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## zlasyf_rk()

subroutine zlasyf_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 )

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

Download ZLASYF_RK + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !> ZLASYF_RK computes a partial factorization of a complex symmetric
      !> 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**T U22**T )
      !>
      !> A  =  ( L11  0 ) (  D   0  ) ( L11**T L21**T )  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.
      !>
      !> ZLASYF_RK is an auxiliary routine called by ZSYTRF_RK. It uses
      !> blocked code (calling Level 3 BLAS) to update the submatrix
      !> A11 (if UPLO = 'U') or A22 (if UPLO = 'L').
      !>

Parameters

  [in]

  UPLO

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

  [in]

  N

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

  [in]

  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.
      !>

  [out]

  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.
      !>

  [in,out]

  A

      !>          A is COMPLEX*16 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, contains:
      !>            a) ONLY diagonal elements of the symmetric 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.
      !>

  [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 superdiagonal (or subdiagonal)
      !>          elements of the symmetric 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.
      !>

  [out]

  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 symmetric 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.
      !>

  [out]

  W

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

  [in]

  LDW

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

  [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
      !>
      !>