version 1.6, 2010/08/13 21:04:05
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version 1.18, 2018/05/29 07:18:18
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*> \brief \b ZHBGVX |
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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 ZHBGVX + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhbgvx.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/zhbgvx.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/zhbgvx.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 ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, |
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* LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, |
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* LDZ, WORK, RWORK, IWORK, IFAIL, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER JOBZ, RANGE, UPLO |
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* INTEGER IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, |
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* $ N |
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* DOUBLE PRECISION ABSTOL, VL, VU |
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* .. |
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* .. Array Arguments .. |
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* INTEGER IFAIL( * ), IWORK( * ) |
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* DOUBLE PRECISION RWORK( * ), W( * ) |
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* COMPLEX*16 AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), |
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* $ WORK( * ), Z( LDZ, * ) |
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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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*> ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors |
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*> of a complex generalized Hermitian-definite banded eigenproblem, of |
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*> the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian |
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*> and banded, and B is also positive definite. Eigenvalues and |
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*> eigenvectors can be selected by specifying either all eigenvalues, |
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*> a range of values or a range of indices for the desired eigenvalues. |
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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] JOBZ |
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*> \verbatim |
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*> JOBZ is CHARACTER*1 |
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*> = 'N': Compute eigenvalues only; |
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*> = 'V': Compute eigenvalues and eigenvectors. |
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*> \endverbatim |
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*> |
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*> \param[in] RANGE |
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*> \verbatim |
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*> RANGE is CHARACTER*1 |
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*> = 'A': all eigenvalues will be found; |
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*> = 'V': all eigenvalues in the half-open interval (VL,VU] |
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*> will be found; |
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*> = 'I': the IL-th through IU-th eigenvalues will be found. |
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*> \endverbatim |
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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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*> = 'U': Upper triangles of A and B are stored; |
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*> = 'L': Lower triangles of A and B are stored. |
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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 matrices A and B. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KA |
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*> \verbatim |
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*> KA is INTEGER |
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*> The number of superdiagonals of the matrix A if UPLO = 'U', |
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*> or the number of subdiagonals if UPLO = 'L'. KA >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KB |
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*> \verbatim |
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*> KB is INTEGER |
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*> The number of superdiagonals of the matrix B if UPLO = 'U', |
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*> or the number of subdiagonals if UPLO = 'L'. KB >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] AB |
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*> \verbatim |
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*> AB is COMPLEX*16 array, dimension (LDAB, N) |
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*> On entry, the upper or lower triangle of the Hermitian band |
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*> matrix A, stored in the first ka+1 rows of the array. The |
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*> j-th column of A is stored in the j-th column of the array AB |
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*> as follows: |
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*> if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; |
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*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). |
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*> |
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*> On exit, the contents of AB are destroyed. |
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*> \endverbatim |
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*> |
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*> \param[in] LDAB |
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*> \verbatim |
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*> LDAB is INTEGER |
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*> The leading dimension of the array AB. LDAB >= KA+1. |
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*> \endverbatim |
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*> |
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*> \param[in,out] BB |
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*> \verbatim |
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*> BB is COMPLEX*16 array, dimension (LDBB, N) |
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*> On entry, the upper or lower triangle of the Hermitian band |
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*> matrix B, stored in the first kb+1 rows of the array. The |
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*> j-th column of B is stored in the j-th column of the array BB |
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*> as follows: |
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*> if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; |
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*> if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). |
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*> |
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*> On exit, the factor S from the split Cholesky factorization |
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*> B = S**H*S, as returned by ZPBSTF. |
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*> \endverbatim |
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*> |
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*> \param[in] LDBB |
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*> \verbatim |
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*> LDBB is INTEGER |
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*> The leading dimension of the array BB. LDBB >= KB+1. |
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*> \endverbatim |
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*> |
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*> \param[out] Q |
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*> \verbatim |
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*> Q is COMPLEX*16 array, dimension (LDQ, N) |
