*> \brief \b ZHETRF_RK computes the factorization of a complex Hermitian indefinite matrix using the bounded Bunch-Kaufman (rook) diagonal pivoting method (BLAS3 blocked algorithm).
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
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*
* Definition:
* ===========
*
* SUBROUTINE ZHETRF_RK( UPLO, N, A, LDA, E, IPIV, WORK, LWORK,
* INFO )
*
* .. Scalar Arguments ..
* CHARACTER UPLO
* INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* COMPLEX*16 A( LDA, * ), E ( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*> ZHETRF_RK computes the factorization of a complex Hermitian matrix A
*> using the bounded Bunch-Kaufman (rook) diagonal pivoting method:
*>
*> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
*>
*> 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.
*>
*> This is the blocked version of the algorithm, calling Level 3 BLAS.
*> For more information see Further Details section.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] UPLO
*> \verbatim
*> 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
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix A. N >= 0.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[out] E
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[out] IPIV
*> \verbatim
*> 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. For more info see Further
*> Details section.
*>
*> 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 matrix A(1:N,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,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 matrix A(1:N,1:N).
*> If -IPIV(k-1) = k-1, no interchange occurred.
*>
*> c) In both cases a) and b), 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 matrix A(1:N,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: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 matrix A(1:N,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 matrix A(1:N,1:N).
*> If -IPIV(k+1) = k+1, no interchange occurred.
*>
*> c) In both cases a) and b), always ABS( IPIV(k) ) >= k.
*>
*> d) NOTE: Any entry IPIV(k) is always NONZERO on output.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 array, dimension ( MAX(1,LWORK) ).
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of WORK. LWORK >=1. For best performance
*> LWORK >= N*NB, where NB is the block size returned
*> by ILAENV.
*>
*> If LWORK = -1, then a workspace query is assumed;
*> the routine only calculates the optimal size of the WORK
*> array, returns this value as the first entry of the WORK
*> array, and no error message related to LWORK is issued
*> by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> 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.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16HEcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*> TODO: put correct description
*> \endverbatim
*
*> \par Contributors:
* ==================
*>
*> \verbatim
*>
*> 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
*>
*> \endverbatim
*
* =====================================================================
SUBROUTINE ZHETRF_RK( UPLO, N, A, LDA, E, IPIV, WORK, LWORK,
$ INFO )
*
* -- LAPACK computational routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LWORK, N
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY, UPPER
INTEGER I, IINFO, IP, IWS, K, KB, LDWORK, LWKOPT,
$ NB, NBMIN
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
EXTERNAL LSAME, ILAENV
* ..
* .. External Subroutines ..
EXTERNAL ZLAHEF_RK, ZHETF2_RK, ZSWAP, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
UPPER = LSAME( UPLO, 'U' )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
INFO = -1
ELSE IF( N.LT.0 ) THEN
INFO = -2
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
INFO = -8
END IF
*
IF( INFO.EQ.0 ) THEN
*
* Determine the block size
*
NB = ILAENV( 1, 'ZHETRF_RK', UPLO, N, -1, -1, -1 )
LWKOPT = N*NB
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZHETRF_RK', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
NBMIN = 2
LDWORK = N
IF( NB.GT.1 .AND. NB.LT.N ) THEN
IWS = LDWORK*NB
IF( LWORK.LT.IWS ) THEN
NB = MAX( LWORK / LDWORK, 1 )
NBMIN = MAX( 2, ILAENV( 2, 'ZHETRF_RK',
$ UPLO, N, -1, -1, -1 ) )
END IF
ELSE
IWS = 1
END IF
IF( NB.LT.NBMIN )
$ NB = N
*
IF( UPPER ) THEN
*
* Factorize A as U*D*U**T using the upper triangle of A
*
* K is the main loop index, decreasing from N to 1 in steps of
* KB, where KB is the number of columns factorized by ZLAHEF_RK;
* KB is either NB or NB-1, or K for the last block
*
K = N
10 CONTINUE
*
* If K < 1, exit from loop
*
IF( K.LT.1 )
$ GO TO 15
*
IF( K.GT.NB ) THEN
*
* Factorize columns k-kb+1:k of A and use blocked code to
* update columns 1:k-kb
*
CALL ZLAHEF_RK( UPLO, K, NB, KB, A, LDA, E,
$ IPIV, WORK, LDWORK, IINFO )
ELSE
*
* Use unblocked code to factorize columns 1:k of A
*
CALL ZHETF2_RK( UPLO, K, A, LDA, E, IPIV, IINFO )
KB = K
END IF
*
* Set INFO on the first occurrence of a zero pivot
*
IF( INFO.EQ.0 .AND. IINFO.GT.0 )
$ INFO = IINFO
*
* No need to adjust IPIV
*
*
* Apply permutations to the leading panel 1:k-1
*
* Read IPIV from the last block factored, i.e.
