version 1.4, 2010/08/06 15:32:41
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version 1.14, 2014/01/27 09:24:36
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*> \brief \b ZHETF2 computes the factorization of a complex Hermitian matrix, using the diagonal pivoting method (unblocked algorithm, calling Level 2 BLAS). |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download ZHETF2 + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhetf2.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhetf2.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhetf2.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE ZHETF2( UPLO, N, A, LDA, IPIV, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER UPLO |
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* INTEGER INFO, LDA, N |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IPIV( * ) |
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* COMPLEX*16 A( LDA, * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> ZHETF2 computes the factorization of a complex Hermitian matrix A |
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*> using the Bunch-Kaufman diagonal pivoting method: |
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*> |
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*> A = U*D*U**H or A = L*D*L**H |
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*> |
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*> where U (or L) is a product of permutation and unit upper (lower) |
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*> triangular matrices, U**H is the conjugate transpose of U, and D is |
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*> Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. |
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*> |
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*> This is the unblocked version of the algorithm, calling Level 2 BLAS. |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] UPLO |
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*> \verbatim |
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*> UPLO is CHARACTER*1 |
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*> Specifies whether the upper or lower triangular part of the |
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*> Hermitian matrix A is stored: |
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*> = 'U': Upper triangular |
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*> = 'L': Lower triangular |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The order of the matrix A. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is COMPLEX*16 array, dimension (LDA,N) |
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*> On entry, the Hermitian matrix A. If UPLO = 'U', the leading |
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*> n-by-n upper triangular part of A contains the upper |
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*> triangular part of the matrix A, and the strictly lower |
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*> triangular part of A is not referenced. If UPLO = 'L', the |
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*> leading n-by-n lower triangular part of A contains the lower |
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*> triangular part of the matrix A, and the strictly upper |
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*> triangular part of A is not referenced. |
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*> |
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*> On exit, the block diagonal matrix D and the multipliers used |
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*> to obtain the factor U or L (see below for further details). |
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*> \endverbatim |
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*> |
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*> \param[in] LDA |
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*> \verbatim |
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*> LDA is INTEGER |
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*> The leading dimension of the array A. LDA >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[out] IPIV |
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*> \verbatim |
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*> IPIV is INTEGER array, dimension (N) |
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*> Details of the interchanges and the block structure of D. |
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*> |
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*> If UPLO = 'U': |
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*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were |
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*> interchanged and D(k,k) is a 1-by-1 diagonal block. |
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*> |
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*> If IPIV(k) = IPIV(k-1) < 0, then rows and columns |
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*> k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) |
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*> is a 2-by-2 diagonal block. |
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*> |
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*> If UPLO = 'L': |
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*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were |
