493 lines
15 KiB
Fortran
493 lines
15 KiB
Fortran
*> \brief <b> SSPEVX 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 SSPEVX + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sspevx.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/sspevx.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/sspevx.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 SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
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* ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
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* 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, LDZ, M, N
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* REAL ABSTOL, VL, VU
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* ..
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* .. Array Arguments ..
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* INTEGER IFAIL( * ), IWORK( * )
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* REAL AP( * ), W( * ), 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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*> SSPEVX computes selected eigenvalues and, optionally, eigenvectors
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*> of a real symmetric matrix A in packed storage. Eigenvalues/vectors
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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,out] AP
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*> \verbatim
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*> AP is REAL array, dimension (N*(N+1)/2)
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*> On entry, the upper or lower triangle of the symmetric matrix
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*> A, packed columnwise in a linear array. The j-th column of A
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*> is stored in the array AP as follows:
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*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
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*> if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
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*>
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*> On exit, AP is overwritten by values generated during the
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*> reduction to tridiagonal form. If UPLO = 'U', the diagonal
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*> and first superdiagonal of the tridiagonal matrix T overwrite
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*> the corresponding elements of A, and if UPLO = 'L', the
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*> diagonal and first subdiagonal of T overwrite the
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*> corresponding elements of A.
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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 REAL
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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 REAL
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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 REAL
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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*SLAMCH('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*SLAMCH('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 REAL array, dimension (N)
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*> If INFO = 0, the selected 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 REAL 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 REAL array, dimension (8*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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*> \ingroup realOTHEReigen
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*
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* =====================================================================
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SUBROUTINE SSPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
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$ ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL,
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$ INFO )
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*
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* -- LAPACK driver routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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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, LDZ, M, N
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REAL ABSTOL, VL, VU
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* ..
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* .. Array Arguments ..
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INTEGER IFAIL( * ), IWORK( * )
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REAL AP( * ), W( * ), WORK( * ), Z( LDZ, * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ZERO, ONE
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PARAMETER ( ZERO = 0.0E0, ONE = 1.0E0 )
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* ..
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* .. Local Scalars ..
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LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
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CHARACTER ORDER
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INTEGER I, IINFO, IMAX, INDD, INDE, INDEE,
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$ INDISP, INDIWO, INDTAU, INDWRK, ISCALE, ITMP1,
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$ J, JJ, NSPLIT
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REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
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$ SIGMA, SMLNUM, TMP1, VLL, VUU
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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REAL SLAMCH, SLANSP
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EXTERNAL LSAME, SLAMCH, SLANSP
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* ..
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* .. External Subroutines ..
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EXTERNAL SCOPY, SOPGTR, SOPMTR, SSCAL, SSPTRD, SSTEBZ,
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$ SSTEIN, SSTEQR, SSTERF, SSWAP, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MAX, MIN, SQRT
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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WANTZ = LSAME( JOBZ, 'V' )
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ALLEIG = LSAME( RANGE, 'A' )
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VALEIG = LSAME( RANGE, 'V' )
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INDEIG = LSAME( RANGE, 'I' )
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*
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INFO = 0
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IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
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INFO = -1
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ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
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INFO = -2
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ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR. LSAME( UPLO, 'U' ) ) )
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$ THEN
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INFO = -3
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ELSE IF( N.LT.0 ) THEN
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INFO = -4
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ELSE
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IF( VALEIG ) THEN
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IF( N.GT.0 .AND. VU.LE.VL )
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$ INFO = -7
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ELSE IF( INDEIG ) THEN
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IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
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INFO = -8
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ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
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INFO = -9
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END IF
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END IF
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END IF
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IF( INFO.EQ.0 ) THEN
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IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
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$ INFO = -14
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END IF
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*
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'SSPEVX', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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M = 0
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IF( N.EQ.0 )
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$ RETURN
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*
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IF( N.EQ.1 ) THEN
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IF( ALLEIG .OR. INDEIG ) THEN
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M = 1
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W( 1 ) = AP( 1 )
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ELSE
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IF( VL.LT.AP( 1 ) .AND. VU.GE.AP( 1 ) ) THEN
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M = 1
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W( 1 ) = AP( 1 )
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END IF
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END IF
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IF( WANTZ )
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$ Z( 1, 1 ) = ONE
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RETURN
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END IF
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*
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* Get machine constants.
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*
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SAFMIN = SLAMCH( 'Safe minimum' )
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EPS = SLAMCH( 'Precision' )
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SMLNUM = SAFMIN / EPS
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BIGNUM = ONE / SMLNUM
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RMIN = SQRT( SMLNUM )
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RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
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*
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* Scale matrix to allowable range, if necessary.
