spteqr.f

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*> \brief \b SPTEQR
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  Definition:
*  ===========
*
*       SUBROUTINE SPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
* 
*       .. Scalar Arguments ..
*       CHARACTER          COMPZ
*       INTEGER            INFO, LDZ, N
*       ..
*       .. Array Arguments ..
*       REAL               D( * ), E( * ), WORK( * ), Z( LDZ, * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SPTEQR computes all eigenvalues and, optionally, eigenvectors of a
*> symmetric positive definite tridiagonal matrix by first factoring the
*> matrix using SPTTRF, and then calling SBDSQR to compute the singular
*> values of the bidiagonal factor.
*>
*> This routine computes the eigenvalues of the positive definite
*> tridiagonal matrix to high relative accuracy.  This means that if the
*> eigenvalues range over many orders of magnitude in size, then the
*> small eigenvalues and corresponding eigenvectors will be computed
*> more accurately than, for example, with the standard QR method.
*>
*> The eigenvectors of a full or band symmetric positive definite matrix
*> can also be found if SSYTRD, SSPTRD, or SSBTRD has been used to
*> reduce this matrix to tridiagonal form. (The reduction to tridiagonal
*> form, however, may preclude the possibility of obtaining high
*> relative accuracy in the small eigenvalues of the original matrix, if
*> these eigenvalues range over many orders of magnitude.)
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] COMPZ
*> \verbatim
*>          COMPZ is CHARACTER*1
*>          = 'N':  Compute eigenvalues only.
*>          = 'V':  Compute eigenvectors of original symmetric
*>                  matrix also.  Array Z contains the orthogonal
*>                  matrix used to reduce the original matrix to
*>                  tridiagonal form.
*>          = 'I':  Compute eigenvectors of tridiagonal matrix also.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The order of the matrix.  N >= 0.
*> \endverbatim
*>
*> \param[in,out] D
*> \verbatim
*>          D is REAL array, dimension (N)
*>          On entry, the n diagonal elements of the tridiagonal
*>          matrix.
*>          On normal exit, D contains the eigenvalues, in descending
*>          order.
*> \endverbatim
*>
*> \param[in,out] E
*> \verbatim
*>          E is REAL array, dimension (N-1)
*>          On entry, the (n-1) subdiagonal elements of the tridiagonal
*>          matrix.
*>          On exit, E has been destroyed.
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*>          Z is REAL array, dimension (LDZ, N)
*>          On entry, if COMPZ = 'V', the orthogonal matrix used in the
*>          reduction to tridiagonal form.
*>          On exit, if COMPZ = 'V', the orthonormal eigenvectors of the
*>          original symmetric matrix;
*>          if COMPZ = 'I', the orthonormal eigenvectors of the
*>          tridiagonal matrix.
*>          If INFO > 0 on exit, Z contains the eigenvectors associated
*>          with only the stored eigenvalues.
*>          If  COMPZ = '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
*>          COMPZ = 'V' or 'I', LDZ >= max(1,N).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*>          WORK is REAL array, dimension (4*N)
*> \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, and i is:
*>                <= N  the Cholesky factorization of the matrix could
*>                      not be performed because the i-th principal minor
*>                      was not positive definite.
*>                > N   the SVD algorithm failed to converge;
*>                      if INFO = N+i, i off-diagonal elements of the
*>                      bidiagonal factor 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 September 2012
*
*> \ingroup realPTcomputational
*
*  =====================================================================
      SUBROUTINE spteqr( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
*
*  -- LAPACK computational routine (version 3.4.2) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     September 2012
*
*     .. Scalar Arguments ..
      CHARACTER          compz
      INTEGER            info, ldz, n
*     ..
*     .. Array Arguments ..
      REAL               d( * ), e( * ), work( * ), z( ldz, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               zero, one
      parameter( zero = 0.0e0, one = 1.0e0 )
*     ..
*     .. External Functions ..
      LOGICAL            lsame
      EXTERNAL           lsame
*     ..
*     .. External Subroutines ..
      EXTERNAL           sbdsqr, slaset, spttrf, xerbla
*     ..
*     .. Local Arrays ..
      REAL               c( 1, 1 ), vt( 1, 1 )
*     ..
*     .. Local Scalars ..
      INTEGER            i, icompz, nru
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          max, sqrt
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
*
      IF( lsame( compz, 'N' ) ) THEN
         icompz = 0
      ELSE IF( lsame( compz, 'V' ) ) THEN
         icompz = 1
      ELSE IF( lsame( compz, 'I' ) ) THEN
         icompz = 2
      ELSE
         icompz = -1
      END IF
      IF( icompz.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.0 ) THEN
         info = -2
      ELSE IF( ( ldz.LT.1 ) .OR. ( icompz.GT.0 .AND. ldz.LT.max( 1,
     $         n ) ) ) THEN
         info = -6
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'SPTEQR', -info )
         return
      END IF
*
*     Quick return if possible
*
      IF( n.EQ.0 )
     $   return
*
      IF( n.EQ.1 ) THEN
         IF( icompz.GT.0 )
     $      z( 1, 1 ) = one
         return
      END IF
      IF( icompz.EQ.2 )
     $   CALL slaset( 'Full', n, n, zero, one, z, ldz )
*
*     Call SPTTRF to factor the matrix.
*
      CALL spttrf( n, d, e, info )
      IF( info.NE.0 )
     $   return
      DO 10 i = 1, n
         d( i ) = sqrt( d( i ) )
   10 continue
      DO 20 i = 1, n - 1
         e( i ) = e( i )*d( i )
   20 continue
*
*     Call SBDSQR to compute the singular values/vectors of the
*     bidiagonal factor.
*
      IF( icompz.GT.0 ) THEN
         nru = n
      ELSE
         nru = 0
      END IF
      CALL sbdsqr( 'Lower', n, 0, nru, 0, d, e, vt, 1, z, ldz, c, 1,
     $             work, info )
*
*     Square the singular values.
*
      IF( info.EQ.0 ) THEN
         DO 30 i = 1, n
            d( i ) = d( i )*d( i )
   30    continue
      ELSE
         info = n + info
      END IF
*
      return
*
*     End of SPTEQR
*
      END