# zsytf2 (l) - Linux Man Pages

## NAME

ZSYTF2 - computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method

## SYNOPSIS

SUBROUTINE ZSYTF2(
UPLO, N, A, LDA, IPIV, INFO )

CHARACTER UPLO

INTEGER INFO, LDA, N

INTEGER IPIV( * )

COMPLEX*16 A( LDA, * )

## PURPOSE

ZSYTF2 computes the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method:
U*D*Uaq  or  L*D*Laq
where U (or L) is a product of permutation and unit upper (lower) triangular matrices, Uaq 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) CHARACTER*1
Specifies whether the upper or lower triangular part of the symmetric matrix A is stored:
= aqUaq: Upper triangular
= aqLaq: Lower triangular
N (input) INTEGER
The order of the matrix A. N >= 0.
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = aqUaq, 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 = aqLaq, 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 = aqUaq 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 = aqLaq 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) INTEGER

= 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

09-29-06 - patch from

Bobby Cheng, MathWorks

Replace l.209 and l.377

IF( MAX( ABSAKK, COLMAX ).EQ.ZERO THEN

by

IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) THEN 1-96 - Based on modifications by J. Lewis, Boeing Computer Services
Company
If UPLO = aqUaq, then A = U*D*Uaq, where

H(n)*H(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
k-s

U(k)              s

n-k

k-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 = aqLaq, then A = L*D*Laq, where

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
k-1

L(k)              s

n-k-s+1

k-1    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).