ssytrf(l) — SunSoft Performance Library

NAME

ssytrf - compute the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS

SUBROUTINE SSYTRF(
UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )

CHARACTER UPLO

INTEGER INFO, LDA, LWORK, N

INTEGER IPIV( ∗ )

REAL A( LDA, ∗ ), WORK( LWORK )

PURPOSE

SSYTRF computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method. The form of the factorization is

A = U∗D∗U∗∗T or A = L∗D∗L∗∗T

where U (or L) is a product of permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Level 3 BLAS.

ARGUMENTS

UPLO (input) CHARACTER∗1
= ’U’: Upper triangle of A is stored;
= ’L’: Lower triangle of A is stored.

N (input) INTEGER
The order of the matrix A. N >= 0.

A (input/output) REAL array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = ’U’, the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A, and the strictly lower triangular part of A is not referenced. If UPLO = ’L’, the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A, and the strictly upper triangular part of A is not referenced.

On exit, the block diagonal matrix D and the multipliers used to obtain the factor U or L (see below for further details).

LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).

IPIV (output) INTEGER array, dimension (N)
Details of the interchanges and the block structure of D. 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 UPLO = ’U’ and IPIV(k) = IPIV(k-1) < 0, then rows and columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) is a 2-by-2 diagonal block. If UPLO = ’L’ and IPIV(k) = IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.

WORK (workspace/output) REAL array, dimension (LWORK)
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK (input) INTEGER
The length of WORK. LWORK >=1. For best performance LWORK >= N∗NB, where NB is the block size returned by ILAENV.

INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: if INFO = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.

FURTHER DETAILS

If UPLO = ’U’, then A = U∗D∗U’, where
   U = P(n)∗U(n)∗ ... ∗P(k)U(k)∗ ...,
i.e., U is a product of terms P(k)∗U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    v    0   )   k-s
   U(k) = (   0    I    0   )   s
           (   0    0    I   )   n-k
              k-s   s   n-k

If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites A(1:k-2,k-1:k).

If UPLO = ’L’, then A = L∗D∗L’, where
   L = P(1)∗L(1)∗ ... ∗P(k)∗L(k)∗ ...,
i.e., L is a product of terms P(k)∗L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by IPIV(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then

           (   I    0     0   ) k-1
   L(k) = (   0    I     0   ) s
           (   0    v     I   ) n-k-s+1
              k-1   s n-k-s+1

If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1).