.TH "SRC/zlahef_rk.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
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.SH NAME
SRC/zlahef_rk.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBzlahef_rk\fP (uplo, n, nb, kb, a, lda, e, ipiv, w, ldw, info)"
.br
.RI "\fBZLAHEF_RK\fP computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&. "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "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)"

.PP
\fBZLAHEF_RK\fP computes a partial factorization of a complex Hermitian indefinite matrix using bounded Bunch-Kaufman (rook) diagonal pivoting method\&.  
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\fBPurpose:\fP
.RS 4

.PP
.nf
!> 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')\&.
!> 
.fi
.PP
 
.RE
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\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
!>          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
!> 
.fi
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\fIN\fP 
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!>          N is INTEGER
!>          The order of the matrix A\&.  N >= 0\&.
!> 
.fi
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\fINB\fP 
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!>          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\&.
!> 
.fi
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\fIKB\fP 
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.nf
!>          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\&.
!> 
.fi
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.br
\fIA\fP 
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.nf
!>          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\&.
!> 
.fi
.PP
.br
\fILDA\fP 
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!>          LDA is INTEGER
!>          The leading dimension of the array A\&.  LDA >= max(1,N)\&.
!> 
.fi
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.br
\fIE\fP 
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.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIIPIV\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIW\fP 
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!>          W is COMPLEX*16 array, dimension (LDW,NB)
!> 
.fi
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\fILDW\fP 
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.nf
!>          LDW is INTEGER
!>          The leading dimension of the array W\&.  LDW >= max(1,N)\&.
!> 
.fi
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.br
\fIINFO\fP 
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.nf
!>          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\&.
!> 
.fi
.PP
 
.RE
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\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

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Univ\&. of California Berkeley 

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Univ\&. of Colorado Denver 

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NAG Ltd\&. 
.RE
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\fBContributors:\fP
.RS 4

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.nf
!>
!>  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
!>
!> 
.fi
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.RE
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Definition at line \fB260\fP of file \fBzlahef_rk\&.f\fP\&.
.SH "Author"
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Generated automatically by Doxygen for LAPACK from the source code\&.