spteqr(3S)

NAME
     SPTEQR - compute 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

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

         CHARACTER      COMPZ

         INTEGER        INFO, LDZ, N

         REAL           D( * ), E( * ), WORK( * ), Z( LDZ, * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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.

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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