Normal use case is strideP >= n.
[out]	info	pointer to rocblas_int. Array of batch_count integers on the GPU. If info[i] = 0, successful exit for factorization of A_i. If info[i] = j > 0, D_i is singular. D_i[j,j] is the first diagonal zero.
[in]	batch_count	rocblas_int. batch_count >= 0. Number of matrices in the batch.