numeric-linalg

Educational material on the SciPy implementation of numerical linear algebra algorithms

Name	Size	Mode
..
lapack/SRC/chpevx.f	16169B	-rw-r--r--

*> \brief <b> CHPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CHPEVX + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chpevx.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chpevx.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chpevx.f">
*> [TXT]</a>
*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE CHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
*                          ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK,
*                          IFAIL, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, RANGE, UPLO
*       INTEGER            IL, INFO, IU, LDZ, M, N
*       REAL               ABSTOL, VL, VU
*       ..
*       .. Array Arguments ..
*       INTEGER            IFAIL( * ), IWORK( * )
*       REAL               RWORK( * ), W( * )
*       COMPLEX            AP( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> CHPEVX computes selected eigenvalues and, optionally, eigenvectors
*> of a complex Hermitian matrix A in packed storage.
*> Eigenvalues/vectors can be selected by specifying either a range of
*> values or a range of indices for the desired eigenvalues.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \endverbatim
*>
*> \param[in] RANGE
*> \verbatim
*>          RANGE is CHARACTER*1
*>          = 'A': all eigenvalues will be found;
*>          = 'V': all eigenvalues in the half-open interval (VL,VU]
*>                 will be found;
*>          = 'I': the IL-th through IU-th eigenvalues will be found.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          = 'U':  Upper triangle of A is stored;
*>          = 'L':  Lower triangle of A is stored.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] AP
*> \verbatim
*>          AP is COMPLEX array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the Hermitian matrix
*>          A, packed columnwise in a linear array.  The j-th column of A
*>          is stored in the array AP as follows:
*>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
*>          if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
*>
*>          On exit, AP is overwritten by values generated during the
*>          reduction to tridiagonal form.  If UPLO = 'U', the diagonal
*>          and first superdiagonal of the tridiagonal matrix T overwrite
*>          the corresponding elements of A, and if UPLO = 'L', the
*>          diagonal and first subdiagonal of T overwrite the
*>          corresponding elements of A.
*> \endverbatim
*>
*> \param[in] VL
*> \verbatim
*>          VL is REAL
*>          If RANGE='V', the lower bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] VU
*> \verbatim
*>          VU is REAL
*>          If RANGE='V', the upper bound of the interval to
*>          be searched for eigenvalues. VL < VU.
*>          Not referenced if RANGE = 'A' or 'I'.
*> \endverbatim
*>
*> \param[in] IL
*> \verbatim
*>          IL is INTEGER
*>          If RANGE='I', the index of the
*>          smallest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] IU
*> \verbatim
*>          IU is INTEGER
*>          If RANGE='I', the index of the
*>          largest eigenvalue to be returned.
*>          1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
*>          Not referenced if RANGE = 'A' or 'V'.
*> \endverbatim
*>
*> \param[in] ABSTOL
*> \verbatim
*>          ABSTOL is REAL
*>          The absolute error tolerance for the eigenvalues.
*>          An approximate eigenvalue is accepted as converged
*>          when it is determined to lie in an interval [a,b]
*>          of width less than or equal to
*>
*>                  ABSTOL + EPS *   max( |a|,|b| ) ,
*>
*>          where EPS is the machine precision.  If ABSTOL is less than
*>          or equal to zero, then  EPS*|T|  will be used in its place,
*>          where |T| is the 1-norm of the tridiagonal matrix obtained
*>          by reducing AP to tridiagonal form.
*>
*>          Eigenvalues will be computed most accurately when ABSTOL is
*>          set to twice the underflow threshold 2*SLAMCH('S'), not zero.
*>          If this routine returns with INFO>0, indicating that some
*>          eigenvectors did not converge, try setting ABSTOL to
*>          2*SLAMCH('S').
*>
*>          See "Computing Small Singular Values of Bidiagonal Matrices
*>          with Guaranteed High Relative Accuracy," by Demmel and
*>          Kahan, LAPACK Working Note #3.
*> \endverbatim
*>
*> \param[out] M
*> \verbatim
*>          M is INTEGER
*>          The total number of eigenvalues found.  0 <= M <= N.
*>          If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
*> \endverbatim
*>
*> \param[out] W
*> \verbatim
*>          W is REAL array, dimension (N)
*>          If INFO = 0, the selected eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is COMPLEX array, dimension (LDZ, max(1,M))
*>          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
*>          contain the orthonormal eigenvectors of the matrix A
*>          corresponding to the selected eigenvalues, with the i-th
*>          column of Z holding the eigenvector associated with W(i).
*>          If an eigenvector fails to converge, then that column of Z
*>          contains the latest approximation to the eigenvector, and
*>          the index of the eigenvector is returned in IFAIL.
*>          If JOBZ = 'N', then Z is not referenced.
