zhetrf (l) - Linux Manuals

zhetrf: computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal pivoting method

Command to display zhetrf manual in Linux: $ man l zhetrf

NAME

ZHETRF - computes 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( * )

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: = aqUaq: Upper triangle of A is stored;
= aqLaq: 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 = aqUaq, 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 = aqLaq, 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 interchanged and D(k,k) is a 1-by-1 diagonal block. If UPLO = aqUaq 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 = aqLaq 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 (MAX(1,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 = aqUaq, then A = U*D*Uaq, where

U = H(n)*H(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 = aqLaq, then A = L*D*Laq, 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).