Man page - hbgvx(3)

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Manual

hbgvx

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine chbgvx (character jobz, character range, character uplo, integern, integer ka, integer kb, complex, dimension( ldab, * ) ab, integerldab, complex, dimension( ldbb, * ) bb, integer ldbb, complex,dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il,integer iu, real abstol, integer m, real, dimension( * ) w, complex,dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, real,dimension( * ) rwork, integer, dimension( * ) iwork, integer,dimension( * ) ifail, integer info)
subroutine dsbgvx (character jobz, character range, character uplo, integern, integer ka, integer kb, double precision, dimension( ldab, * ) ab,integer ldab, double precision, dimension( ldbb, * ) bb, integer ldbb,double precision, dimension( ldq, * ) q, integer ldq, double precisionvl, double precision vu, integer il, integer iu, double precisionabstol, integer m, double precision, dimension( * ) w, doubleprecision, dimension( ldz, * ) z, integer ldz, double precision,dimension( * ) work, integer, dimension( * ) iwork, integer, dimension(* ) ifail, integer info)
subroutine ssbgvx (character jobz, character range, character uplo, integern, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab,real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( ldq, * )q, integer ldq, real vl, real vu, integer il, integer iu, real abstol,integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integerldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer,dimension( * ) ifail, integer info)
subroutine zhbgvx (character jobz, character range, character uplo, integern, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integerldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, complex*16,dimension( ldq, * ) q, integer ldq, double precision vl, doubleprecision vu, integer il, integer iu, double precision abstol, integerm, double precision, dimension( * ) w, complex*16, dimension( ldz, * )z, integer ldz, complex*16, dimension( * ) work, double precision,dimension( * ) rwork, integer, dimension( * ) iwork, integer,dimension( * ) ifail, integer info)
Author

NAME

hbgvx - {hb,sb}gvx: eig, bisection

SYNOPSIS

Functions

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)
CHBGVX
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)
DSBGVX
subroutine ssbgvx (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)
SSBGVX
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)
ZHBGVX

Detailed Description

Function Documentation

subroutine chbgvx (character jobz, character range, character uplo, integern, integer ka, integer kb, complex, dimension( ldab, * ) ab, integerldab, complex, dimension( ldbb, * ) bb, integer ldbb, complex,dimension( ldq, * ) q, integer ldq, real vl, real vu, integer il,integer iu, real abstol, integer m, real, dimension( * ) w, complex,dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, real,dimension( * ) rwork, integer, dimension( * ) iwork, integer,dimension( * ) ifail, integer info)

CHBGVX

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.

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 is INTEGER
The order of the matrices A and B. N >= 0.

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 is INTEGER
The number of superdiagonals of the matrix B if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KB >= 0.

AB is COMPLEX 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 is COMPLEX 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 CPBSTF.

LDBB

LDBB is INTEGER
The leading dimension of the array BB. LDBB >= KB+1.

Q is COMPLEX 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 is REAL

If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

VU is REAL

If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

IL is INTEGER

If RANGE=’I’, the index of the
smallest eigenvalue 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 is INTEGER

If RANGE=’I’, the index of the
largest eigenvalue 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 REAL
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 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 is REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

Z is COMPLEX 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 array, dimension (N)

RWORK

RWORK is REAL 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 CPBSTF
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

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

subroutine dsbgvx (character jobz, character range, character uplo, integern, integer ka, integer kb, double precision, dimension( ldab, * ) ab,integer ldab, double precision, dimension( ldbb, * ) bb, integer ldbb,double precision, dimension( ldq, * ) q, integer ldq, double precisionvl, double precision vu, integer il, integer iu, double precisionabstol, integer m, double precision, dimension( * ) w, doubleprecision, dimension( ldz, * ) z, integer ldz, double precision,dimension( * ) work, integer, dimension( * ) iwork, integer, dimension(* ) ifail, integer info)

DSBGVX

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.

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 is INTEGER
The order of the matrices A and B. N >= 0.

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 is INTEGER
The number of superdiagonals of the matrix B if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KB >= 0.

AB is DOUBLE PRECISION array, dimension (LDAB, N)
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

LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KA+1.

BB is DOUBLE PRECISION array, dimension (LDBB, N)
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 DPBSTF.

