NAME

dspgvx - compute selected eigenvalues, and optionally, 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

SYNOPSIS

  SUBROUTINE DSPGVX( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL, 
 *      IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  INTEGER ITYPE, N, IL, IU, M, LDZ, INFO
  INTEGER IWORK(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABSTOL
  DOUBLE PRECISION AP(*), BP(*), W(*), Z(LDZ,*), WORK(*)

  SUBROUTINE DSPGVX_64( ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, 
 *      IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
  CHARACTER * 1 JOBZ, RANGE, UPLO
  INTEGER*8 ITYPE, N, IL, IU, M, LDZ, INFO
  INTEGER*8 IWORK(*), IFAIL(*)
  DOUBLE PRECISION VL, VU, ABSTOL
  DOUBLE PRECISION AP(*), BP(*), W(*), Z(LDZ,*), WORK(*)

F95 INTERFACE

  SUBROUTINE SPGVX( ITYPE, JOBZ, RANGE, UPLO, [N], AP, BP, VL, VU, IL, 
 *       IU, ABSTOL, M, W, Z, [LDZ], [WORK], [IWORK], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  INTEGER :: ITYPE, N, IL, IU, M, LDZ, INFO
  INTEGER, DIMENSION(:) :: IWORK, IFAIL
  REAL(8) :: VL, VU, ABSTOL
  REAL(8), DIMENSION(:) :: AP, BP, W, WORK
  REAL(8), DIMENSION(:,:) :: Z

  SUBROUTINE SPGVX_64( ITYPE, JOBZ, RANGE, UPLO, [N], AP, BP, VL, VU, 
 *       IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], [IWORK], IFAIL, [INFO])
  CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
  INTEGER(8) :: ITYPE, N, IL, IU, M, LDZ, INFO
  INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
  REAL(8) :: VL, VU, ABSTOL
  REAL(8), DIMENSION(:) :: AP, BP, W, WORK
  REAL(8), DIMENSION(:,:) :: Z

void dspgvx(int itype, char jobz, char range, char uplo, int n, double *ap, double *bp, double vl, double vu, int il, int iu, double abstol, int *m, double *w, double *z, int ldz, int *ifail, int *info);

void dspgvx_64(long itype, char jobz, char range, char uplo, long n, double *ap, double *bp, double vl, double vu, long il, long iu, double abstol, long *m, double *w, double *z, long ldz, long *ifail, long *info);

PURPOSE

dspgvx computes selected eigenvalues, and optionally, 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 storage, 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.

ARGUMENTS

ITYPE (input)
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 (input)

 = 'N':  Compute eigenvalues only;

 = 'V':  Compute eigenvalues and eigenvectors.

RANGE (input)

 = '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 (input)

 = 'U':  Upper triangle of A and B are stored;

 = 'L':  Lower triangle of A and B are stored.

N (input)
The order of the matrix pencil (A,B). N > = 0.
AP (input/output)
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 (input/output)
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.
VL (input)
If RANGE ='V', the lower and upper bounds of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.
VU (input)
See the description of VL.
IL (input)
If RANGE ='I', the indices (in ascending order) of the smallest and largest eigenvalues 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 (input)
See the description of IL.
ABSTOL (input)
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 (output)
The total number of eigenvalues found. 0 < = M < = N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
W (output)
On normal exit, the first M elements contain the selected eigenvalues in ascending order.
Z (output)
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**T*B*Z = I; if ITYPE = 3, Z**T*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 (input)
The leading dimension of the array Z. LDZ > = 1, and if JOBZ = 'V', LDZ > = max(1,N).
WORK (workspace)
dimension(8*N)
IWORK (workspace)
dimension(5*N)
IFAIL (output)
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 (output)

 = 0:  successful exit

 < 0:  if INFO  = -i, the i-th argument had an illegal value

 > 0:  SPPTRF or SSPEVX returned an error code:

 < = N:  if INFO  = i, SSPEVX 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.

FURTHER DETAILS

Based on contributions by