NAME

chbgvx - compute all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x

SYNOPSIS

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

  SUBROUTINE CHBGVX_64( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, 
 *      LDBB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, 
 *      IWORK, IFAIL, INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  COMPLEX AB(LDAB,*), BB(LDBB,*), Q(LDQ,*), Z(LDZ,*), WORK(*)
  INTEGER*8 N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
  INTEGER*8 IWORK(*), IFAIL(*)
  REAL VL, VU, ABSTOL
  REAL W(*), RWORK(*)

F95 INTERFACE

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

  SUBROUTINE HBGVX_64( JOBZ, RANGE, UPLO, [N], KA, KB, AB, [LDAB], BB, 
 *       [LDBB], Q, [LDQ], VL, VU, IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], 
 *       [RWORK], [IWORK], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  COMPLEX, DIMENSION(:) :: WORK
  COMPLEX, DIMENSION(:,:) :: AB, BB, Q, Z
  INTEGER(8) :: N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
  INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
  REAL :: VL, VU, ABSTOL
  REAL, DIMENSION(:) :: W, RWORK

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

void chbgvx_64(char jobz, char range, char uplo, long n, long ka, long kb, complex *ab, long ldab, complex *bb, long ldbb, complex *q, long ldq, float vl, float vu, long il, long iu, float abstol, long *m, float *w, complex *z, long ldz, long *ifail, long *info);

PURPOSE

chbgvx computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized Hermitian-definite banded eigenproblem, of the form A*x=(lambda)*B*x. Here A and B are assumed to be Hermitian 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 Hermitian 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 Hermitian 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(kb+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**H*S, as returned by CPBSTF.
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)
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)
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)
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)
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 AP 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 that Z**H*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 > = N.
WORK (workspace)
dimension(N)
RWORK (workspace)
dimension(7*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)

 = 0:  successful exit

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

 > 0:  if INFO  = i, and i is:

 < = N:  then i eigenvectors failed to converge.  Their
indices are stored in array IFAIL.
 > N:   if INFO  = N + i, for 1  < = i  < = N, then CPBSTF

returned INFO = i: 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