hbgv(3)

NAME
hbgv - {hb,sb}gv: eig, QR iteration

SYNOPSIS

  Functions

subroutine chbgv (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, rwork, info)
CHBGV
subroutine dsbgv (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, info)
DSBGV
subroutine ssbgv (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, info)
SSBGV
subroutine zhbgv (jobz, uplo, n, ka, kb, ab, ldab, bb, ldbb, w, z, ldz, work, rwork, info)
ZHBGV

Detailed Description

Function Documentation

  subroutine chbgv (character jobz, character uplo, integer n, integer ka, integer kb, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( ldbb, * ) bb, integer ldbb, real, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, real, dimension( * ) rwork, integer info)

CHBGV

Purpose:

    !>
    !> CHBGV 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.
    !>

Parameters
    JOBZ

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

    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.
    !>

    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 (3*N)
    !>

    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:  the algorithm failed to converge:
    !>                    i off-diagonal elements of an intermediate
    !>                    tridiagonal form did not converge to zero;
    !>             > 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.

  subroutine dsbgv (character jobz, 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( * ) w, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer info)

DSBGV

Purpose:

    !>
    !> DSBGV computes all the eigenvalues, and optionally, the 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.
    !>

Parameters
    JOBZ

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

    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(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**T*S, as returned by DPBSTF.
    !>

    LDBB

    !>          LDBB is INTEGER
    !>          The leading dimension of the array BB.  LDBB >= KB+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 that 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 >= N.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (3*N)
    !>

    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:  the algorithm failed to converge:
    !>                    i off-diagonal elements of an intermediate
    !>                    tridiagonal form did not converge to zero;
    !>             > N:   if INFO = N + i, for 1 <= i <= N, then DPBSTF
    !>                    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.

  subroutine ssbgv (character jobz, character uplo, integer n, integer ka, integer kb, real, dimension( ldab, * ) ab, integer ldab, real, dimension( ldbb, * ) bb, integer ldbb, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer info)

SSBGV

Purpose:

    !>
    !> SSBGV computes all the eigenvalues, and optionally, the 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.
    !>

Parameters
    JOBZ

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

    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(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**T*S, as returned by SPBSTF.
    !>

    LDBB

    !>          LDBB is INTEGER
    !>          The leading dimension of the array BB.  LDBB >= KB+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 that 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 >= N.
    !>

    WORK

    !>          WORK is REAL array, dimension (3*N)
    !>

    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:  the algorithm failed to converge:
    !>                    i off-diagonal elements of an intermediate
    !>                    tridiagonal form did not converge to zero;
    !>             > N:   if INFO = N + i, for 1 <= i <= N, then SPBSTF
    !>                    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.

  subroutine zhbgv (character jobz, character uplo, integer n, integer ka, integer kb, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( ldbb, * ) bb, integer ldbb, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, double precision, dimension( * ) rwork, integer info)

ZHBGV

Purpose:

    !>
    !> ZHBGV 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.
    !>

Parameters
    JOBZ

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

    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.
    !>

    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 (3*N)
    !>

    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:  the algorithm failed to converge:
    !>                    i off-diagonal elements of an intermediate
    !>                    tridiagonal form did not converge to zero;
    !>             > 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.

Author

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