chpgvx.f - Man Page

subroutine chpgvx (ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
CHPGVX

CHPGVX  

Purpose:

 CHPGVX computes selected eigenvalues and, optionally, 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.  Eigenvalues and eigenvectors can be selected by
 specifying either a range of values or a range of indices for the
 desired eigenvalues.

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.

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.

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.

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)
          On normal exit, the first M elements contain the selected
          eigenvalues in ascending order.

Z

          Z is COMPLEX array, dimension (LDZ, N)
          If JOBZ = 'N', then Z is not referenced.
          If JOBZ = 'V', then if INFO = 0, the first M columns of Z
          contain the orthonormal eigenvectors of the matrix A
          corresponding to the selected eigenvalues, with the i-th
          column of Z holding the eigenvector associated with W(i).
          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 an eigenvector fails to converge, then that column of Z
          contains the latest approximation to the eigenvector, and the
          index of the eigenvector is returned in IFAIL.
          Note: the user must ensure that at least max(1,M) columns are
          supplied in the array Z; if RANGE = 'V', the exact value of M
          is not known in advance and an upper bound must be used.

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 (2*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:  CPPTRF or CHPEVX returned an error code:
             <= N:  if INFO = i, CHPEVX failed to converge;
                    i eigenvectors failed to converge.  Their indices
                    are stored in array IFAIL.
             > N:   if INFO = N + i, for 1 <= i <= n, then the leading
                    minor of order i of 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.

Date:

June 2016

Contributors:

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

Definition at line 279 of file chpgvx.f.