*>          Note: the user must ensure that at least max(1,M) columns are
*>          supplied in the array Z; if RANGE = 'V', the exact value of M
*>          is not known in advance and an upper bound must be used.
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*>          LDZ is INTEGER
*>          The leading dimension of the array Z.  LDZ >= 1, and if
*>          JOBZ = 'V', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is COMPLEX array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*>          RWORK is REAL array, dimension (7*N)
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] IFAIL
*> \verbatim
*>          IFAIL is INTEGER array, dimension (N)
*>          If JOBZ = 'V', then if INFO = 0, the first M elements of
*>          IFAIL are zero.  If INFO > 0, then IFAIL contains the
*>          indices of the eigenvectors that failed to converge.
*>          If JOBZ = 'N', then IFAIL is not referenced.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*>          INFO is INTEGER
*>          = 0:  successful exit
*>          < 0:  if INFO = -i, the i-th argument had an illegal value
*>          > 0:  if INFO = i, then i eigenvectors failed to converge.
*>                Their indices are stored in array IFAIL.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup hpevx
*
*  =====================================================================
      SUBROUTINE CHPEVX( JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU,
     $                   ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK,
     $                   IFAIL, INFO )
*
*  -- LAPACK driver routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, RANGE, UPLO
      INTEGER            IL, INFO, IU, LDZ, M, N
      REAL               ABSTOL, VL, VU
*     ..
*     .. Array Arguments ..
      INTEGER            IFAIL( * ), IWORK( * )
      REAL               RWORK( * ), W( * )
      COMPLEX            AP( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0 )
      COMPLEX            CONE
      PARAMETER          ( CONE = ( 1.0E0, 0.0E0 ) )
*     ..
*     .. Local Scalars ..
      LOGICAL            ALLEIG, INDEIG, TEST, VALEIG, WANTZ
      CHARACTER          ORDER
      INTEGER            I, IINFO, IMAX, INDD, INDE, INDEE,
     $                   INDISP, INDIWK, INDRWK, INDTAU, INDWRK, ISCALE,
     $                   ITMP1, J, JJ, NSPLIT
      REAL               ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
     $                   SIGMA, SMLNUM, TMP1, VLL, VUU
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               CLANHP, SLAMCH
      EXTERNAL           LSAME, CLANHP, SLAMCH
*     ..
*     .. External Subroutines ..
      EXTERNAL           CHPTRD, CSSCAL, CSTEIN, CSTEQR, CSWAP,
     $                   CUPGTR,
     $                   CUPMTR, SCOPY, SSCAL, SSTEBZ, SSTERF, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          MAX, MIN, REAL, SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      ALLEIG = LSAME( RANGE, 'A' )
      VALEIG = LSAME( RANGE, 'V' )
      INDEIG = LSAME( RANGE, 'I' )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
         INFO = -2
      ELSE IF( .NOT.( LSAME( UPLO, 'L' ) .OR.
     $         LSAME( UPLO, 'U' ) ) )
     $          THEN
         INFO = -3
      ELSE IF( N.LT.0 ) THEN
         INFO = -4
      ELSE
         IF( VALEIG ) THEN
            IF( N.GT.0 .AND. VU.LE.VL )
     $         INFO = -7
         ELSE IF( INDEIG ) THEN
            IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
               INFO = -8
            ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
               INFO = -9
            END IF
         END IF
      END IF
      IF( INFO.EQ.0 ) THEN
         IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
     $      INFO = -14
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'CHPEVX', -INFO )
         RETURN
      END IF
*
*     Quick return if possible
*
      M = 0
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         IF( ALLEIG .OR. INDEIG ) THEN
            M = 1
            W( 1 ) = REAL( AP( 1 ) )
         ELSE
            IF( VL.LT.REAL( AP( 1 ) ) .AND. VU.GE.REAL( AP( 1 ) ) ) THEN
               M = 1
               W( 1 ) = REAL( AP( 1 ) )
            END IF
         END IF
         IF( WANTZ )
     $      Z( 1, 1 ) = CONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = SLAMCH( 'Safe minimum' )
      EPS = SLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
*
*     Scale matrix to allowable range, if necessary.
*
      ISCALE = 0
      ABSTLL = ABSTOL
      IF ( VALEIG ) THEN
         VLL = VL
         VUU = VU
      ELSE
         VLL = ZERO
         VUU = ZERO
      ENDIF
      ANRM = CLANHP( 'M', UPLO, N, AP, RWORK )
      IF( ANRM.GT.ZERO .AND. ANRM.LT.RMIN ) THEN
         ISCALE = 1
         SIGMA = RMIN / ANRM
      ELSE IF( ANRM.GT.RMAX ) THEN
         ISCALE = 1
         SIGMA = RMAX / ANRM
      END IF
      IF( ISCALE.EQ.1 ) THEN
         CALL CSSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
         IF( ABSTOL.GT.0 )
     $      ABSTLL = ABSTOL*SIGMA
         IF( VALEIG ) THEN
            VLL = VL*SIGMA
            VUU = VU*SIGMA
         END IF
      END IF
*
*     Call CHPTRD to reduce Hermitian packed matrix to tridiagonal form.
