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

NAME

ssbevx - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A

SYNOPSIS

  SUBROUTINE SSBEVX( JOBZ, RANGE, UPLO, N, NDIAG, A, LDA, Q, LDQ, VL, 
 *      VU, IL, IU, ABTOL, NFOUND, W, Z, LDZ, WORK, IWORK2, IFAIL, INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  INTEGER N, NDIAG, LDA, LDQ, IL, IU, NFOUND, LDZ, INFO
  INTEGER IWORK2(*), IFAIL(*)
  REAL VL, VU, ABTOL
  REAL A(LDA,*), Q(LDQ,*), W(*), Z(LDZ,*), WORK(*)

  SUBROUTINE SSBEVX_64( JOBZ, RANGE, UPLO, N, NDIAG, A, LDA, Q, LDQ, 
 *      VL, VU, IL, IU, ABTOL, NFOUND, W, Z, LDZ, WORK, IWORK2, IFAIL, 
 *      INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  INTEGER*8 N, NDIAG, LDA, LDQ, IL, IU, NFOUND, LDZ, INFO
  INTEGER*8 IWORK2(*), IFAIL(*)
  REAL VL, VU, ABTOL
  REAL A(LDA,*), Q(LDQ,*), W(*), Z(LDZ,*), WORK(*)

F95 INTERFACE

  SUBROUTINE SBEVX( JOBZ, RANGE, UPLO, [N], NDIAG, A, [LDA], Q, [LDQ], 
 *       VL, VU, IL, IU, ABTOL, NFOUND, W, Z, [LDZ], [WORK], [IWORK2], 
 *       IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  INTEGER :: N, NDIAG, LDA, LDQ, IL, IU, NFOUND, LDZ, INFO
  INTEGER, DIMENSION(:) :: IWORK2, IFAIL
  REAL :: VL, VU, ABTOL
  REAL, DIMENSION(:) :: W, WORK
  REAL, DIMENSION(:,:) :: A, Q, Z

  SUBROUTINE SBEVX_64( JOBZ, RANGE, UPLO, [N], NDIAG, A, [LDA], Q, 
 *       [LDQ], VL, VU, IL, IU, ABTOL, NFOUND, W, Z, [LDZ], [WORK], 
 *       [IWORK2], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  INTEGER(8) :: N, NDIAG, LDA, LDQ, IL, IU, NFOUND, LDZ, INFO
  INTEGER(8), DIMENSION(:) :: IWORK2, IFAIL
  REAL :: VL, VU, ABTOL
  REAL, DIMENSION(:) :: W, WORK
  REAL, DIMENSION(:,:) :: A, Q, Z

C INTERFACE

#include <sunperf.h>

void ssbevx(char jobz, char range, char uplo, int n, int ndiag, float *a, int lda, float *q, int ldq, float vl, float vu, int il, int iu, float abtol, int *nfound, float *w, float *z, int ldz, int *ifail, int *info);

void ssbevx_64(char jobz, char range, char uplo, long n, long ndiag, float *a, long lda, float *q, long ldq, float vl, float vu, long il, long iu, float abtol, long *nfound, float *w, float *z, long ldz, long *ifail, long *info);

PURPOSE

ssbevx computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A. Eigenvalues and eigenvectors can be selected by specifying either 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 triangle of A is stored;

   = 'L':  Lower triangle of A is stored.

• N (input)
  The order of the matrix A. N > = 0.

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

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

  On exit, A is overwritten by values generated during the reduction to tridiagonal form. If UPLO = 'U', the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows NDIAG and NDIAG+1 of A, and if UPLO = 'L', the diagonal and first subdiagonal of T are returned in the first two rows of A.

• LDA (input)
  The leading dimension of the array A. LDA > = NDIAG + 1.

• Q (output)
  If JOBZ = 'V', the N-by-N orthogonal matrix used in the reduction to tridiagonal form. If JOBZ = 'N', the array Q is not referenced.

• LDQ (input)
  The leading dimension of the array Q. If JOBZ = 'V', then 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.

• ABTOL (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

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

  where EPS is the machine precision. If ABTOL 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 ABTOL 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 ABTOL to 2*SLAMCH('S').

  See ``Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy,'' by Demmel and Kahan, LAPACK Working Note #3.

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

• W (output)
  The first NFOUND elements contain the selected eigenvalues in ascending order.

• Z (output)
  If JOBZ = 'V', then if INFO = 0, the first NFOUND 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). 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. If JOBZ = 'N', then Z is not referenced. Note: the user must ensure that at least max(1,NFOUND) columns are supplied in the array Z; if RANGE = 'V', the exact value of NFOUND 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(7*N)

• IWORK2 (workspace)
  
• IFAIL (output)
  If JOBZ = 'V', then if INFO = 0, the first NFOUND 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)
  

   = 0:  successful exit.

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

   > 0:  if INFO  = i, then i eigenvectors failed to converge.
  Their indices are stored in array IFAIL.