dspgvx

NAME

dspgvx - 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 DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, 
 *      IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  INTEGER ITYPE, N, IL, IU, M, LDZ, INFO
  INTEGER IWORK(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABSTOL
  DOUBLE PRECISION AP(*), BP(*), W(*), Z(LDZ,*), WORK(*)
 
  SUBROUTINE DSPGVX_64( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, 
 *      IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  INTEGER*8 ITYPE, N, IL, IU, M, LDZ, INFO
  INTEGER*8 IWORK(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABSTOL
  DOUBLE PRECISION AP(*), BP(*), W(*), Z(LDZ,*), WORK(*)

F95 INTERFACE 

  SUBROUTINE SPGVX( ITYPE, JOBZ, RANGE, UPLO, [N], AP, BP, VL, VU, IL, 
 *       IU, ABSTOL, M, W, Z, [LDZ], [WORK], [IWORK], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  INTEGER :: ITYPE, N, IL, IU, M, LDZ, INFO
  INTEGER, DIMENSION(:) :: IWORK, IFAIL
  REAL(8) :: VL, VU, ABSTOL
  REAL(8), DIMENSION(:) :: AP, BP, W, WORK
  REAL(8), DIMENSION(:,:) :: Z
 
  SUBROUTINE SPGVX_64( ITYPE, JOBZ, RANGE, UPLO, [N], AP, BP, VL, VU, 
 *       IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], [IWORK], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  INTEGER(8) :: ITYPE, N, IL, IU, M, LDZ, INFO
  INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
  REAL(8) :: VL, VU, ABSTOL
  REAL(8), DIMENSION(:) :: AP, BP, W, WORK
  REAL(8), DIMENSION(:,:) :: Z

C INTERFACE

#include <sunperf.h>

void dspgvx(int itype, char jobz, char range, char uplo, int n, double *ap, double *bp, double vl, double vu, int il, int iu, double abstol, int *m, double *w, double *z, int ldz, int *ifail, int *info);

void dspgvx_64(long itype, char jobz, char range, char uplo, long n, double *ap, double *bp, double vl, double vu, long il, long iu, double abstol, long *m, double *w, double *z, long ldz, long *ifail, long *info);

PURPOSE

dspgvx 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)
Specifies the problem type to be solved:

* JOBZ (input)

* RANGE (input)

* UPLO (input)

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

* AP (input/output)
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)
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)
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'.

* VU (input)
See the description of VL.

* IL (input)
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'.

* IU (input)
See the description of IL.

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

* W (output)
On normal exit, the first M elements contain the selected eigenvalues in ascending order.

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

* WORK (workspace)
dimension(8*N)

* IWORK (workspace)
dimension(5*N)

* IFAIL (output)
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)