.TH ZSYTRF 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
ZSYTRF - the factorization of a complex symmetric matrix A using the Bunch-Kaufman diagonal pivoting method
.SH SYNOPSIS
.TP 19
SUBROUTINE ZSYTRF(
UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
.TP 19
.ti +4
CHARACTER
UPLO
.TP 19
.ti +4
INTEGER
INFO, LDA, LWORK, N
.TP 19
.ti +4
INTEGER
IPIV( * )
.TP 19
.ti +4
COMPLEX*16
A( LDA, * ), WORK( * )
.SH PURPOSE
ZSYTRF computes the factorization of a complex symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method. The form of the
factorization is
.br
A = U*D*U**T or A = L*D*L**T
.br
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is symmetric and block diagonal with
with 1-by-1 and 2-by-2 diagonal blocks.
.br
This is the blocked version of the algorithm, calling Level 3 BLAS.
.SH ARGUMENTS
.TP 8
UPLO (input) CHARACTER*1
= \(aqU\(aq: Upper triangle of A is stored;
.br
= \(aqL\(aq: Lower triangle of A is stored.
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
A (input/output) COMPLEX*16 array, dimension (LDA,N)
On entry, the symmetric matrix A. If UPLO = \(aqU\(aq, 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 = \(aqL\(aq, 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).
.TP 8
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
.TP 8
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 = \(aqU\(aq 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 = \(aqL\(aq 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.
.TP 8
WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
.TP 8
LWORK (input) INTEGER
The length of WORK. LWORK >=1. For best performance
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.
.TP 8
INFO (output) INTEGER
= 0: successful exit
.br
< 0: if INFO = -i, the i-th argument had an illegal value
.br
> 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.
.SH FURTHER DETAILS
If UPLO = \(aqU\(aq, then A = U*D*U\(aq, where
.br
U = P(n)*U(n)* ... *P(k)U(k)* ...,
.br
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
.br
U(k) = ( 0 I 0 ) s
.br
( 0 0 I ) n-k
.br
k-s s n-k
.br
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).
.br
If UPLO = \(aqL\(aq, then A = L*D*L\(aq, where
.br
L = P(1)*L(1)* ... *P(k)*L(k)* ...,
.br
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
.br
L(k) = ( 0 I 0 ) s
.br
( 0 v I ) n-k-s+1
.br
k-1 s n-k-s+1
.br
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).
.br