sspgvx(3S)

NAME
     SSPGVX - compute selected eigenvalues, and optionally, eigenvectors of a
     real generalized symmetric-definite eigenproblem, of the form
     A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x

SYNOPSIS
     SUBROUTINE SSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU,
                        ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )

         CHARACTER      JOBZ, RANGE, UPLO

         INTEGER        IL, INFO, ITYPE, IU, LDZ, M, N

         REAL           ABSTOL, VL, VU

         INTEGER        IFAIL( * ), IWORK( * )

         REAL           AP( * ), BP( * ), W( * ), 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
     SSPGVX computes selected eigenvalues, and optionally, eigenvectors of a
     real generalized symmetric-definite eigenproblem, of the form
     A*x=(lambda)*B*x, A*Bx=(lambda)*x, or B*A*x=(lambda)*x. Here A and B are
     assumed to be symmetric, stored in packed storage, and B is also positive
     definite.  Eigenvalues and eigenvectors can be selected by specifying
     either a range of values or a range of indices for the desired
     eigenvalues.


ARGUMENTS
     ITYPE   (input) INTEGER
             Specifies the problem type to be solved:
             = 1:  A*x = (lambda)*B*x
             = 2:  A*B*x = (lambda)*x
             = 3:  B*A*x = (lambda)*x

     JOBZ    (input) CHARACTER*1
             = 'N':  Compute eigenvalues only;
             = 'V':  Compute eigenvalues and eigenvectors.
     RANGE   (input) CHARACTER*1
             = 'A': all eigenvalues will be found.
             = 'V': all eigenvalues in the half-open interval (VL,VU] will be
             found.  = 'I': the IL-th through IU-th eigenvalues will be found.

     UPLO    (input) CHARACTER*1
             = 'U':  Upper triangle of A and B are stored;
             = 'L':  Lower triangle of A and B are stored.

     N       (input) INTEGER
             The order of the matrix pencil (A,B).  N >= 0.

     AP      (input/output) 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, the contents of AP are destroyed.

     BP      (input/output) REAL array, dimension (N*(N+1)/2)
             On entry, the upper or lower triangle of the symmetric matrix B,
             packed columnwise in a linear array.  The j-th column of B is
             stored in the array BP as follows:  if UPLO = 'U', BP(i + (j-
             1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-
             j)/2) = B(i,j) for j<=i<=n.

             On exit, the triangular factor U or L from the Cholesky
             factorization B = U**T*U or B = L*L**T, in the same storage
             format as B.

     VL      (input) REAL
             VU      (input) REAL If RANGE='V', the lower and upper bounds of
             the interval to be searched for eigenvalues. VL < VU.  Not
             referenced if RANGE = 'A' or 'I'.

     IL      (input) INTEGER
             IU      (input) INTEGER If RANGE='I', the indices (in ascending
             order) of the smallest and largest eigenvalues to be returned.  1
             <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.  Not
             referenced if RANGE = 'A' or 'V'.

     ABSTOL  (input) REAL
             The absolute error tolerance for the eigenvalues.  An approximate
             eigenvalue is accepted as converged when it is determined to lie
             in an interval [a,b] of width less than or equal to

             ABSTOL + EPS *   max( |a|,|b| ) ,

             where EPS is the machine precision.  If ABSTOL is less than or
             equal to zero, then  EPS*|T|  will be used in its place, where
             |T| is the 1-norm of the tridiagonal matrix obtained by reducing
             A to tridiagonal form.

             Eigenvalues will be computed most accurately when ABSTOL is set
             to twice the underflow threshold 2*SLAMCH('S'), not zero.  If
             this routine returns with INFO>0, indicating that some
             eigenvectors did not converge, try setting ABSTOL to
             2*SLAMCH('S').

     M       (output) INTEGER
             The total number of eigenvalues found.  0 <= M <= N.  If RANGE =
             'A', M = N, and if RANGE = 'I', M = IU-IL+1.

     W       (output) REAL array, dimension (N)
             On normal exit, the first M elements contain the selected
             eigenvalues in ascending order.

     Z       (output) REAL array, dimension (LDZ, max(1,M))
             If JOBZ = 'N', then Z is not referenced.  If JOBZ = 'V', then if
             INFO = 0, the first M columns of Z contain the orthonormal
             eigenvectors of the matrix A corresponding to the selected
             eigenvalues, with the i-th column of Z holding the eigenvector
             associated with W(i).  The eigenvectors are normalized as
             follows:  if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3,
             Z**T*inv(B)*Z = I.

             If an eigenvector fails to converge, then that column of Z
             contains the latest approximation to the eigenvector, and the
             index of the eigenvector is returned in IFAIL.  Note: the user
             must ensure that at least max(1,M) columns are supplied in the
             array Z; if RANGE = 'V', the exact value of M is not known in
             advance and an upper bound must be used.

     LDZ     (input) INTEGER
             The leading dimension of the array Z.  LDZ >= 1, and if JOBZ =
             'V', LDZ >= max(1,N).

     WORK    (workspace) REAL array, dimension (8*N)

     IWORK   (workspace) INTEGER array, dimension (5*N)

     IFAIL   (output) INTEGER array, dimension (N)
             If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL
             are zero.  If INFO > 0, then IFAIL contains the indices of the
             eigenvectors that failed to converge.  If JOBZ = 'N', then IFAIL
             is not referenced.

     INFO    (output) INTEGER
             = 0:  successful exit
             < 0:  if INFO = -i, the i-th argument had an illegal value
             > 0:  SPPTRF or SSPEVX returned an error code:
             <= N:  if INFO = i, SSPEVX failed to converge; i eigenvectors
             failed to converge.  Their indices are stored in array IFAIL.  >
             N:   if INFO = N + i, for 1 <= i <= N, then the leading minor of
             order i of B is not positive definite.  The factorization of B
             could not be completed and no eigenvalues or eigenvectors were
             computed.

FURTHER DETAILS
     Based on contributions by
        Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA


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

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