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*> interchanged and D(k,k) is a 1-by-1 diagonal block. |
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*> |
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*> If IPIV(k) = IPIV(k+1) < 0, then rows and columns |
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*> k+1 and -IPIV(k) were interchanged and D(k:k+1,k:k+1) |
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*> is a 2-by-2 diagonal block. |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit |
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*> < 0: if INFO = -k, the k-th argument had an illegal value |
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*> > 0: if INFO = k, D(k,k) is exactly zero. The factorization |
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*> has been completed, but the block diagonal matrix D is |
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*> exactly singular, and division by zero will occur if it |
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*> is used to solve a system of equations. |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \date November 2013 |
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* |
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*> \ingroup complex16HEcomputational |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> If UPLO = 'U', then A = U*D*U**H, where |
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*> U = P(n)*U(n)* ... *P(k)U(k)* ..., |
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*> i.e., U is a product of terms P(k)*U(k), where k decreases from n to |
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*> 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 |
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*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as |
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*> defined by IPIV(k), and U(k) is a unit upper triangular matrix, such |
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*> that if the diagonal block D(k) is of order s (s = 1 or 2), then |
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*> |
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*> ( I v 0 ) k-s |
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*> U(k) = ( 0 I 0 ) s |
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*> ( 0 0 I ) n-k |
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*> k-s s n-k |
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*> |
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*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). |
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*> If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), |
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*> and A(k,k), and v overwrites A(1:k-2,k-1:k). |
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*> |
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*> If UPLO = 'L', then A = L*D*L**H, where |
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*> L = P(1)*L(1)* ... *P(k)*L(k)* ..., |
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*> i.e., L is a product of terms P(k)*L(k), where k increases from 1 to |
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*> n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 |
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*> and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as |
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*> defined by IPIV(k), and L(k) is a unit lower triangular matrix, such |
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*> that if the diagonal block D(k) is of order s (s = 1 or 2), then |
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*> |
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*> ( I 0 0 ) k-1 |
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*> L(k) = ( 0 I 0 ) s |
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*> ( 0 v I ) n-k-s+1 |
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*> k-1 s n-k-s+1 |
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*> |
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*> If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). |
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*> If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), |
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*> and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). |
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*> \endverbatim |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> \verbatim |
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*> 09-29-06 - patch from |
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*> Bobby Cheng, MathWorks |
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*> |
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*> Replace l.210 and l.393 |
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*> IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
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*> by |
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*> IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN |
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*> |
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*> 01-01-96 - Based on modifications by |
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*> J. Lewis, Boeing Computer Services Company |
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*> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA |
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*> \endverbatim |
