version 1.4, 2010/08/06 15:32:41
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version 1.17, 2018/05/29 07:18:18
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*> \brief <b> ZHBEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b> |
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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 ZHBEVX + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhbevx.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/zhbevx.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/zhbevx.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 ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, |
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* VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, |
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* 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, KD, LDAB, LDQ, LDZ, M, 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, * ), Q( LDQ, * ), WORK( * ), |
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* $ 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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*> ZHBEVX computes selected eigenvalues and, optionally, eigenvectors |
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*> of a complex Hermitian band matrix A. Eigenvalues and eigenvectors |
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*> can be selected by specifying either a range of values or a range of |
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*> 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 triangle of A is stored; |
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*> = 'L': Lower triangle of A is 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 matrix A. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] KD |
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*> \verbatim |
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*> KD 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'. KD >= 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 KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; |
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*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). |
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*> |
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*> On exit, AB is overwritten by values generated during the |
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*> reduction to tridiagonal form. |
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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 >= KD + 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 unitary matrix used in the |
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*> reduction 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 = 'V', then |
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*> 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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*> 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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*> 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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*> 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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*> 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 AB 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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*> See "Computing Small Singular Values of Bidiagonal Matrices |
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*> with Guaranteed High Relative Accuracy," by Demmel and |
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*> Kahan, LAPACK Working Note #3. |
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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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*> The first M elements contain the selected eigenvalues in |
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*> 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, max(1,M)) |
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*> If JOBZ = 'V', then if INFO = 0, the first M columns of Z |
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*> contain the orthonormal eigenvectors of the matrix A |
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*> corresponding to the selected eigenvalues, with the i-th |
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*> column of Z holding the eigenvector associated with W(i). |
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*> If an eigenvector fails to converge, then that column of Z |
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*> contains the latest approximation to the eigenvector, and the |
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*> index of the eigenvector is returned in IFAIL. |
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*> If JOBZ = 'N', then Z is not referenced. |
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*> Note: the user must ensure that at least max(1,M) columns are |
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*> supplied in the array Z; if RANGE = 'V', the exact value of M |
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*> is not known in advance and an upper bound must be used. |
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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 >= max(1,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, then i eigenvectors failed to converge. |
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*> Their indices are stored in array IFAIL. |
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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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* ===================================================================== |
SUBROUTINE ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, |
SUBROUTINE ZHBEVX( JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, |
$ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, |
$ VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, |
$ IWORK, IFAIL, INFO ) |
$ 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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$ Z( LDZ, * ) |
$ Z( LDZ, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZHBEVX computes selected eigenvalues and, optionally, eigenvectors |
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* of a complex Hermitian band matrix A. Eigenvalues and eigenvectors |
|
* can be selected by specifying either 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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* 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 triangle of A is stored; |
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* = 'L': Lower triangle of A is stored. |
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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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* KD (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'. KD >= 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 KD+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(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; |
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* if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). |
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* |
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* On exit, AB is overwritten by values generated during the |
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* reduction to tridiagonal form. |
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* |
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* LDAB (input) INTEGER |
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* The leading dimension of the array AB. LDAB >= KD + 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 unitary matrix used in the |
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* reduction 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 = 'V', then |
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* 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 |
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* by reducing AB 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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* See "Computing Small Singular Values of Bidiagonal Matrices |
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* with Guaranteed High Relative Accuracy," by Demmel and |
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* Kahan, LAPACK Working Note #3. |
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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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* The first M elements contain the selected eigenvalues in |
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* ascending order. |
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* |
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* Z (output) COMPLEX*16 array, dimension (LDZ, max(1,M)) |
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* If JOBZ = 'V', then if INFO = 0, the first M columns of Z |
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* contain the orthonormal eigenvectors of the matrix A |
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* corresponding to the selected eigenvalues, with the i-th |
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* column of Z holding the eigenvector associated with W(i). |
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* If an eigenvector fails to converge, then that column of Z |
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* contains the latest approximation to the eigenvector, and the |
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* index of the eigenvector is returned in IFAIL. |
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* If JOBZ = 'N', then Z is not referenced. |
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* Note: the user must ensure that at least max(1,M) columns are |
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* supplied in the array Z; if RANGE = 'V', the exact value of M |
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* is not known in advance and an upper bound must be used. |
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* |
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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 >= max(1,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, then i eigenvectors failed to converge. |
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* Their indices are stored in array IFAIL. |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |