Purpose
=======
LA_SBGVX and LA_HBGVX compute selected eigenvalues and, optionally,
the corresponding eigenvectors of the generalized eigenvalue problem
A*z = lambda*B*z,
where A and B are real symmetric in the case of LA_SBGVX and complex
Hermitian in the case of LA_HBGVX. In both cases B is positive
definite. Matrices A and B are stored in a band format. Eigenvalues
and eigenvectors can be selected by specifying either a range of
values or a range of indices for the desired eigenvalues.
=========
SUBROUTINE LA_SBGVX / LA_HBGVX( AB, BB, W, UPLO=uplo, Z=z, &
VL=vl, VU=vu, IL=il, IU=iu, M=m, IFAIL=ifail, Q=q, &
ABSTOL=abstol, INFO=info )
(), INTENT(INOUT) :: AB(:,:), BB(:,:)
REAL(), INTENT(OUT) :: W(:)
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(:)
(), INTENT(OUT), OPTIONAL :: Q(:,:)
REAL(), INTENT(IN), OPTIONAL :: ABSTOL
INTEGER, INTENT(OUT), OPTIONAL :: INFO
where
::= REAL | COMPLEX
::= KIND(1.0) | KIND(1.0D0)
Arguments
=========
AB (input/output) REAL or COMPLEX array, shape (:,:) with
size(AB,1) = ka + 1 and size(AB,2) = n, where ka is the number
of subdiagonals or superdiagonals in the band of A and n is
the order of A and B.
On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
triangle of A in band storage. The ka + 1 diagonals of A are
stored in the rows of AB so that the j-th column of A is
stored in the j-th column of AB as follows:
if UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for max(1,j-ka)<=i<=j,
1<=j<=n
if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+ka),
1<=j<=n.
On exit, the contents of AB are destroyed.
BB (input/output) REAL or COMPLEX array, shape (:,:) with
size(BB,1) = kb + 1 and size(BB,2) = n, where kb is the number
of subdiagonals or superdiagonals in the band of B.
On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
triangle of matrix B in band storage. The kb + 1 diagonals of
B are stored in the rows of BB so that the j-th column of B
is stored in the j-th column of BB as follows:
if UPLO = 'U', BB(kb+1+i-j,j) = B(i,j) for max(1,j-kb)<=i<=j,
1<=j<=n
if UPLO = 'L', BB(1+i-j,j) = B(i,j) for j<=i<=min(n,j+kb),
1<=j<=n.
On exit, the factor S from the split Cholesky factorization
B = S^H*S.
W (output) REAL array, shape (:) with size(W) = n.
The first M elements contain the selected eigenvalues in
ascending order.
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 square array, shape (:,:)
with size(Z,1) = n.
The first M columns of Z contain the orthonormal eigenvectors
corresponding to the selected eigenvalues, with the i-th
column of Z containing the eigenvector associated with the
eigenvalue in W(i). The eigenvectors are normalized so that
Z^H*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.
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)=n.
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.
Q Optional, (Output) REAL or COMPLEX square array, shape(:,:)
with size(Q,1) = n.
If Z is present, the matrix used in the reduction of
A*z = lambda*B*z to tridiagonal form.
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.