spteqr.f

          SUBROUTINE SPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          COMPZ
          INTEGER            INFO, LDZ, N
    *     ..
    *     .. Array Arguments ..
          REAL               D( * ), E( * ), WORK( * ), Z( LDZ, * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  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.)
    *
    *  Arguments
    *  =========
    *
    *  COMPZ   (input) 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.
    *
    *  N       (input) INTEGER
    *          The order of the matrix.  N >= 0.
    *
    *  D       (input/output) REAL array, dimension (N)
    *          On entry, the n diagonal elements of the tridiagonal
    *          matrix.
    *          On normal exit, D contains the eigenvalues, in descending
    *          order.
    *
    *  E       (input/output) REAL array, dimension (N-1)
    *          On entry, the (n-1) subdiagonal elements of the tridiagonal
    *          matrix.
    *          On exit, E has been destroyed.
    *
    *  Z       (input/output) 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.
    *
    *  LDZ     (input) INTEGER
    *          The leading dimension of the array Z.  LDZ >= 1, and if
    *          COMPZ = 'V' or 'I', LDZ >= max(1,N).
    *
    *  WORK    (workspace) REAL array, dimension (4*N)
    *
    *  INFO    (output) 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.
    *
    *  =====================================================================
    *
    *     .. 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