CSYTRF(l) ) CSYTRF(l)NAME
CSYTRF - compute the factorization of a complex symmetric matrix A
using the Bunch-Kaufman diagonal pivoting method
SYNOPSIS
SUBROUTINE CSYTRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
INTEGER IPIV( * )
COMPLEX A( LDA, * ), WORK( * )
PURPOSE
CSYTRF computes the factorization of a complex symmetric matrix A using
the Bunch-Kaufman diagonal pivoting method. The form of the factoriza‐
tion 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) tri‐
angular 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 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 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) 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 inter‐
changed 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 diag‐
onal 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 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 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.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal 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 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).
LAPACK version 3.0 15 June 2000 CSYTRF(l)