hpgvd(3)

NAME
hpgvd - {hp,sp}gvd: eig, divide and conquer

SYNOPSIS

  Functions
subroutine chpgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)
CHPGVD
subroutine dspgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, iwork, liwork, info)
DSPGVD
subroutine sspgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, iwork, liwork, info)
SSPGVD
subroutine zhpgvd (itype, jobz, uplo, n, ap, bp, w, z, ldz, work, lwork, rwork, lrwork, iwork, liwork, info)
ZHPGVD

Detailed Description

Function Documentation

  subroutine chpgvd (integer itype, character jobz, character uplo, integer n, complex, dimension( * ) ap, complex, dimension( * ) bp, real, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, integer lwork, real, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

CHPGVD

Purpose:

     CHPGVD computes all the eigenvalues and, optionally, the eigenvectors
     of a complex generalized Hermitian-definite eigenproblem, of the form
     A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
     B are assumed to be Hermitian, stored in packed format, and B is also
     positive definite.
     If eigenvectors are desired, it uses a divide and conquer algorithm.

Parameters
    ITYPE

              ITYPE is INTEGER
              Specifies the problem type to be solved:
              = 1:  A*x = (lambda)*B*x
              = 2:  A*B*x = (lambda)*x
              = 3:  B*A*x = (lambda)*x

    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.

    AP

              AP is COMPLEX array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the Hermitian matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, the contents of AP are destroyed.

    BP

              BP is COMPLEX array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the Hermitian matrix
              B, packed columnwise in a linear array.  The j-th column of B
              is stored in the array BP as follows:
              if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
              if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
              On exit, the triangular factor U or L from the Cholesky
              factorization B = U**H*U or B = L*L**H, in the same storage
              format as B.

    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.  The eigenvectors are normalized as follows:
              if ITYPE = 1 or 2, Z**H*B*Z = I;
              if ITYPE = 3, Z**H*inv(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 COMPLEX array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the required LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of array WORK.
              If N <= 1,               LWORK >= 1.
              If JOBZ = 'N' and N > 1, LWORK >= N.
              If JOBZ = 'V' and N > 1, LWORK >= 2*N.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the required sizes of the WORK, RWORK and
              IWORK arrays, returns these values as the first entries of
              the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    RWORK

              RWORK is REAL array, dimension (MAX(1,LRWORK))
              On exit, if INFO = 0, RWORK(1) returns the required LRWORK.

    LRWORK

              LRWORK is INTEGER
              The dimension of array RWORK.
              If N <= 1,               LRWORK >= 1.
              If JOBZ = 'N' and N > 1, LRWORK >= N.
              If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
              If LRWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of array IWORK.
              If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
              If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  CPPTRF or CHPEVD returned an error code:
                 <= N:  if INFO = i, CHPEVD failed to converge;
                        i off-diagonal elements of an intermediate
                        tridiagonal form did not convergeto zero;
                 > N:   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 223 of file chpgvd.f.

  subroutine dspgvd (integer itype, character jobz, character uplo, integer n, double precision, dimension( * ) ap, double precision, dimension( * ) bp, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

DSPGVD

Purpose:

     DSPGVD computes all the eigenvalues, and optionally, the eigenvectors
     of a real generalized symmetric-definite eigenproblem, of the form
     A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
     B are assumed to be symmetric, stored in packed format, and B is also
     positive definite.
     If eigenvectors are desired, it uses a divide and conquer algorithm.

Parameters
    ITYPE

              ITYPE is INTEGER
              Specifies the problem type to be solved:
              = 1:  A*x = (lambda)*B*x
              = 2:  A*B*x = (lambda)*x
              = 3:  B*A*x = (lambda)*x

    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.

    AP

              AP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, the contents of AP are destroyed.

    BP

              BP is DOUBLE PRECISION array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              B, packed columnwise in a linear array.  The j-th column of B
              is stored in the array BP as follows:
              if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
              if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
              On exit, the triangular factor U or L from the Cholesky
              factorization B = U**T*U or B = L*L**T, in the same storage
              format as B.

    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.  The eigenvectors are normalized as follows:
              if ITYPE = 1 or 2, Z**T*B*Z = I;
              if ITYPE = 3, Z**T*inv(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 (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the required LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N <= 1,               LWORK >= 1.
              If JOBZ = 'N' and N > 1, LWORK >= 2*N.
              If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the required sizes of the WORK and IWORK
              arrays, returns these values as the first entries of the WORK
              and IWORK arrays, and no error message related to LWORK or
              LIWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
              If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK and
              IWORK arrays, returns these values as the first entries of
              the WORK and IWORK arrays, and no error message related to
              LWORK or LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  DPPTRF or DSPEVD returned an error code:
                 <= N:  if INFO = i, DSPEVD 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 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 202 of file dspgvd.f.

  subroutine sspgvd (integer itype, character jobz, character uplo, integer n, real, dimension( * ) ap, real, dimension( * ) bp, real, dimension( * ) w, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer liwork, integer info)

SSPGVD

Purpose:

     SSPGVD computes all the eigenvalues, and optionally, the eigenvectors
     of a real generalized symmetric-definite eigenproblem, of the form
     A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
     B are assumed to be symmetric, stored in packed format, and B is also
     positive definite.
     If eigenvectors are desired, it uses a divide and conquer algorithm.