LDBB

LDBB is INTEGER
The leading dimension of the array BB. LDBB >= KB+1.

Q is DOUBLE PRECISION 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 is DOUBLE PRECISION

If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

VU is DOUBLE PRECISION

If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

IL is INTEGER

If RANGE=’I’, the index of the
smallest eigenvalue 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 is INTEGER

If RANGE=’I’, the index of the
largest eigenvalue 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| ) ,

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 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 is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

Z is DOUBLE PRECISION 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 Z**T*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 >= max(1,N).

WORK

WORK is DOUBLE PRECISION array, dimension (7*N)

IWORK

IWORK is INTEGER array, dimension (5*N)

IFAIL

IFAIL is INTEGER array, dimension (M)
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

INFO is INTEGER
= 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: DPBSTF returned an error code; i.e.,
if INFO = N + i, for 1 <= i <= N, then the leading
principal minor of order i of B is not positive.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

subroutine ssbgvx (character jobz, character range, character uplo, integern, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab,real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( ldq, * )q, integer ldq, real vl, real vu, integer il, integer iu, real abstol,integer m, real, dimension( * ) w, real, dimension( ldz, * ) z, integerldz, real, dimension( * ) work, integer, dimension( * ) iwork, integer,dimension( * ) ifail, integer info)

SSBGVX

Purpose:

SSBGVX 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.

Parameters

JOBZ

JOBZ is CHARACTER*1
= ’N’: Compute eigenvalues only;
= ’V’: Compute eigenvalues and eigenvectors.

RANGE

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangles of A and B are stored;
= ’L’: Lower triangles of A and B are stored.

N is INTEGER
The order of the matrices A and B. N >= 0.

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 is INTEGER
The number of superdiagonals of the matrix B if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KB >= 0.

AB is REAL array, dimension (LDAB, N)
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

LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KA+1.

BB is REAL array, dimension (LDBB, N)
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

LDBB is INTEGER
The leading dimension of the array BB. LDBB >= KB+1.

Q is REAL 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 is REAL

If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

VU is REAL

If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

IL is INTEGER

If RANGE=’I’, the index of the
smallest eigenvalue 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 is INTEGER

If RANGE=’I’, the index of the
largest eigenvalue 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 REAL
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| ) ,

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 is REAL array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

Z is REAL 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 Z**T*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 >= max(1,N).

WORK

WORK is REAL array, dimension (7*N)

IWORK

IWORK is INTEGER array, dimension (5*N)

IFAIL

INFO

INFO is INTEGER
= 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
principal minor of order i of B is not positive.
The factorization of B could not be completed and
no eigenvalues or eigenvectors were computed.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

subroutine zhbgvx (character jobz, character range, character uplo, integern, integer ka, integer kb, complex16, dimension( ldab, ) ab, integerldab, complex16, dimension( ldbb, ) bb, integer ldbb, complex16,dimension( ldq, ) q, integer ldq, double precision vl, doubleprecision vu, integer il, integer iu, double precision abstol, integerm, double precision, dimension( * ) w, complex16, dimension( ldz, )z, integer ldz, complex16, dimension( ) work, double precision,dimension( * ) rwork, integer, dimension( * ) iwork, integer,dimension( * ) ifail, integer info)

ZHBGVX

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

UPLO

UPLO is CHARACTER*1
= ’U’: Upper triangles of A and B are stored;
= ’L’: Lower triangles of A and B are stored.

N is INTEGER
The order of the matrices A and B. N >= 0.

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 is INTEGER
The number of superdiagonals of the matrix B if UPLO = ’U’,
or the number of subdiagonals if UPLO = ’L’. KB >= 0.

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 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 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 is DOUBLE PRECISION

If RANGE=’V’, the lower bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

VU is DOUBLE PRECISION

If RANGE=’V’, the upper bound of the interval to
be searched for eigenvalues. VL < VU.
Not referenced if RANGE = ’A’ or ’I’.

IL is INTEGER

If RANGE=’I’, the index of the
smallest eigenvalue 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 is INTEGER

If RANGE=’I’, the index of the
largest eigenvalue 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 + EPS * max( |a|,|b| ) ,

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 is DOUBLE PRECISION array, dimension (N)
If INFO = 0, the eigenvalues in ascending order.

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

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

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Author

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