Purpose
=======
LA_SPGVX and LA_HPGVX compute selected eigenvalues and, optionally,
the corresponding eigenvectors of generalized eigenvalue problems of
the form
A*z = lambda*B*z, A*B*z = lambda*z, and B*A*z = lambda*z,
where A and B are real symmetric in the case of LA_SPGVX and complex
Hermitian in the case of LA_HPGVX. In both cases B is positive
definite. Eigenvalues and eigenvectors can be selected by specifying
either a range of values or a range of indices for the desired
eigenvalues. Matrices A and B are stored in a packed format.
=========
SUBROUTINE LA_SPGVX / LA_HPGVX( AP, BP, W, ITYPE= itype, &
UPLO= uplo, Z= z, VL= vl, VU= vu, IL= il, IU= iu, M= m, &
IFAIL= ifail, ABSTOL= abstol, INFO= info )
(), INTENT(INOUT) :: AP(:), BP(:)
REAL(), INTENT(OUT) :: W(:)
INTEGER, INTENT(IN), OPTIONAL :: ITYPE
CHARACTER(LEN=1), INTENT(IN), OPTIONAL :: UPLO
(), INTENT(OUT), OPTIONAL :: Z(:,:)
REAL(), INTENT(IN), OPTIONAL :: VL, VU
INTEGER, INTENT(IN), OPTIONAL :: IL, IU
INTEGER, INTENT(OUT), OPTIONAL :: M
INTEGER, INTENT(OUT), OPTIONAL :: IFAIL(:)
REAL(), INTENT(IN), OPTIONAL :: ABSTOL
INTEGER, INTENT(OUT), OPTIONAL :: INFO
where
::= REAL | COMPLEX
::= KIND(1.0) | KIND(1.0D0)
Arguments
=========
AP (input/output) REAL or COMPLEX array, shape (:) with
size(AP) = n*(n + 1)/2, where n is the order of A and B.
On entry, the upper or lower triangle of matrix A in packed
storage. The elements are stored columnwise as follows:
if UPLO = 'U', AP(i +(j-1)*j/2) = A(i,j) for 1<=i<=j<=n;
if UPLO = 'L', AP(i +(j-1)*(2*n-j)/2) = A(i,j) for 1<=j<=i<=n.
On exit, the contents of AP are destroyed.
BP (input/output) REAL or COMPLEX array, shape (:) with
size(BP) = size(AP).
On entry, the upper or lower triangle of matrix B in packed
storage. The elements are stored columnwise as follows:
if UPLO = 'U', BP(i +(j-1)*j/2) = B(i,j) for 1<=i<=j<=n;
if UPLO = 'L', BP(i +(j-1)*(2*n-j)/2) = B(i,j) for 1<=j<=i<=n.
On exit, the triangular factor U or L of the Cholesky
factorization B = U^T*U or B = L*L^T, in the same storage
format as B.
W (output) REAL array, shape (:) with size(W) = n.
The eigenvalues in ascending order.
ITYPE Optional (input) INTEGER.
Specifies the problem type to be solved:
= 1: A*z = lambda*B*z
= 2: A*B*z = lambda*z
= 3: B*A*z = lambda*z
Default value: 1.
UPLO Optional (input) CHARACTER(LEN=1).
= 'U': Upper triangles of A and B are stored;
= 'L': Lower triangles of A and B are stored.
Default value: 'U'.
Z Optional (output) REAL or COMPLEX rectangular array, shape
(:,:) with size(Z,1) = n and size(Z,2) = M.
The first M columns of Z contain the orthonormal eigenvectors
corresponding to the selected eigenvalues, with the i-th
column of Z holding the eigenvector associated with the
eigenvalue in W(i). The eigenvectors are normalized as
follows:
if ITYPE = 1 or 2: Z^H * B * Z = I ,
if ITYPE = 3: Z^H * B^-1 * 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.
VL,VU Optional (input) REAL.
The lower and upper bounds of the interval to be searched for
eigenvalues. VL < VU.
Default values: VL = -HUGE() and VU = HUGE(), where
::= KIND(1.0) | KIND(1.0D0).
Note: Neither VL nor VU may be present if IL and/or IU is
present.
IL,IU Optional (input) INTEGER.
The indices of the smallest and largest eigenvalues to be
returned. The IL-th through IU-th eigenvalues will be found.
1 <= IL <= IU <= size(A,1).
Default values: IL = 1 and IU = size(A,1).
Note: Neither IL nor IU may be present if VL and/or VU is
present.
Note: All eigenvalues are calculated if none of the arguments
VL, VU, IL and IU are present.
M Optional (output) INTEGER.
The total number of eigenvalues found. 0 <= M <= size(A,1).
Note: If IL and IU are present then M = IU - IL + 1.
IFAIL Optional (output) INTEGER array, shape (:) with size(IFAIL) =
size(A,1).
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.
Note: If Z is present then IFAIL should also be present.
ABSTOL Optional (input) 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 + EPSILON(1.0_) * max(|a|,|b|),
where is the working precision. If ABSTOL <= 0, then
EPSILON(1.0_) * ||T||1 will be used in its place, where
||T||1 is the l1 norm of the tridiagonal matrix obtained by
reducing the generalized eigenvalue problem to tridiagonal
form. Eigenvalues will be computed most accurately when
ABSTOL is set to twice the underflow threshold
2 * LA_LAMCH(1.0_, 'S'), not zero.
Default value: 0.0_.
Note: If this routine returns with 0 < INFO <= n, then some
eigenvectors did not converge.
Try setting ABSTOL to 2 * LA_LAMCH(1.0_, 'S').
INFO Optional (output) INTEGER.
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value
> 0: the algorithm failed to converge or matrix B is not
positive definite:
<= n: the algorithm failed to converge; if INFO = i, then
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.
If INFO is not present and an error occurs, then the program
is terminated with an error message.