zsytrf(3)

NAME

ZSYTRF - compute the factorization of a complex symmetric

matrix A using the Bunch-Kaufman diagonal pivoting method

SYNOPSIS

SUBROUTINE ZSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
    CHARACTER      UPLO
    INTEGER        INFO, LDA, LWORK, N
    INTEGER        IPIV( * )
    COMPLEX*16     A( LDA, * ), WORK( * )

PURPOSE

ZSYTRF computes the factorization of a complex 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 diago

nal with with 1-by-1 and 2-by-2 diagonal blocks.

This is the blocked version of the algorithm, calling Lev

el 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) COMPLEX*16 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 triangu

lar 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 mul

tipliers 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 struc

ture 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 diago

nal 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) COMPLEX*16 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 perfor

mance LWORK >= N*NB, where NB is the block size returned by

ILAENV.

If LWORK = -1, then a workspace query is assumed;

the routine only calculates the optimal size of the WORK array,

returns this value as the first entry of the WORK array, and no

error message related to LWORK is issued by XERBLA.

INFO (output) INTEGER

= 0: successful exit
< 0: if INFO = -i, the i-th argument had an ille

gal 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 permuta

tion matrix as defined by IPIV(k), and U(k) is a unit upper tri

angular 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 permuta

tion matrix as defined by IPIV(k), and L(k) is a unit lower tri

angular 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).

LAPACK version 3.0 15 June 2000

ZSYTRF(3)