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* |
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* ===================================================================== |
SUBROUTINE ZHETF2( UPLO, N, A, LDA, IPIV, INFO ) |
SUBROUTINE ZHETF2( UPLO, N, A, LDA, IPIV, INFO ) |
* |
* |
* -- LAPACK routine (version 3.2) -- |
* -- LAPACK computational routine (version 3.5.0) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* November 2013 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER UPLO |
CHARACTER UPLO |
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COMPLEX*16 A( LDA, * ) |
COMPLEX*16 A( LDA, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
|
* |
|
* ZHETF2 computes the factorization of a complex Hermitian matrix A |
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* using the Bunch-Kaufman diagonal pivoting method: |
|
* |
|
* A = U*D*U' or A = L*D*L' |
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* |
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* where U (or L) is a product of permutation and unit upper (lower) |
|
* triangular matrices, U' is the conjugate transpose of U, and D is |
|
* Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks. |
|
* |
|
* This is the unblocked version of the algorithm, calling Level 2 BLAS. |
|
* |
|
* Arguments |
|
* ========= |
|
* |
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* UPLO (input) CHARACTER*1 |
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* Specifies whether the upper or lower triangular part of the |
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* Hermitian matrix A is stored: |
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* = 'U': Upper triangular |
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* = 'L': Lower triangular |
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* |
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* N (input) INTEGER |
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* The order of the matrix A. N >= 0. |
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* |
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* A (input/output) COMPLEX*16 array, dimension (LDA,N) |
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* On entry, the Hermitian matrix A. If UPLO = 'U', the leading |
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* n-by-n upper triangular part of A contains the upper |
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* triangular part of the matrix A, and the strictly lower |
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* triangular part of A is not referenced. If UPLO = 'L', the |
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* leading n-by-n lower triangular part of A contains the lower |
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* triangular part of the matrix A, and the strictly upper |
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* triangular part of A is not referenced. |
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* |
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* On exit, the block diagonal matrix D and the multipliers used |
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* to obtain the factor U or L (see below for further details). |
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* |
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* LDA (input) INTEGER |
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* The leading dimension of the array A. LDA >= max(1,N). |
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* |
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* IPIV (output) INTEGER array, dimension (N) |
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* Details of the interchanges and the block structure of D. |
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* If IPIV(k) > 0, then rows and columns k and IPIV(k) were |
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* interchanged and D(k,k) is a 1-by-1 diagonal block. |
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* If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and |
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* columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k) |
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* is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) = |
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* IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were |
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* interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block. |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -k, the k-th argument had an illegal value |
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* > 0: if INFO = k, D(k,k) is exactly zero. The factorization |
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* has been completed, but the block diagonal matrix D is |
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* exactly singular, and division by zero will occur if it |
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* is used to solve a system of equations. |
|
* |
|
* Further Details |
|
* =============== |
|
* |
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* 09-29-06 - patch from |
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* Bobby Cheng, MathWorks |
|
* |
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* Replace l.210 and l.393 |
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* IF( MAX( ABSAKK, COLMAX ).EQ.ZERO ) THEN |
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* by |
