OpenBLAS/lapack-netlib/SRC/sspevd.f

*> \brief <b> SSPEVD 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
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*> \endhtmlonly
*
*  Definition:
*  ===========
*
*       SUBROUTINE SSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
*                          IWORK, LIWORK, INFO )
*
*       .. Scalar Arguments ..
*       CHARACTER          JOBZ, UPLO
*       INTEGER            INFO, LDZ, LIWORK, LWORK, N
*       ..
*       .. Array Arguments ..
*       INTEGER            IWORK( * )
*       REAL               AP( * ), W( * ), WORK( * ), Z( LDZ, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SSPEVD computes all the eigenvalues and, optionally, eigenvectors
*> of a real symmetric matrix A in packed storage. If eigenvectors are
*> desired, it uses a divide and conquer algorithm.
*>
*> The divide and conquer algorithm makes very mild assumptions about
*> floating point arithmetic. It will work on machines with a guard
*> digit in add/subtract, or on those binary machines without guard
*> digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
*> Cray-2. It could conceivably fail on hexadecimal or decimal machines
*> without guard digits, but we know of none.
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] JOBZ
*> \verbatim
*>          JOBZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only;
*>          = 'V':  Compute eigenvalues and eigenvectors.
*> \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 REAL array, dimension (N*(N+1)/2)
*>          On entry, the upper or lower triangle of the symmetric 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[out] W
*> \verbatim
*>          W is REAL array, dimension (N)
*>          If INFO = 0, the eigenvalues in ascending order.
*> \endverbatim
*>
*> \param[out] Z
*> \verbatim
*>          Z is REAL array, dimension (LDZ, N)
*>          If JOBZ = 'V', then if INFO = 0, Z contains the orthonormal
*>          eigenvectors of the matrix A, with the i-th column of Z
*>          holding the eigenvector associated with W(i).
*>          If JOBZ = 'N', then Z is not referenced.
*> \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 REAL array, dimension (MAX(1,LWORK))
*>          On exit, if INFO = 0, WORK(1) returns the required LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*>          LWORK is INTEGER
*>          The dimension of the array WORK.
*>          If N <= 1,               LWORK must be at least 1.
*>          If JOBZ = 'N' and N > 1, LWORK must be at least 2*N.
*>          If JOBZ = 'V' and N > 1, LWORK must be at least
*>                                                 1 + 6*N + N**2.
*>
*>          If LWORK = -1, then a workspace query is assumed; the routine
*>          only calculates the required sizes of the WORK and IWORK
*>          arrays, returns these values as the first entries of the WORK
*>          and IWORK arrays, and no error message related to LWORK or
*>          LIWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] IWORK
*> \verbatim
*>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
*>          On exit, if INFO = 0, IWORK(1) returns the required LIWORK.
*> \endverbatim
*>
*> \param[in] LIWORK
*> \verbatim
*>          LIWORK is INTEGER
*>          The dimension of the array IWORK.
*>          If JOBZ  = 'N' or N <= 1, LIWORK must be at least 1.
*>          If JOBZ  = 'V' and N > 1, LIWORK must be at least 3 + 5*N.
*>
*>          If LIWORK = -1, then a workspace query is assumed; the
*>          routine only calculates the required sizes of the WORK and
*>          IWORK arrays, returns these values as the first entries of
*>          the WORK and IWORK arrays, and no error message related to
*>          LWORK or LIWORK is issued by XERBLA.
*> \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, the algorithm failed to converge; i
*>                off-diagonal elements of an intermediate tridiagonal
*>                form did not converge to zero.
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup realOTHEReigen
*
*  =====================================================================
      SUBROUTINE SSPEVD( JOBZ, UPLO, N, AP, W, Z, LDZ, WORK, LWORK,
     $                   IWORK, LIWORK, INFO )
*
*  -- LAPACK driver routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      CHARACTER          JOBZ, UPLO
      INTEGER            INFO, LDZ, LIWORK, LWORK, N
*     ..
*     .. Array Arguments ..