Parameters
    ITYPE

              ITYPE is INTEGER
              Specifies the problem type to be solved:
              = 1:  A*x = (lambda)*B*x
              = 2:  A*B*x = (lambda)*x
              = 3:  B*A*x = (lambda)*x

    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.

    AP

              AP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, the contents of AP are destroyed.

    BP

              BP is REAL array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the symmetric matrix
              B, packed columnwise in a linear array.  The j-th column of B
              is stored in the array BP as follows:
              if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
              if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
              On exit, the triangular factor U or L from the Cholesky
              factorization B = U**T*U or B = L*L**T, in the same storage
              format as B.

    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.  The eigenvectors are normalized as follows:
              if ITYPE = 1 or 2, Z**T*B*Z = I;
              if ITYPE = 3, Z**T*inv(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 (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the required LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N <= 1,               LWORK >= 1.
              If JOBZ = 'N' and N > 1, LWORK >= 2*N.
              If JOBZ = 'V' and N > 1, LWORK >= 1 + 6*N + 2*N**2.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the required sizes of the WORK and IWORK
              arrays, returns these values as the first entries of the WORK
              and IWORK arrays, and no error message related to LWORK or
              LIWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of the array IWORK.
              If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
              If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK and
              IWORK arrays, returns these values as the first entries of
              the WORK and IWORK arrays, and no error message related to
              LWORK or LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  SPPTRF or SSPEVD returned an error code:
                 <= N:  if INFO = i, SSPEVD 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 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 202 of file sspgvd.f.

  subroutine zhpgvd (integer itype, character jobz, character uplo, integer n, complex*16, dimension( * ) ap, complex*16, dimension( * ) bp, double precision, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, double precision, dimension( * ) rwork, integer lrwork, integer, dimension( * ) iwork, integer liwork, integer info)

ZHPGVD

Purpose:

     ZHPGVD computes all the eigenvalues and, optionally, the eigenvectors
     of a complex generalized Hermitian-definite eigenproblem, of the form
     A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and
     B are assumed to be Hermitian, stored in packed format, and B is also
     positive definite.
     If eigenvectors are desired, it uses a divide and conquer algorithm.

Parameters
    ITYPE

              ITYPE is INTEGER
              Specifies the problem type to be solved:
              = 1:  A*x = (lambda)*B*x
              = 2:  A*B*x = (lambda)*x
              = 3:  B*A*x = (lambda)*x

    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.

    AP

              AP is COMPLEX*16 array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the Hermitian matrix
              A, packed columnwise in a linear array.  The j-th column of A
              is stored in the array AP as follows:
              if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
              if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n.
              On exit, the contents of AP are destroyed.

    BP

              BP is COMPLEX*16 array, dimension (N*(N+1)/2)
              On entry, the upper or lower triangle of the Hermitian matrix
              B, packed columnwise in a linear array.  The j-th column of B
              is stored in the array BP as follows:
              if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j;
              if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n.
              On exit, the triangular factor U or L from the Cholesky
              factorization B = U**H*U or B = L*L**H, in the same storage
              format as B.

    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.  The eigenvectors are normalized as follows:
              if ITYPE = 1 or 2, Z**H*B*Z = I;
              if ITYPE = 3, Z**H*inv(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 COMPLEX*16 array, dimension (MAX(1,LWORK))
              On exit, if INFO = 0, WORK(1) returns the required LWORK.

    LWORK

              LWORK is INTEGER
              The dimension of the array WORK.
              If N <= 1,               LWORK >= 1.
              If JOBZ = 'N' and N > 1, LWORK >= N.
              If JOBZ = 'V' and N > 1, LWORK >= 2*N.
              If LWORK = -1, then a workspace query is assumed; the routine
              only calculates the required sizes of the WORK, RWORK and
              IWORK arrays, returns these values as the first entries of
              the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    RWORK

              RWORK is DOUBLE PRECISION array, dimension (MAX(1,LRWORK))
              On exit, if INFO = 0, RWORK(1) returns the required LRWORK.

    LRWORK

              LRWORK is INTEGER
              The dimension of array RWORK.
              If N <= 1,               LRWORK >= 1.
              If JOBZ = 'N' and N > 1, LRWORK >= N.
              If JOBZ = 'V' and N > 1, LRWORK >= 1 + 5*N + 2*N**2.
              If LRWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    IWORK

              IWORK is INTEGER array, dimension (MAX(1,LIWORK))
              On exit, if INFO = 0, IWORK(1) returns the required LIWORK.

    LIWORK

              LIWORK is INTEGER
              The dimension of array IWORK.
              If JOBZ  = 'N' or N <= 1, LIWORK >= 1.
              If JOBZ  = 'V' and N > 1, LIWORK >= 3 + 5*N.
              If LIWORK = -1, then a workspace query is assumed; the
              routine only calculates the required sizes of the WORK, RWORK
              and IWORK arrays, returns these values as the first entries
              of the WORK, RWORK and IWORK arrays, and no error message
              related to LWORK or LRWORK or LIWORK is issued by XERBLA.

    INFO

              INFO is INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal value
              > 0:  ZPPTRF or ZHPEVD returned an error code:
                 <= N:  if INFO = i, ZHPEVD failed to converge;
                        i off-diagonal elements of an intermediate
                        tridiagonal form did not convergeto zero;
                 > N:   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 223 of file zhpgvd.f.

Author

Generated automatically by Doxygen for LAPACK from the source code.