ZHETRF(3F)ZHETRF(3F)NAME
ZHETRF - compute the factorization of a complex Hermitian matrix A using
the Bunch-Kaufman diagonal pivoting method
SYNOPSIS
SUBROUTINE ZHETRF( UPLO, N, A, LDA, IPIV, WORK, LWORK, INFO )
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * ), WORK( LWORK )
PURPOSE
ZHETRF computes the factorization of a complex Hermitian matrix A using
the Bunch-Kaufman diagonal pivoting method. The form of the
factorization is
A = U*D*U**H or A = L*D*L**H
where U (or L) is a product of permutation and unit upper (lower)
triangular matrices, and D is Hermitian and block diagonal 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 Hermitian 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).
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ZHETRF(3F)ZHETRF(3F)
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 =
'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 performance LWORK >=
N*NB, where NB is the block size returned by ILAENV.
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
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ZHETRF(3F)ZHETRF(3F)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).
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