SYTF2 computes the factorization of a symmetric indefinite matrix $A$ using Bunch-Kaufman diagonal pivoting.

(This is the unblocked version of the algorithm).

The factorization has the form

$$\begin{array}{cl} A = U D U^T & \: \text{or}\newline A = L D L^T & \end{array}$$

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

Specifically, $U$ and $L$ are computed as

$$\begin{array}{cl} U = P(n) U(n) \cdots P(k) U(k) \cdots & \: \text{and}\newline L = P(1) L(1) \cdots P(k) L(k) \cdots & \end{array}$$

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

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

and

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

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

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,out]	A	pointer to type. Array on the GPU of dimension lda*n. On entry, the symmetric matrix A to be factored. On exit, the block diagonal matrix D and the multipliers needed to compute U or L.
[in]	lda	rocblas_int. lda >= n. Specifies the leading dimension of A.
[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[k] > 0 then rows and columns k and ipiv[k] were interchanged and D[k,k] is a 1-by-1 diagonal block. If, instead, ipiv[k] = ipiv[k-1] < 0 and uplo is upper (or ipiv[k] = ipiv[k+1] < 0 and uplo is lower), then rows and columns k-1 and -ipiv[k] (or rows and columns k+1 and -ipiv[k]) were interchanged and D[k-1,k-1] to D[k,k] (or D[k,k] to D[k+1,k+1]) is a 2-by-2 diagonal block.
[out]	info	pointer to a rocblas_int on the GPU. If info = 0, successful exit. If info[i] = j > 0, D is singular. D[j,j] is the first diagonal zero.