Contents


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, eigen-
     vectors  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 posi-
     tive 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  sym-
               metric  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  sym-
               metric  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 fac-
               torization 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  tri-
               diagonal  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 tri-
               diagonal 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 (input) 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/output)
               dimension(5*N)

     IFAIL (input)
               If JOBZ = 'V', then if INFO = 0, the first M  ele-
               ments  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 ille-
               gal 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  fac-
               torization  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