SRC/zlahef_rk.f(3)

NAME
SRC/zlahef_rk.f

SYNOPSIS

  Functions/Subroutines
subroutine zlahef_rk (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)
ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.

Function/Subroutine Documentation

  subroutine zlahef_rk (character uplo, integer n, integer nb, integer kb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( * ) e, integer, dimension( * ) ipiv, complex*16, dimension( ldw, * ) w, integer ldw, integer info)

ZLAHEF_RK computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method.

Purpose:

    !> ZLAHEF_RK computes a partial factorization of a complex Hermitian
    !> matrix A using the bounded Bunch-Kaufman (rook) diagonal
    !> pivoting method. The partial factorization has the form:
    !>
    !> A  =  ( I  U12 ) ( A11  0  ) (  I       0    )  if UPLO = 'U', or:
    !>       ( 0  U22 ) (  0   D  ) ( U12**H U22**H )
    !>
    !> A  =  ( L11  0 ) (  D   0  ) ( L11**H L21**H )  if UPLO = 'L',
    !>       ( L21  I ) (  0  A22 ) (  0       I    )
    !>
    !> where the order of D is at most NB. The actual order is returned in
    !> the argument KB, and is either NB or NB-1, or N if N <= NB.
    !>
    !> ZLAHEF_RK is an auxiliary routine called by ZHETRF_RK. It uses
    !> blocked code (calling Level 3 BLAS) to update the submatrix
    !> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
    !>

Parameters
    UPLO

    !>          UPLO is CHARACTER*1
    !>          Specifies whether the upper or lower triangular part of the
    !>          Hermitian matrix A is stored:
    !>          = 'U':  Upper triangular
    !>          = 'L':  Lower triangular
    !>

    N

    !>          N is INTEGER
    !>          The order of the matrix A.  N >= 0.
    !>

    NB

    !>          NB is INTEGER
    !>          The maximum number of columns of the matrix A that should be
    !>          factored.  NB should be at least 2 to allow for 2-by-2 pivot
    !>          blocks.
    !>

    KB

    !>          KB is INTEGER
    !>          The number of columns of A that were actually factored.
    !>          KB is either NB-1 or NB, or N if N <= NB.
    !>

    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, contains:
    !>            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.
    !>

    LDA

    !>          LDA is INTEGER
    !>          The leading dimension of the array A.  LDA >= max(1,N).
    !>

    E

    !>          E is COMPLEX*16 array, dimension (N)
    !>          On exit, contains 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.
    !>

    IPIV

    !>          IPIV is INTEGER array, dimension (N)
    !>          IPIV describes the permutation matrix P in the factorization
    !>          of matrix A as follows. The absolute value of IPIV(k)
    !>          represents the index of row and column that were
    !>          interchanged with the k-th row and column. The value of UPLO
    !>          describes the order in which the interchanges were applied.
    !>          Also, the sign of IPIV represents the block structure of
    !>          the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2
    !>          diagonal blocks which correspond to 1 or 2 interchanges
    !>          at each factorization step.
    !>
    !>          If UPLO = 'U',
    !>          ( in factorization order, k decreases from N to 1 ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the submatrix A(1:N,N-KB+1:N);
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k-1) < 0 means:
    !>               D(k-1:k,k-1:k) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the matrix A(1:N,N-KB+1:N).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k-1) != k-1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the submatrix A(1:N,N-KB+1:N).
    !>                  If -IPIV(k-1) = k-1, no interchange occurred.
    !>
    !>            c) In both cases a) and b) is always ABS( IPIV(k) ) <= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>
    !>          If UPLO = 'L',
    !>          ( in factorization order, k increases from 1 to N ):
    !>            a) A single positive entry IPIV(k) > 0 means:
    !>               D(k,k) is a 1-by-1 diagonal block.
    !>               If IPIV(k) != k, rows and columns k and IPIV(k) were
    !>               interchanged in the submatrix A(1:N,1:KB).
    !>               If IPIV(k) = k, no interchange occurred.
    !>
    !>            b) A pair of consecutive negative entries
    !>               IPIV(k) < 0 and IPIV(k+1) < 0 means:
    !>               D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
    !>               (NOTE: negative entries in IPIV appear ONLY in pairs).
    !>               1) If -IPIV(k) != k, rows and columns
    !>                  k and -IPIV(k) were interchanged
    !>                  in the submatrix A(1:N,1:KB).
    !>                  If -IPIV(k) = k, no interchange occurred.
    !>               2) If -IPIV(k+1) != k+1, rows and columns
    !>                  k-1 and -IPIV(k-1) were interchanged
    !>                  in the submatrix A(1:N,1:KB).
    !>                  If -IPIV(k+1) = k+1, no interchange occurred.
    !>
    !>            c) In both cases a) and b) is always ABS( IPIV(k) ) >= k.
    !>
    !>            d) NOTE: Any entry IPIV(k) is always NONZERO on output.
    !>

    W

    !>          W is COMPLEX*16 array, dimension (LDW,NB)
    !>

    LDW

    !>          LDW is INTEGER
    !>          The leading dimension of the array W.  LDW >= max(1,N).
    !>

    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
    !>
    !>

Definition at line 260 of file zlahef_rk.f.

Author

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