sspgvx(3P) Sun Performance Library sspgvx(3P)NAMEsspgvx - 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 SSPGVX(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(*)
REAL VL, VU, ABSTOL
REAL AP(*), BP(*), W(*), Z(LDZ,*), WORK(*)
SUBROUTINE SSPGVX_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(*)
REAL VL, VU, ABSTOL
REAL 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 :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: AP, BP, W, WORK
REAL, 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 :: VL, VU, ABSTOL
REAL, DIMENSION(:) :: AP, BP, W, WORK
REAL, DIMENSION(:,:) :: Z
C INTERFACE
#include <sunperf.h>
void sspgvx(int itype, char jobz, char range, char uplo, int n, float
*ap, float *bp, float vl, float vu, int il, int iu, float
abstol, int *m, float *w, float *z, int ldz, int *ifail, int
*info);
void sspgvx_64(long itype, char jobz, char range, char uplo, long n,
float *ap, float *bp, float vl, float vu, long il, long iu,
float abstol, long *m, float *w, float *z, long ldz, long
*ifail, long *info);
PURPOSEsspgvx 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)
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 (input/output)
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 fac‐
torization B = U**T*U or B = L*L**T, in the same storage for‐
mat 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 small‐
est 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 approx‐
imate eigenvalue is accepted as converged when it is deter‐
mined 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)
Real array, dimension (N) On normal exit, the first M ele‐
ments contain the selected eigenvalues in ascending order.
Z (output)
Real array, dimension (LDZ, max(1,M)) 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)
Real array, dimension(8*N)
IWORK (workspace)
INTEGER array, dimension(5*N)
IFAIL (output)
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 (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 factor‐
ization of B could not be completed and no eigenvalues or
eigenvectors were computed.
FURTHER DETAILS
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA
6 Mar 2009 sspgvx(3P)