*
      INDD = 1
      INDE = INDD + N
      INDRWK = INDE + N
      INDTAU = 1
      INDWRK = INDTAU + N
      CALL CHPTRD( UPLO, N, AP, RWORK( INDD ), RWORK( INDE ),
     $             WORK( INDTAU ), IINFO )
*
*     If all eigenvalues are desired and ABSTOL is less than or equal
*     to zero, then call SSTERF or CUPGTR and CSTEQR.  If this fails
*     for some eigenvalue, then try SSTEBZ.
*
      TEST = .FALSE.
      IF (INDEIG) THEN
         IF (IL.EQ.1 .AND. IU.EQ.N) THEN
            TEST = .TRUE.
         END IF
      END IF
      IF ((ALLEIG .OR. TEST) .AND. (ABSTOL.LE.ZERO)) THEN
         CALL SCOPY( N, RWORK( INDD ), 1, W, 1 )
         INDEE = INDRWK + 2*N
         IF( .NOT.WANTZ ) THEN
            CALL SCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
            CALL SSTERF( N, W, RWORK( INDEE ), INFO )
         ELSE
            CALL CUPGTR( UPLO, N, AP, WORK( INDTAU ), Z, LDZ,
     $                   WORK( INDWRK ), IINFO )
            CALL SCOPY( N-1, RWORK( INDE ), 1, RWORK( INDEE ), 1 )
            CALL CSTEQR( JOBZ, N, W, RWORK( INDEE ), Z, LDZ,
     $                   RWORK( INDRWK ), INFO )
            IF( INFO.EQ.0 ) THEN
               DO 10 I = 1, N
                  IFAIL( I ) = 0
   10          CONTINUE
            END IF
         END IF
         IF( INFO.EQ.0 ) THEN
            M = N
            GO TO 20
         END IF
         INFO = 0
      END IF
*
*     Otherwise, call SSTEBZ and, if eigenvectors are desired, CSTEIN.
*
      IF( WANTZ ) THEN
         ORDER = 'B'
      ELSE
         ORDER = 'E'
      END IF
      INDISP = 1 + N
      INDIWK = INDISP + N
      CALL SSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTLL,
     $             RWORK( INDD ), RWORK( INDE ), M, NSPLIT, W,
     $             IWORK( 1 ), IWORK( INDISP ), RWORK( INDRWK ),
     $             IWORK( INDIWK ), INFO )
*
      IF( WANTZ ) THEN
         CALL CSTEIN( N, RWORK( INDD ), RWORK( INDE ), M, W,
     $                IWORK( 1 ), IWORK( INDISP ), Z, LDZ,
     $                RWORK( INDRWK ), IWORK( INDIWK ), IFAIL, INFO )
*
*        Apply unitary matrix used in reduction to tridiagonal
*        form to eigenvectors returned by CSTEIN.
*
         INDWRK = INDTAU + N
         CALL CUPMTR( 'L', UPLO, 'N', N, M, AP, WORK( INDTAU ), Z,
     $                LDZ,
     $                WORK( INDWRK ), IINFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
   20 CONTINUE
      IF( ISCALE.EQ.1 ) THEN
         IF( INFO.EQ.0 ) THEN
            IMAX = M
         ELSE
            IMAX = INFO - 1
         END IF
         CALL SSCAL( IMAX, ONE / SIGMA, W, 1 )
      END IF
*
*     If eigenvalues are not in order, then sort them, along with
*     eigenvectors.
*
      IF( WANTZ ) THEN
         DO 40 J = 1, M - 1
            I = 0
            TMP1 = W( J )
            DO 30 JJ = J + 1, M
               IF( W( JJ ).LT.TMP1 ) THEN
                  I = JJ
                  TMP1 = W( JJ )
               END IF
   30       CONTINUE
*
            IF( I.NE.0 ) THEN
               ITMP1 = IWORK( 1 + I-1 )
               W( I ) = W( J )
               IWORK( 1 + I-1 ) = IWORK( 1 + J-1 )
               W( J ) = TMP1
               IWORK( 1 + J-1 ) = ITMP1
               CALL CSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
               IF( INFO.NE.0 ) THEN
                  ITMP1 = IFAIL( I )
                  IFAIL( I ) = IFAIL( J )
                  IFAIL( J ) = ITMP1
               END IF
            END IF
   40    CONTINUE
      END IF
*
      RETURN
*
*     End of CHPEVX
*
      END