NAME

dsytf2 - compute the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS

  SUBROUTINE DSYTF2( UPLO, N, A, LDA, IPIV, INFO)
  CHARACTER * 1 UPLO
  INTEGER N, LDA, INFO
  INTEGER IPIV(*)
  DOUBLE PRECISION A(LDA,*)

  SUBROUTINE DSYTF2_64( UPLO, N, A, LDA, IPIV, INFO)
  CHARACTER * 1 UPLO
  INTEGER*8 N, LDA, INFO
  INTEGER*8 IPIV(*)
  DOUBLE PRECISION A(LDA,*)

F95 INTERFACE

  SUBROUTINE SYTF2( UPLO, [N], A, [LDA], IPIV, [INFO])
  CHARACTER(LEN=1) :: UPLO
  INTEGER :: N, LDA, INFO
  INTEGER, DIMENSION(:) :: IPIV
  REAL(8), DIMENSION(:,:) :: A

  SUBROUTINE SYTF2_64( UPLO, [N], A, [LDA], IPIV, [INFO])
  CHARACTER(LEN=1) :: UPLO
  INTEGER(8) :: N, LDA, INFO
  INTEGER(8), DIMENSION(:) :: IPIV
  REAL(8), DIMENSION(:,:) :: A

C INTERFACE

#include <sunperf.h>

void dsytf2(char uplo, int n, double *a, int lda, int *ipiv, int *info);

void dsytf2_64(char uplo, long n, double *a, long lda, long *ipiv, long *info);

PURPOSE

dsytf2 computes the factorization of a real symmetric matrix A using the Bunch-Kaufman diagonal pivoting method:

   A = U*D*U'  or  A = L*D*L'

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

This is the unblocked version of the algorithm, calling Level 2 BLAS.

ARGUMENTS

UPLO (input)
Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
```
 = 'U':  Upper triangular
```
```
 = 'L':  Lower triangular
```
N (input)
The order of the matrix A. N > = 0.
A (input/output)
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)
The leading dimension of the array A. LDA > = max(1,N).
IPIV (output)
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.

INFO (output)

 = 0: successful exit

 < 0: if INFO  = -k, the k-th argument had an illegal value

 > 0: if INFO  = k, D(k,k) 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

1-96 - Based on modifications by J. Lewis, Boeing Computer Services Company

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).