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*> If JOBZ = 'V', the n-by-n matrix used in the reduction of |
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*> A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, |
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*> and consequently C to tridiagonal form. |
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*> If JOBZ = 'N', the array Q is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDQ |
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*> \verbatim |
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*> LDQ is INTEGER |
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*> The leading dimension of the array Q. If JOBZ = 'N', |
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*> LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). |
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*> \endverbatim |
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*> |
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*> \param[in] VL |
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*> \verbatim |
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*> VL is DOUBLE PRECISION |
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*> |
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*> If RANGE='V', the lower bound of the interval to |
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*> be searched for eigenvalues. VL < VU. |
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*> Not referenced if RANGE = 'A' or 'I'. |
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*> \endverbatim |
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*> |
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*> \param[in] VU |
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*> \verbatim |
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*> VU is DOUBLE PRECISION |
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*> |
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*> If RANGE='V', the upper bound of the interval to |
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*> be searched for eigenvalues. VL < VU. |
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*> Not referenced if RANGE = 'A' or 'I'. |
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*> \endverbatim |
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*> |
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*> \param[in] IL |
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*> \verbatim |
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*> IL is INTEGER |
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*> |
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*> If RANGE='I', the index of the |
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*> smallest eigenvalue to be returned. |
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*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. |
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*> Not referenced if RANGE = 'A' or 'V'. |
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*> \endverbatim |
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*> |
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*> \param[in] IU |
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*> \verbatim |
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*> IU is INTEGER |
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*> |
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*> If RANGE='I', the index of the |
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*> largest eigenvalue to be returned. |
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*> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. |
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*> Not referenced if RANGE = 'A' or 'V'. |
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*> \endverbatim |
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*> |
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*> \param[in] ABSTOL |
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*> \verbatim |
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*> ABSTOL is DOUBLE PRECISION |
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*> The absolute error tolerance for the eigenvalues. |
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*> An approximate eigenvalue is accepted as converged |
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*> when it is determined to lie in an interval [a,b] |
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*> of width less than or equal to |
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*> |
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*> ABSTOL + EPS * max( |a|,|b| ) , |
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*> |
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*> where EPS is the machine precision. If ABSTOL is less than |
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*> or equal to zero, then EPS*|T| will be used in its place, |
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*> where |T| is the 1-norm of the tridiagonal matrix obtained |
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*> by reducing AP to tridiagonal form. |
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*> |
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*> Eigenvalues will be computed most accurately when ABSTOL is |
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*> set to twice the underflow threshold 2*DLAMCH('S'), not zero. |
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*> If this routine returns with INFO>0, indicating that some |
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*> eigenvectors did not converge, try setting ABSTOL to |
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*> 2*DLAMCH('S'). |
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*> \endverbatim |
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*> |
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*> \param[out] M |
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*> \verbatim |
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*> M is INTEGER |
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*> The total number of eigenvalues found. 0 <= M <= N. |
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*> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. |
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*> \endverbatim |
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*> |
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*> \param[out] W |
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*> \verbatim |
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*> W is DOUBLE PRECISION array, dimension (N) |
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*> If INFO = 0, the eigenvalues in ascending order. |
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*> \endverbatim |
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*> |
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*> \param[out] Z |
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*> \verbatim |
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*> Z is COMPLEX*16 array, dimension (LDZ, N) |
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*> If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of |
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*> eigenvectors, with the i-th column of Z holding the |
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*> eigenvector associated with W(i). The eigenvectors are |
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*> normalized so that Z**H*B*Z = I. |
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*> If JOBZ = 'N', then Z is not referenced. |
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*> \endverbatim |
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*> |
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*> \param[in] LDZ |
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*> \verbatim |
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*> LDZ is INTEGER |
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*> The leading dimension of the array Z. LDZ >= 1, and if |
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*> JOBZ = 'V', LDZ >= N. |
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*> \endverbatim |
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*> |
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*> \param[out] WORK |
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*> \verbatim |