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* IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN |
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* |
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* 01-01-96 - Based on modifications by |
|
* J. Lewis, Boeing Computer Services Company |
|
* A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA |
|
* |
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* If UPLO = 'U', then A = U*D*U', where |
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* U = P(n)*U(n)* ... *P(k)U(k)* ..., |
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* i.e., U is a product of terms P(k)*U(k), where k decreases from n to |
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* 1 in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 |
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* and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as |
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* defined by IPIV(k), and U(k) is a unit upper triangular matrix, such |
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* that if the diagonal block D(k) is of order s (s = 1 or 2), then |
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* |
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* ( I v 0 ) k-s |
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* U(k) = ( 0 I 0 ) s |
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* ( 0 0 I ) n-k |
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* k-s s n-k |
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* |
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* If s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k). |
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* If s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), |
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* and A(k,k), and v overwrites A(1:k-2,k-1:k). |
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* |
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* If UPLO = 'L', then A = L*D*L', where |
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* L = P(1)*L(1)* ... *P(k)*L(k)* ..., |
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* i.e., L is a product of terms P(k)*L(k), where k increases from 1 to |
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* n in steps of 1 or 2, and D is a block diagonal matrix with 1-by-1 |
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* and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as |
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* defined by IPIV(k), and L(k) is a unit lower triangular matrix, such |
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* that if the diagonal block D(k) is of order s (s = 1 or 2), then |
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* |
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* ( I 0 0 ) k-1 |
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* L(k) = ( 0 I 0 ) s |
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* ( 0 v I ) n-k-s+1 |
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* k-1 s n-k-s+1 |
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* |
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* If s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k). |
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* If s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), |
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* and A(k+1,k+1), and v overwrites A(k+2:n,k:k+1). |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* |
* |
IF( UPPER ) THEN |
IF( UPPER ) THEN |
* |
* |
* Factorize A as U*D*U' using the upper triangle of A |
* Factorize A as U*D*U**H using the upper triangle of A |
* |
* |
* K is the main loop index, decreasing from N to 1 in steps of |
* K is the main loop index, decreasing from N to 1 in steps of |
* 1 or 2 |
* 1 or 2 |
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ABSAKK = ABS( DBLE( A( K, K ) ) ) |
ABSAKK = ABS( DBLE( A( K, K ) ) ) |
* |
* |
* IMAX is the row-index of the largest off-diagonal element in |
* IMAX is the row-index of the largest off-diagonal element in |
* column K, and COLMAX is its absolute value |
* column K, and COLMAX is its absolute value. |
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* Determine both COLMAX and IMAX. |
* |
* |
IF( K.GT.1 ) THEN |
IF( K.GT.1 ) THEN |
IMAX = IZAMAX( K-1, A( 1, K ), 1 ) |
IMAX = IZAMAX( K-1, A( 1, K ), 1 ) |
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* |
* |
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN |
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN |
* |
* |
* Column K is zero or contains a NaN: set INFO and continue |
* Column K is zero or underflow, or contains a NaN: |
|
* set INFO and continue |
* |
* |
IF( INFO.EQ.0 ) |
IF( INFO.EQ.0 ) |
$ INFO = K |
$ INFO = K |
KP = K |
KP = K |
A( K, K ) = DBLE( A( K, K ) ) |
A( K, K ) = DBLE( A( K, K ) ) |
ELSE |
ELSE |
|
* |
|
* ============================================================ |
|
* |
|
* Test for interchange |
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* |
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN |
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN |
* |
* |
* no interchange, use 1-by-1 pivot block |
* no interchange, use 1-by-1 pivot block |
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ELSE |
ELSE |
* |
* |
* JMAX is the column-index of the largest off-diagonal |
* JMAX is the column-index of the largest off-diagonal |
* element in row IMAX, and ROWMAX is its absolute value |
