hbgvx(3)

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, integer n, integer ka, integer kb, complex, dimension( ldab, * ) ab, integer ldab, 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

              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 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 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

              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

              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

              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

              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

              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

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

    Z

              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

Definition at line 297 of file chbgvx.f.

  subroutine dsbgvx (character jobz, character range, character uplo, integer n, 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 precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, double precision, 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

              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 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

              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

              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

              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

              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

              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

              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| ) ,
              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 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 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

Definition at line 291 of file dsbgvx.f.

  subroutine ssbgvx (character jobz, character range, character uplo, integer n, 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, integer ldz, 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

              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 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

              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

              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

              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

              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

              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

              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 A 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

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

    Z

              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

              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:  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

Definition at line 291 of file ssbgvx.f.

  subroutine zhbgvx (character jobz, character range, character uplo, integer n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, complex*16, dimension( ldq, * ) q, integer ldq, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, 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)

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

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

    VU

              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

              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

              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| ) ,
              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

    Univ. of Colorado Denver

    NAG Ltd.

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

Definition at line 297 of file zhbgvx.f.

Author

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