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*> WORK is COMPLEX*16 array, dimension (N) |
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*> \endverbatim |
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*> |
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*> \param[out] RWORK |
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*> \verbatim |
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*> RWORK is DOUBLE PRECISION array, dimension (7*N) |
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*> \endverbatim |
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*> |
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*> \param[out] IWORK |
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*> \verbatim |
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*> IWORK is INTEGER array, dimension (5*N) |
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*> \endverbatim |
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*> |
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*> \param[out] IFAIL |
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*> \verbatim |
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*> IFAIL is INTEGER array, dimension (N) |
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*> If JOBZ = 'V', then if INFO = 0, the first M elements of |
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*> IFAIL are zero. If INFO > 0, then IFAIL contains the |
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*> indices of the eigenvectors that failed to converge. |
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*> If JOBZ = 'N', then IFAIL is not referenced. |
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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 = -i, the i-th argument had an illegal value |
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*> > 0: if INFO = i, and i is: |
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*> <= N: then i eigenvectors failed to converge. Their |
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*> indices are stored in array IFAIL. |
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*> > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF |
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*> returned INFO = i: B is not positive definite. |
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*> The factorization of B could not be completed and |
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*> no eigenvalues or eigenvectors were computed. |
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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 June 2016 |
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* |
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*> \ingroup complex16OTHEReigen |
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* |
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*> \par Contributors: |
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* ================== |
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*> |
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*> Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA |
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* |
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* ===================================================================== |
SUBROUTINE ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, |
SUBROUTINE ZHBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, |
$ LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, |
$ LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, |
$ LDZ, WORK, RWORK, IWORK, IFAIL, INFO ) |
$ LDZ, WORK, RWORK, IWORK, IFAIL, INFO ) |
* |
* |
* -- LAPACK driver routine (version 3.2) -- |
* -- LAPACK driver routine (version 3.7.0) -- |
* -- 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 |
* June 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER JOBZ, RANGE, UPLO |
CHARACTER JOBZ, RANGE, UPLO |
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$ WORK( * ), Z( LDZ, * ) |
$ WORK( * ), Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZHBGVX computes all the eigenvalues, and optionally, the eigenvectors |
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* of a complex generalized Hermitian-definite banded eigenproblem, of |
|
* the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian |
|
* and banded, and B is also positive definite. Eigenvalues and |
|
* eigenvectors can be selected by specifying either all eigenvalues, |
|
* a range of values or a range of indices for the desired eigenvalues. |
|
* |
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* Arguments |
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* ========= |
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* |
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* JOBZ (input) CHARACTER*1 |
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* = 'N': Compute eigenvalues only; |
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* = 'V': Compute eigenvalues and eigenvectors. |
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* |
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* RANGE (input) CHARACTER*1 |
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* = 'A': all eigenvalues will be found; |
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* = 'V': all eigenvalues in the half-open interval (VL,VU] |
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* will be found; |
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* = 'I': the IL-th through IU-th eigenvalues will be found. |
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* |
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* UPLO (input) CHARACTER*1 |
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* = 'U': Upper triangles of A and B are stored; |
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* = 'L': Lower triangles of A and B are stored. |
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* |
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* N (input) INTEGER |
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* The order of the matrices A and B. N >= 0. |
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* |
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* KA (input) INTEGER |
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* The number of superdiagonals of the matrix A if UPLO = 'U', |
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* or the number of subdiagonals if UPLO = 'L'. KA >= 0. |
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* |
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* KB (input) INTEGER |
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* The number of superdiagonals of the matrix B if UPLO = 'U', |
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* or the number of subdiagonals if UPLO = 'L'. KB >= 0. |
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* |
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* AB (input/output) COMPLEX*16 array, dimension (LDAB, N) |
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* On entry, the upper or lower triangle of the Hermitian band |
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* matrix A, stored in the first ka+1 rows of the array. The |
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* j-th column of A is stored in the j-th column of the array AB |
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* as follows: |
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* if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j; |
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* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka). |
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* |
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* On exit, the contents of AB are destroyed. |
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* |
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* LDAB (input) INTEGER |
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* The leading dimension of the array AB. LDAB >= KA+1. |
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* |
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* BB (input/output) COMPLEX*16 array, dimension (LDBB, N) |
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* On entry, the upper or lower triangle of the Hermitian band |
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* matrix B, stored in the first kb+1 rows of the array. The |
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* j-th column of B is stored in the j-th column of the array BB |