* indices k-kb+1:k and apply row permutations to the
* last k+1 colunms k+1:N after that block
* (We can do the simple loop over IPIV with decrement -1,
* since the ABS value of IPIV( I ) represents the row index
* of the interchange with row i in both 1x1 and 2x2 pivot cases)
*
IF( K.LT.N ) THEN
DO I = K, ( K - KB + 1 ), -1
IP = ABS( IPIV( I ) )
IF( IP.NE.I ) THEN
CALL ZSWAP( N-K, A( I, K+1 ), LDA,
$ A( IP, K+1 ), LDA )
END IF
END DO
END IF
*
* Decrease K and return to the start of the main loop
*
K = K - KB
GO TO 10
*
* This label is the exit from main loop over K decreasing
* from N to 1 in steps of KB
*
15 CONTINUE
*
ELSE
*
* Factorize A as L*D*L**T using the lower triangle of A
*
* K is the main loop index, increasing from 1 to N in steps of
* KB, where KB is the number of columns factorized by ZLAHEF_RK;
* KB is either NB or NB-1, or N-K+1 for the last block
*
K = 1
20 CONTINUE
*
* If K > N, exit from loop
*
IF( K.GT.N )
$ GO TO 35
*
IF( K.LE.N-NB ) THEN
*
* Factorize columns k:k+kb-1 of A and use blocked code to
* update columns k+kb:n
*
CALL ZLAHEF_RK( UPLO, N-K+1, NB, KB, A( K, K ), LDA, E( K ),
$ IPIV( K ), WORK, LDWORK, IINFO )
ELSE
*
* Use unblocked code to factorize columns k:n of A
*
CALL ZHETF2_RK( UPLO, N-K+1, A( K, K ), LDA, E( K ),
$ IPIV( K ), IINFO )
KB = N - K + 1
*
END IF
*
* Set INFO on the first occurrence of a zero pivot
*
IF( INFO.EQ.0 .AND. IINFO.GT.0 )
$ INFO = IINFO + K - 1
*
* Adjust IPIV
*
DO I = K, K + KB - 1
IF( IPIV( I ).GT.0 ) THEN
IPIV( I ) = IPIV( I ) + K - 1
ELSE
IPIV( I ) = IPIV( I ) - K + 1
END IF
END DO
*
* Apply permutations to the leading panel 1:k-1
*
* Read IPIV from the last block factored, i.e.
* indices k:k+kb-1 and apply row permutations to the
* first k-1 colunms 1:k-1 before that block
* (We can do the simple loop over IPIV with increment 1,
* since the ABS value of IPIV( I ) represents the row index
* of the interchange with row i in both 1x1 and 2x2 pivot cases)
*
IF( K.GT.1 ) THEN
DO I = K, ( K + KB - 1 ), 1
IP = ABS( IPIV( I ) )
IF( IP.NE.I ) THEN
CALL ZSWAP( K-1, A( I, 1 ), LDA,
$ A( IP, 1 ), LDA )
END IF
END DO
END IF
*
* Increase K and return to the start of the main loop
*
K = K + KB
GO TO 20
*
* This label is the exit from main loop over K increasing
* from 1 to N in steps of KB
*
35 CONTINUE
*
* End Lower
*
END IF
*
WORK( 1 ) = LWKOPT
RETURN
*
* End of ZHETRF_RK
*
END