* element in row IMAX, and ROWMAX is its absolute value. |
|
* Determine only ROWMAX. |
* |
* |
JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA ) |
JMAX = IMAX + IZAMAX( K-IMAX, A( IMAX, IMAX+1 ), LDA ) |
ROWMAX = CABS1( A( IMAX, JMAX ) ) |
ROWMAX = CABS1( A( IMAX, JMAX ) ) |
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* no interchange, use 1-by-1 pivot block |
* no interchange, use 1-by-1 pivot block |
* |
* |
KP = K |
KP = K |
|
* |
ELSE IF( ABS( DBLE( A( IMAX, IMAX ) ) ).GE.ALPHA*ROWMAX ) |
ELSE IF( ABS( DBLE( A( IMAX, IMAX ) ) ).GE.ALPHA*ROWMAX ) |
$ THEN |
$ THEN |
* |
* |
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KP = IMAX |
KP = IMAX |
KSTEP = 2 |
KSTEP = 2 |
END IF |
END IF |
|
* |
END IF |
END IF |
* |
* |
|
* ============================================================ |
|
* |
KK = K - KSTEP + 1 |
KK = K - KSTEP + 1 |
IF( KP.NE.KK ) THEN |
IF( KP.NE.KK ) THEN |
* |
* |
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|
* |
* |
* Perform a rank-1 update of A(1:k-1,1:k-1) as |
* Perform a rank-1 update of A(1:k-1,1:k-1) as |
* |
* |
* A := A - U(k)*D(k)*U(k)' = A - W(k)*1/D(k)*W(k)' |
* A := A - U(k)*D(k)*U(k)**H = A - W(k)*1/D(k)*W(k)**H |
* |
* |
R1 = ONE / DBLE( A( K, K ) ) |
R1 = ONE / DBLE( A( K, K ) ) |
CALL ZHER( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA ) |
CALL ZHER( UPLO, K-1, -R1, A( 1, K ), 1, A, LDA ) |
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* |
* |
* Perform a rank-2 update of A(1:k-2,1:k-2) as |
* Perform a rank-2 update of A(1:k-2,1:k-2) as |
* |
* |
* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )' |
* A := A - ( U(k-1) U(k) )*D(k)*( U(k-1) U(k) )**H |
* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )' |
* = A - ( W(k-1) W(k) )*inv(D(k))*( W(k-1) W(k) )**H |
* |
* |
IF( K.GT.2 ) THEN |
IF( K.GT.2 ) THEN |
* |
* |
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* |
* |
ELSE |
ELSE |
* |
* |
* Factorize A as L*D*L' using the lower triangle of A |
* Factorize A as L*D*L**H using the lower triangle of A |
* |
* |
* K is the main loop index, increasing from 1 to N in steps of |
* K is the main loop index, increasing from 1 to N in steps of |
* 1 or 2 |
* 1 or 2 |
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ABSAKK = ABS( DBLE( A( K, K ) ) ) |
ABSAKK = ABS( DBLE( A( K, K ) ) ) |
* |
* |
* IMAX is the row-index of the largest off-diagonal element in |
* IMAX is the row-index of the largest off-diagonal element in |
* column K, and COLMAX is its absolute value |
* column K, and COLMAX is its absolute value. |
|
* Determine both COLMAX and IMAX. |
* |
* |
IF( K.LT.N ) THEN |
IF( K.LT.N ) THEN |
IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 ) |
IMAX = K + IZAMAX( N-K, A( K+1, K ), 1 ) |
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* |
* |
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN |
IF( (MAX( ABSAKK, COLMAX ).EQ.ZERO) .OR. DISNAN(ABSAKK) ) THEN |
* |
* |
* Column K is zero or contains a NaN: set INFO and continue |
* Column K is zero or underflow, or contains a NaN: |
|
* set INFO and continue |
* |
* |
IF( INFO.EQ.0 ) |
IF( INFO.EQ.0 ) |
$ INFO = K |
$ INFO = K |
KP = K |
KP = K |
A( K, K ) = DBLE( A( K, K ) ) |
A( K, K ) = DBLE( A( K, K ) ) |
ELSE |
ELSE |
|
* |
|
* ============================================================ |
|
* |
|
* Test for interchange |
|
* |
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN |
IF( ABSAKK.GE.ALPHA*COLMAX ) THEN |
* |
* |
* no interchange, use 1-by-1 pivot block |
* no interchange, use 1-by-1 pivot block |
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ELSE |
ELSE |
* |
* |
* JMAX is the column-index of the largest off-diagonal |
* JMAX is the column-index of the largest off-diagonal |
* element in row IMAX, and ROWMAX is its absolute value |
* element in row IMAX, and ROWMAX is its absolute value. |
|
* Determine only ROWMAX. |
* |
* |
JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA ) |
JMAX = K - 1 + IZAMAX( IMAX-K, A( IMAX, K ), LDA ) |
ROWMAX = CABS1( A( IMAX, JMAX ) ) |
ROWMAX = CABS1( A( IMAX, JMAX ) ) |
Line 422
|
Line 525
|
* no interchange, use 1-by-1 pivot block |
* no interchange, use 1-by-1 pivot block |
* |
* |
KP = K |
KP = K |
|
* |
ELSE IF( ABS( DBLE( A( IMAX, IMAX ) ) ).GE.ALPHA*ROWMAX ) |
ELSE IF( ABS( DBLE( A( IMAX, IMAX ) ) ).GE.ALPHA*ROWMAX ) |
$ THEN |
$ THEN |
* |
* |
Line 437
|
Line 541
|
KP = IMAX |
KP = IMAX |
KSTEP = 2 |
KSTEP = 2 |
END IF |
END IF |
|
* |
END IF |
END IF |
* |
* |
|
* ============================================================ |
|
* |
KK = K + KSTEP - 1 |
KK = K + KSTEP - 1 |
IF( KP.NE.KK ) THEN |
IF( KP.NE.KK ) THEN |
* |
* |
Line 482
|
Line 589
|
* |
* |
* Perform a rank-1 update of A(k+1:n,k+1:n) as |
* Perform a rank-1 update of A(k+1:n,k+1:n) as |
* |
* |
* A := A - L(k)*D(k)*L(k)' = A - W(k)*(1/D(k))*W(k)' |
* A := A - L(k)*D(k)*L(k)**H = A - W(k)*(1/D(k))*W(k)**H |
* |
* |
R1 = ONE / DBLE( A( K, K ) ) |
R1 = ONE / DBLE( A( K, K ) ) |
CALL ZHER( UPLO, N-K, -R1, A( K+1, K ), 1, |
CALL ZHER( UPLO, N-K, -R1, A( K+1, K ), 1, |
Line 500
|
Line 607
|
* |
* |
* Perform a rank-2 update of A(k+2:n,k+2:n) as |
* Perform a rank-2 update of A(k+2:n,k+2:n) as |
* |
* |
* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )' |
* A := A - ( L(k) L(k+1) )*D(k)*( L(k) L(k+1) )**H |
* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )' |
* = A - ( W(k) W(k+1) )*inv(D(k))*( W(k) W(k+1) )**H |
* |
* |
* where L(k) and L(k+1) are the k-th and (k+1)-th |
* where L(k) and L(k+1) are the k-th and (k+1)-th |
* columns of L |
* columns of L |