ZPTEQR (3lapack)

compute all eigenvalues and, optionally, eigenvectors of a symmetric positive definite tridiagonal matrix by first factoring the matrix using DPTTRF and then calling ZBDSQR to compute the singular values of the bidiagonal factor

SYNOPSIS

  SUBROUTINE ZPTEQR( COMPZ, N, D, E, Z, LDZ, WORK, INFO )

      CHARACTER      COMPZ

      INTEGER        INFO, LDZ, N

      DOUBLE         PRECISION D( * ), E( * ), WORK( * )

      COMPLEX*16     Z( LDZ, * )

PURPOSE

  ZPTEQR computes all eigenvalues and, optionally, eigenvectors of a
  symmetric positive definite tridiagonal matrix by first factoring the
  matrix using DPTTRF and then calling ZBDSQR 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 positive definite Hermitian matrix can
  also be found if ZHETRD, ZHPTRD, or ZHBTRD 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 Hermitian matrix also.
          Array Z contains the unitary 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N-1)
          On entry, the (n-1) subdiagonal elements of the tridiagonal matrix.
          On exit, E has been destroyed.

  Z       (input/output) COMPLEX*16 array, dimension (LDZ, N)
          On entry, if COMPZ = 'V', the unitary matrix used in the reduction
          to tridiagonal form.  On exit, if COMPZ = 'V', the orthonormal
          eigenvectors of the original Hermitian 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) DOUBLE PRECISION array, dimension (LWORK)
          If  COMPZ = 'N', then LWORK = 2*N If  COMPZ = 'V' or 'I', then
          LWORK = MAX(1,4*N-4)

  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.