zhesv_rk()

subroutine zhesv_rk	(	character	uplo,
		integer	n,
		integer	nrhs,
		complex*16, dimension( lda, * )	a,
		integer	lda,
		complex*16, dimension( * )	e,
		integer, dimension( * )	ipiv,
		complex*16, dimension( ldb, * )	b,
		integer	ldb,
		complex*16, dimension( * )	work,
		integer	lwork,
		integer	info )

ZHESV_RK computes the solution to system of linear equations A * X = B for SY matrices

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Purpose:

!> ZHESV_RK computes the solution to a complex system of linear
!> equations A * X = B, where A is an N-by-N Hermitian matrix
!> and X and B are N-by-NRHS matrices.
!>
!> The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
!> to factor A as
!>    A = P*U*D*(U**H)*(P**T),  if UPLO = 'U', or
!>    A = P*L*D*(L**H)*(P**T),  if UPLO = 'L',
!> where U (or L) is unit upper (or lower) triangular matrix,
!> U**H (or L**H) is the conjugate of U (or L), P is a permutation
!> matrix, P**T is the transpose of P, and D is Hermitian and block
!> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
!>
!> ZHETRF_RK is called to compute the factorization of a complex
!> Hermitian matrix.  The factored form of A is then used to solve
!> the system of equations A * X = B by calling BLAS3 routine ZHETRS_3.
!> 

Parameters

[in]	UPLO	!> UPLO is CHARACTER*1 !> Specifies whether the upper or lower triangular part of the !> Hermitian matrix A is stored: !> = 'U': Upper triangle of A is stored; !> = 'L': Lower triangle of A is stored. !>
[in]	N	!> N is INTEGER !> The number of linear equations, i.e., the order of the !> matrix A. N >= 0. !>
[in]	NRHS	!> NRHS is INTEGER !> The number of right hand sides, i.e., the number of columns !> of the matrix B. NRHS >= 0. !>
[in,out]	A	!> A is 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, if INFO = 0, diagonal of the block diagonal !> matrix D and factors U or L as computed by ZHETRF_RK: !> a) ONLY diagonal elements of the Hermitian block diagonal !> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k); !> (superdiagonal (or subdiagonal) elements of D !> are stored on exit in array E), and !> b) If UPLO = 'U': factor U in the superdiagonal part of A. !> If UPLO = 'L': factor L in the subdiagonal part of A. !> !> For more info see the description of ZHETRF_RK routine. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !>
[out]	E	!> E is COMPLEX*16 array, dimension (N) !> On exit, contains the output computed by the factorization !> routine ZHETRF_RK, i.e. the superdiagonal (or subdiagonal) !> elements of the Hermitian block diagonal matrix D !> with 1-by-1 or 2-by-2 diagonal blocks, where !> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0; !> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0. !> !> NOTE: For 1-by-1 diagonal block D(k), where !> 1 <= k <= N, the element E(k) is set to 0 in both !> UPLO = 'U' or UPLO = 'L' cases. !> !> For more info see the description of ZHETRF_RK routine. !>
[out]	IPIV	!> IPIV is INTEGER array, dimension (N) !> Details of the interchanges and the block structure of D, !> as determined by ZHETRF_RK. !> !> For more info see the description of ZHETRF_RK routine. !>
[in,out]	B	!> B is COMPLEX*16 array, dimension (LDB,NRHS) !> On entry, the N-by-NRHS right hand side matrix B. !> On exit, if INFO = 0, the N-by-NRHS solution matrix X. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[out]	WORK	!> WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ). !> Work array used in the factorization stage. !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> LWORK is INTEGER !> The length of WORK. LWORK >= 1. For best performance !> of factorization stage LWORK >= max(1,N*NB), where NB is !> the optimal blocksize for ZHETRF_RK. !> !> If LWORK = -1, then a workspace query is assumed; !> the routine only calculates the optimal size of the WORK !> array for factorization stage, returns this value as !> the first entry of the WORK array, and no error message !> related to LWORK is issued by XERBLA. !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> !> < 0: If INFO = -k, the k-th argument had an illegal value !> !> > 0: If INFO = k, the matrix A is singular, because: !> If UPLO = 'U': column k in the upper !> triangular part of A contains all zeros. !> If UPLO = 'L': column k in the lower !> triangular part of A contains all zeros. !> !> Therefore D(k,k) is exactly zero, and superdiagonal !> elements of column k of U (or subdiagonal elements of !> column k of L ) are all zeros. 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. !> !> NOTE: INFO only stores the first occurrence of !> a singularity, any subsequent occurrence of singularity !> is not stored in INFO even though the factorization !> always completes. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

!>
!>  December 2016,  Igor Kozachenko,
!>                  Computer Science Division,
!>                  University of California, Berkeley
!>
!>  September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
!>                  School of Mathematics,
!>                  University of Manchester
!>
!>