Contents
zhpgvx - compute 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
SUBROUTINE ZHPGVX(ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL,
IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
DOUBLE COMPLEX AP(*), BP(*), Z(LDZ,*), WORK(*)
INTEGER ITYPE, N, IL, IU, M, LDZ, INFO
INTEGER IWORK(*), IFAIL(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION W(*), RWORK(*)
SUBROUTINE ZHPGVX_64(ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL,
IU, ABSTOL, M, W, Z, LDZ, WORK, RWORK, IWORK, IFAIL, INFO)
CHARACTER * 1 JOBZ, RANGE, UPLO
DOUBLE COMPLEX AP(*), BP(*), Z(LDZ,*), WORK(*)
INTEGER*8 ITYPE, N, IL, IU, M, LDZ, INFO
INTEGER*8 IWORK(*), IFAIL(*)
DOUBLE PRECISION VL, VU, ABSTOL
DOUBLE PRECISION W(*), RWORK(*)
F95 INTERFACE
SUBROUTINE HPGVX(ITYPE, JOBZ, RANGE, UPLO, [N], AP, BP, VL, VU, IL,
IU, ABSTOL, M, W, Z, [LDZ], [WORK], [RWORK], [IWORK], IFAIL,
[INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
COMPLEX(8), DIMENSION(:) :: AP, BP, WORK
COMPLEX(8), DIMENSION(:,:) :: Z
INTEGER :: ITYPE, N, IL, IU, M, LDZ, INFO
INTEGER, DIMENSION(:) :: IWORK, IFAIL
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: W, RWORK
SUBROUTINE HPGVX_64(ITYPE, JOBZ, RANGE, UPLO, [N], AP, BP, VL, VU,
IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], [RWORK], [IWORK], IFAIL,
[INFO])
CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
COMPLEX(8), DIMENSION(:) :: AP, BP, WORK
COMPLEX(8), DIMENSION(:,:) :: Z
INTEGER(8) :: ITYPE, N, IL, IU, M, LDZ, INFO
INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
REAL(8) :: VL, VU, ABSTOL
REAL(8), DIMENSION(:) :: W, RWORK
C INTERFACE
#include <sunperf.h>
void zhpgvx(int itype, char jobz, char range, char uplo, int
n, doublecomplex *ap, doublecomplex *bp, double
vl, double vu, int il, int iu, double abstol, int
*m, double *w, doublecomplex *z, int ldz, int
*ifail, int *info);
void zhpgvx_64(long itype, char jobz, char range, char uplo,
long n, doublecomplex *ap, doublecomplex *bp, dou-
ble vl, double vu, long il, long iu, double
abstol, long *m, double *w, doublecomplex *z, long
ldz, long *ifail, long *info);
zhpgvx computes selected eigenvalues and, optionally, eigen-
vectors of a complex generalized Hermitian-definite eigen-
problem, 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.
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 triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
N (input) The order of the matrices A and B. N >= 0.
AP (input/output) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Her-
mitian 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) COMPLEX*16 array, dimension (N*(N+1)/2)
On entry, the upper or lower triangle of the Her-
mitian 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 (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)
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'.
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)
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'.
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 AP to tri-
diagonal 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) DOUBLE PRECISION array, dimension (N)
On normal exit, the first M elements contain the
selected eigenvalues in ascending order.
Z (input) COMPLEX*16 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 asso-
ciated 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 (input)
The leading dimension of the array Z. LDZ >= 1,
and if JOBZ = 'V', LDZ >= max(1,N).
WORK (workspace)
COMPLEX*16 array, dimension(2*N)
RWORK (workspace)
DOUBLE PRECISION array, dimension(7*N)
IWORK (workspace)
INTEGER array, dimension(5*N)
IFAIL (output)
If JOBZ = 'V', then if INFO = 0, the first M ele-
ments 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 ille-
gal value
> 0: CPPTRF or ZHPEVX returned an error code:
<= N: if INFO = i, ZHPEVX 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 fac-
torization of B could not be completed and no
eigenvalues or eigenvectors were computed.
Based on contributions by
Mark Fahey, Department of Mathematics, Univ. of Kentucky,
USA