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* as follows: |
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* if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j; |
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* if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb). |
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* |
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* On exit, the factor S from the split Cholesky factorization |
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* B = S**H*S, as returned by ZPBSTF. |
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* |
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* LDBB (input) INTEGER |
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* The leading dimension of the array BB. LDBB >= KB+1. |
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* |
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* Q (output) COMPLEX*16 array, dimension (LDQ, N) |
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* If JOBZ = 'V', the n-by-n matrix used in the reduction of |
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* A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, |
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* and consequently C to tridiagonal form. |
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* If JOBZ = 'N', the array Q is not referenced. |
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* |
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* LDQ (input) INTEGER |
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* The leading dimension of the array Q. If JOBZ = 'N', |
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* LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N). |
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* |
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* VL (input) DOUBLE PRECISION |
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* VU (input) DOUBLE PRECISION |
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* If RANGE='V', the lower and upper bounds of the interval to |
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* be searched for eigenvalues. VL < VU. |
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* Not referenced if RANGE = 'A' or 'I'. |
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* |
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* IL (input) INTEGER |
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* IU (input) INTEGER |
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* If RANGE='I', the indices (in ascending order) of the |
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* smallest and largest eigenvalues to be returned. |
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* 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. |
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* Not referenced if RANGE = 'A' or 'V'. |
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* |
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* ABSTOL (input) DOUBLE PRECISION |
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* The absolute error tolerance for the eigenvalues. |
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* An approximate eigenvalue is accepted as converged |
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* when it is determined to lie in an interval [a,b] |
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* of width less than or equal to |
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* |
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* ABSTOL + EPS * max( |a|,|b| ) , |
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* |
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* where EPS is the machine precision. If ABSTOL is less than |
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* or equal to zero, then EPS*|T| will be used in its place, |
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* where |T| is the 1-norm of the tridiagonal matrix obtained |
|
* by reducing AP to tridiagonal form. |
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* |
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* Eigenvalues will be computed most accurately when ABSTOL is |
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* set to twice the underflow threshold 2*DLAMCH('S'), not zero. |
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* If this routine returns with INFO>0, indicating that some |
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* eigenvectors did not converge, try setting ABSTOL to |
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* 2*DLAMCH('S'). |
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* |
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* M (output) INTEGER |
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* The total number of eigenvalues found. 0 <= M <= N. |
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* If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1. |
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* |
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* W (output) DOUBLE PRECISION array, dimension (N) |
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* If INFO = 0, the eigenvalues in ascending order. |
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* |
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* Z (output) COMPLEX*16 array, dimension (LDZ, N) |
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* If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of |
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* eigenvectors, with the i-th column of Z holding the |
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* eigenvector associated with W(i). The eigenvectors are |
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* normalized so that Z**H*B*Z = I. |
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* If JOBZ = 'N', then Z is not referenced. |
|
* |
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* LDZ (input) INTEGER |
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* The leading dimension of the array Z. LDZ >= 1, and if |
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* JOBZ = 'V', LDZ >= N. |
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* |
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* WORK (workspace) COMPLEX*16 array, dimension (N) |
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* |
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* RWORK (workspace) DOUBLE PRECISION array, dimension (7*N) |
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* |
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* IWORK (workspace) INTEGER array, dimension (5*N) |
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* |
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* IFAIL (output) INTEGER array, dimension (N) |
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* If JOBZ = 'V', then if INFO = 0, the first M elements of |
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* IFAIL are zero. If INFO > 0, then IFAIL contains the |
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* indices of the eigenvectors that failed to converge. |
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* If JOBZ = 'N', then IFAIL is not referenced. |
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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 = -i, the i-th argument had an illegal value |
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* > 0: if INFO = i, and i is: |
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* <= N: then i eigenvectors failed to converge. Their |
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* indices are stored in array IFAIL. |
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* > N: if INFO = N + i, for 1 <= i <= N, then ZPBSTF |
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* returned INFO = i: B is not positive definite. |
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* The factorization of B could not be completed and |
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* no eigenvalues or eigenvectors were computed. |
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* |
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* Further Details |
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* =============== |
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* |
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* Based on contributions by |
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* Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |