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 zhbgvx.f(3) LAPACK zhbgvx.f(3)

zhbgvx.f -

# SYNOPSIS

## Functions/Subroutines

subroutine zhbgvx (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)

ZHBGST

# Function/Subroutine Documentation

## subroutine zhbgvx (characterJOBZ, characterRANGE, characterUPLO, integerN, integerKA, integerKB, complex*16, dimension( ldab, * )AB, integerLDAB, complex*16, dimension( ldbb, * )BB, integerLDBB, complex*16, dimension( ldq, * )Q, integerLDQ, double precisionVL, double precisionVU, integerIL, integerIU, double precisionABSTOL, integerM, double precision, dimension( * )W, complex*16, dimension( ldz, * )Z, integerLDZ, complex*16, dimension( * )WORK, double precision, dimension( * )RWORK, integer, dimension( * )IWORK, integer, dimension( * )IFAIL, integerINFO)

ZHBGST
Purpose:
``` ZHBGVX 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.
```
Parameters:
JOBZ
```          JOBZ is CHARACTER*1
= 'N':  Compute eigenvalues only;
= 'V':  Compute eigenvalues and eigenvectors.
```
RANGE
```          RANGE is CHARACTER*1
= '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
```          UPLO is CHARACTER*1
= 'U':  Upper triangles of A and B are stored;
= 'L':  Lower triangles of A and B are stored.
```
N
```          N is INTEGER
The order of the matrices A and B.  N >= 0.
```
KA
```          KA is INTEGER
The number of superdiagonals of the matrix A if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KA >= 0.
```
KB
```          KB is INTEGER
The number of superdiagonals of the matrix B if UPLO = 'U',
or the number of subdiagonals if UPLO = 'L'. KB >= 0.
```
AB
```          AB is COMPLEX*16 array, dimension (LDAB, N)
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
```          LDAB is INTEGER
The leading dimension of the array AB.  LDAB >= KA+1.
```
BB
```          BB is COMPLEX*16 array, dimension (LDBB, N)
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 ZPBSTF.
```
LDBB
```          LDBB is INTEGER
The leading dimension of the array BB.  LDBB >= KB+1.
```
Q
```          Q is COMPLEX*16 array, dimension (LDQ, N)
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
```          LDQ is INTEGER
The leading dimension of the array Q.  If JOBZ = 'N',
LDQ >= 1. If JOBZ = 'V', LDQ >= max(1,N).
```
VL
```          VL is DOUBLE PRECISION
```
VU
```          VU is DOUBLE PRECISION

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
```          IL is INTEGER
```
IU
```          IU is INTEGER

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
```          ABSTOL is DOUBLE PRECISION
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*DLAMCH('S'), not zero.
If this routine returns with INFO>0, indicating that some
eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH('S').
```
M
```          M is INTEGER
The total number of eigenvalues found.  0 <= M <= N.
If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
```
W
```          W is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.
```
Z
```          Z is COMPLEX*16 array, dimension (LDZ, N)
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
```          LDZ is INTEGER
The leading dimension of the array Z.  LDZ >= 1, and if
JOBZ = 'V', LDZ >= N.
```
WORK
```          WORK is COMPLEX*16 array, dimension (N)
```
RWORK
```          RWORK is DOUBLE PRECISION array, dimension (7*N)
```
IWORK
```          IWORK is INTEGER array, dimension (5*N)
```
IFAIL
```          IFAIL is INTEGER array, dimension (N)
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
```          INFO is INTEGER
= 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 ZPBSTF
returned INFO = i: B is not positive definite.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.
```
Author:
Univ. of Tennessee
Univ. of California Berkeley
NAG Ltd.
Date:
November 2011
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
Definition at line 290 of file zhbgvx.f.

# Author

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