• NAME
• SYNOPSIS
• F95 INTERFACE
• C INTERFACE
• PURPOSE
• ARGUMENTS
• FURTHER DETAILS

NAME

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

SYNOPSIS

  SUBROUTINE DSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, 
 *      Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, 
 *      INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  INTEGER N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
  INTEGER IWORK(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABSTOL
  DOUBLE PRECISION AB(LDAB,*), BB(LDBB,*), Q(LDQ,*), W(*), Z(LDZ,*), WORK(*)

  SUBROUTINE DSBGVX_64( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, 
 *      LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, 
 *      IFAIL, INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  INTEGER*8 N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
  INTEGER*8 IWORK(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABSTOL
  DOUBLE PRECISION AB(LDAB,*), BB(LDBB,*), Q(LDQ,*), W(*), Z(LDZ,*), WORK(*)

F95 INTERFACE

  SUBROUTINE SBGVX( JOBZ, RANGE, UPLO, [N], KA, KB, AB, [LDAB], BB, 
 *       [LDBB], Q, [LDQ], VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], 
 *       [IWORK], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  INTEGER :: N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
  INTEGER, DIMENSION(:) :: IWORK, IFAIL
  REAL(8) :: VL, VU, ABSTOL
  REAL(8), DIMENSION(:) :: W, WORK
  REAL(8), DIMENSION(:,:) :: AB, BB, Q, Z

  SUBROUTINE SBGVX_64( JOBZ, RANGE, UPLO, [N], KA, KB, AB, [LDAB], BB, 
 *       [LDBB], Q, [LDQ], VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], 
 *       [IWORK], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  INTEGER(8) :: N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
  INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
  REAL(8) :: VL, VU, ABSTOL
  REAL(8), DIMENSION(:) :: W, WORK
  REAL(8), DIMENSION(:,:) :: AB, BB, Q, Z

C INTERFACE

#include <sunperf.h>

void dsbgvx(char jobz, char range, char uplo, int n, int ka, int kb, double *ab, int ldab, double *bb, int ldbb, double *q, int ldq, double vl, double vu, int il, int iu, double abstol, int *m, double *w, double *z, int ldz, int *ifail, int *info);

void dsbgvx_64(char jobz, char range, char uplo, long n, long ka, long kb, double *ab, long ldab, double *bb, long ldbb, double *q, long ldq, double vl, double vu, long il, long iu, double abstol, long *m, double *w, double *z, long ldz, long *ifail, long *info);

PURPOSE

dsbgvx computes selected eigenvalues, and optionally, eigenvectors of a real generalized symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be symmetric and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.

ARGUMENTS

• JOBZ (input)
  

   = 'N':  Compute eigenvalues only;

   = 'V':  Compute eigenvalues and eigenvectors.

• RANGE (input)
  

   = '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)
  

   = 'U':  Upper triangles of A and B are stored;

   = 'L':  Lower triangles of A and B are stored.

• N (input)
  The order of the matrices A and B. N > = 0.

• KA (input)
  The number of superdiagonals of the matrix A if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KA > = 0.

• KB (input)
  The number of superdiagonals of the matrix B if UPLO = 'U', or the number of subdiagonals if UPLO = 'L'. KB > = 0.

• AB (input/output)
  On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first ka+1 rows of the array. The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka) < =i < =j; if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j < =i < =min(n,j+ka).

  On exit, the contents of AB are destroyed.

• LDAB (input)
  The leading dimension of the array AB. LDAB > = KA+1.

• BB (input/output)
  On entry, the upper or lower triangle of the symmetric band matrix B, stored in the first kb+1 rows of the array. The j-th column of B is stored in the j-th column of the array BB as follows: if UPLO = 'U', BB(ka+1+i-j,j) = B(i,j) for max(1,j-kb) < =i < =j; if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j < =i < =min(n,j+kb).

  On exit, the factor S from the split Cholesky factorization B = S**T*S, as returned by SPBSTF.

• LDBB (input)
  The leading dimension of the array BB. LDBB > = KB+1.

• Q (output)
  If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x = (lambda)*B*x to standard form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.

• LDQ (input)
  The leading dimension of the array Q. If JOBZ = 'N', LDQ > = 1. If JOBZ = 'V', LDQ > = max(1,N).

• 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)
  If INFO = 0, the eigenvalues in ascending order.

• Z (output)
  If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized so Z**T*B*Z = I. If JOBZ = 'N', then Z is not referenced.

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

• WORK (workspace)
  dimension(7*N)

• IWORK (workspace)
  dimension(5*N)

• IFAIL (input)
  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 eigenvalues that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.

• INFO (output)
  

   = 0 : successful exit

   < 0 : if INFO  = -i, the i-th argument had an illegal value

   < = N: if INFO  = i, then i eigenvectors failed to converge.
  Their indices are stored in IFAIL.
   > N : SPBSTF returned an error code; i.e.,
  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