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*
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ISCALE = 0
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ABSTLL = ABSTOL
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IF ( VALEIG ) THEN
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VLL = VL
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VUU = VU
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ELSE
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VLL = ZERO
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VUU = ZERO
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ENDIF
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ANRM = SLANSP( 'M', UPLO, N, AP, WORK )
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IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
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ISCALE = 1
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SIGMA = RMIN / ANRM
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ELSE IF( ANRM.GT.RMAX ) THEN
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ISCALE = 1
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SIGMA = RMAX / ANRM
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END IF
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IF( ISCALE.EQ.1 ) THEN
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CALL SSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
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IF( ABSTOL.GT.0 )
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$ ABSTLL = ABSTOL*SIGMA
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IF( VALEIG ) THEN
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VLL = VL*SIGMA
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VUU = VU*SIGMA
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END IF
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END IF
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*
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* Call SSPTRD to reduce symmetric packed matrix to tridiagonal form.
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*
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INDTAU = 1
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INDE = INDTAU + N
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INDD = INDE + N
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INDWRK = INDD + N
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CALL SSPTRD( UPLO, N, AP, WORK( INDD ), WORK( INDE ),
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$ WORK( INDTAU ), IINFO )
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*
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* If all eigenvalues are desired and ABSTOL is less than or equal
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* to zero, then call SSTERF or SOPGTR and SSTEQR. If this fails
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* for some eigenvalue, then try SSTEBZ.
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*
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TEST = .FALSE.
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IF (INDEIG) THEN
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IF (IL.EQ.1 .AND. IU.EQ.N) THEN
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TEST = .TRUE.
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END IF
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END IF
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IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
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CALL SCOPY( N, WORK( INDD ), 1, W, 1 )
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INDEE = INDWRK + 2*N
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IF( .NOT.WANTZ ) THEN
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CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
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CALL SSTERF( N, W, WORK( INDEE ), INFO )
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ELSE
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CALL SOPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
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$ WORK( INDWRK ), IINFO )
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CALL SCOPY( N-1, WORK( INDE ), 1, WORK( INDEE ), 1 )
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CALL SSTEQR( JOBZ, N, W, WORK( INDEE ), Z, LDZ,
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$ WORK( INDWRK ), INFO )
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IF( INFO.EQ.0 ) THEN
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DO 10 I = 1, N
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IFAIL( I ) = 0
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10 CONTINUE
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END IF
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END IF
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IF( INFO.EQ.0 ) THEN
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M = N
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GO TO 20
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END IF
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INFO = 0
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END IF
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*
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* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
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*
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IF( WANTZ ) THEN
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ORDER = 'B'
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ELSE
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ORDER = 'E'
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END IF
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INDISP = 1 + N
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INDIWO = INDISP + N
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CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
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$ WORK( INDD ), WORK( INDE ), M, NSPLIT, W,
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$ IWORK( 1 ), IWORK( INDISP ), WORK( INDWRK ),
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$ IWORK( INDIWO ), INFO )
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*
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IF( WANTZ ) THEN
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CALL SSTEIN( N, WORK( INDD ), WORK( INDE ), M, W,
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$ IWORK( 1 ), IWORK( INDISP ), Z, LDZ,
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$ WORK( INDWRK ), IWORK( INDIWO ), IFAIL, INFO )
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*
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* Apply orthogonal matrix used in reduction to tridiagonal
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* form to eigenvectors returned by SSTEIN.
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*
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CALL SOPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z, LDZ,
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$ WORK( INDWRK ), IINFO )
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END IF
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*
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* If matrix was scaled, then rescale eigenvalues appropriately.
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*
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20 CONTINUE
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IF( ISCALE.EQ.1 ) THEN
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IF( INFO.EQ.0 ) THEN
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IMAX = M
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ELSE
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IMAX = INFO - 1
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END IF
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CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
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END IF
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*
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* If eigenvalues are not in order, then sort them, along with
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* eigenvectors.
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*
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IF( WANTZ ) THEN
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DO 40 J = 1, M - 1
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I = 0
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TMP1 = W( J )
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DO 30 JJ = J + 1, M
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IF( W( JJ ).LT.TMP1 ) THEN
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I = JJ
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TMP1 = W( JJ )
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END IF
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30 CONTINUE
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*
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IF( I.NE.0 ) THEN
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ITMP1 = IWORK( 1 + I-1 )
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W( I ) = W( J )
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IWORK( 1 + I-1 ) = IWORK( 1 + J-1 )
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W( J ) = TMP1
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IWORK( 1 + J-1 ) = ITMP1
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CALL SSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
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IF( INFO.NE.0 ) THEN
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ITMP1 = IFAIL( I )
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IFAIL( I ) = IFAIL( J )
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IFAIL( J ) = ITMP1
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END IF
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END IF
|
|
40 CONTINUE
|
|
END IF
|
|
*
|
|
RETURN
|
|
*
|
|
* End of SSPEVX
|
|
*
|
|
END
|