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
diagonal with 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) 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 con
tains 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 tri
angular 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).
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 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) 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 divi
sion 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
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) over
writes 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) over
writes A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites
A(k+2:n,k:k+1).
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