      INTEGER            IWORK( * )
      REAL               AP( * ), W( * ), WORK( * ), Z( LDZ, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            LQUERY, WANTZ
      INTEGER            IINFO, INDE, INDTAU, INDWRK, ISCALE, LIWMIN,
     $                   LLWORK, LWMIN
      REAL               ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA,
     $                   SMLNUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      REAL               SLAMCH, SLANSP
      EXTERNAL           LSAME, SLAMCH, SLANSP
*     ..
*     .. External Subroutines ..
      EXTERNAL           SOPMTR, SSCAL, SSPTRD, SSTEDC, SSTERF, XERBLA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          SQRT
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      WANTZ = LSAME( JOBZ, 'V' )
      LQUERY = ( LWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
*
      INFO = 0
      IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
         INFO = -1
      ELSE IF( .NOT.( LSAME( UPLO, 'U' ) .OR. LSAME( UPLO, 'L' ) ) )
     $          THEN
         INFO = -2
      ELSE IF( N.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) ) THEN
         INFO = -7
      END IF
*
      IF( INFO.EQ.0 ) THEN
         IF( N.LE.1 ) THEN
            LIWMIN = 1
            LWMIN = 1
         ELSE
            IF( WANTZ ) THEN
               LIWMIN = 3 + 5*N
               LWMIN = 1 + 6*N + N**2
            ELSE
               LIWMIN = 1
               LWMIN = 2*N
            END IF
         END IF
         IWORK( 1 ) = LIWMIN
         WORK( 1 ) = LWMIN
*
         IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -9
         ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
            INFO = -11
         END IF
      END IF
*
      IF( INFO.NE.0 ) THEN
         CALL XERBLA( 'SSPEVD', -INFO )
         RETURN
      ELSE IF( LQUERY ) THEN
         RETURN
      END IF
*
*     Quick return if possible
*
      IF( N.EQ.0 )
     $   RETURN
*
      IF( N.EQ.1 ) THEN
         W( 1 ) = AP( 1 )
         IF( WANTZ )
     $      Z( 1, 1 ) = ONE
         RETURN
      END IF
*
*     Get machine constants.
*
      SAFMIN = SLAMCH( 'Safe minimum' )
      EPS = SLAMCH( 'Precision' )
      SMLNUM = SAFMIN / EPS
      BIGNUM = ONE / SMLNUM
      RMIN = SQRT( SMLNUM )
      RMAX = SQRT( BIGNUM )
*
*     Scale matrix to allowable range, if necessary.
*
      ANRM = SLANSP( 'M', UPLO, N, AP, WORK )
      ISCALE = 0
      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 SSCAL( ( N*( N+1 ) ) / 2, SIGMA, AP, 1 )
      END IF
*
*     Call SSPTRD to reduce symmetric packed matrix to tridiagonal form.
*
      INDE = 1
      INDTAU = INDE + N
      CALL SSPTRD( UPLO, N, AP, W, WORK( INDE ), WORK( INDTAU ), IINFO )
*
*     For eigenvalues only, call SSTERF.  For eigenvectors, first call
*     SSTEDC to generate the eigenvector matrix, WORK(INDWRK), of the
*     tridiagonal matrix, then call SOPMTR to multiply it by the
*     Householder transformations represented in AP.
*
      IF( .NOT.WANTZ ) THEN
         CALL SSTERF( N, W, WORK( INDE ), INFO )
      ELSE
         INDWRK = INDTAU + N
         LLWORK = LWORK - INDWRK + 1
         CALL SSTEDC( 'I', N, W, WORK( INDE ), Z, LDZ, WORK( INDWRK ),
     $                LLWORK, IWORK, LIWORK, INFO )
         CALL SOPMTR( 'L', UPLO, 'N', N, N, AP, WORK( INDTAU ), Z, LDZ,
     $                WORK( INDWRK ), IINFO )
      END IF
*
*     If matrix was scaled, then rescale eigenvalues appropriately.
*
      IF( ISCALE.EQ.1 )
     $   CALL SSCAL( N, ONE / SIGMA, W, 1 )
*
      WORK( 1 ) = LWMIN
      IWORK( 1 ) = LIWMIN
      RETURN
*
*     End of SSPEVD
*
      END