Purpose
=======
LA_SPEVX / LA_HPEVX compute selected eigenvalues and, optionally,
the corresponding eigenvectors of a real symmetric/complex hermitian
matrix A in packed storage. Eigenvalues and eigenvectors can be
selected by specifying either a range of values or a range of indices
for the desired eigenvalues.
=========
SUBROUTINE LA_SPEVX / LA_HPEVX( AP, W, UPLO=uplo, Z=z, &
VL=vl, VU=vu, IL=il, IU=iu, M=m, IFAIL=ifail, &
ABSTOL=abstol, INFO=info )
(), INTENT(INOUT) :: AP(:)
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(:)
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.
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, AP is overwritten by values generated during the
reduction of A to a tridiagonal matrix T . If UPLO = 'U', the
diagonal and first superdiagonal of T overwrite the correspond-
ing diagonals of A. If UPLO = 'L', the diagonal and first
subdiagonal of T overwrite the corresponding diagonals of A.
W (output) REAL array, shape (:) with size(W) = n.
The eigenvalues in ascending order.
UPLO Optional (input) CHARACTER(LEN=1).
= 'U': Upper triangle of A is stored;
= 'L': Lower triangle of A is stored.
Default value: 'U'.
Z Optional (output) REAL or COMPLEX array, shape (:,:) with
size(Z,1) = n and size(Z,2) = M.
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 containing the eigenvector associated with
the eigenvalue in W(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 M columns are supplied
in the array Z. When the exact value of M is not known in
advance, an upper bound must be used. In all cases M <= n.
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.
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 A to tridiagonal form. Eigenvalues will be computed
most accurately when ABSTOL is set to twice the underflow
threshold 2*LA_LAMCH(1.0_, 'Save minimum'), not zero.
Default value: 0.0_.
Note: If this routine returns with INFO > 0, then some
eigenvectors did not converge. Try setting ABSTOL to
2*LA_LAMCH(1.0_, 'Save minimum').
INFO Optional (output) INTEGER.
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, then i eigenvectors failed to converge. Their
indices are stored in array IFAIL.
If INFO is not present and an error occurs, then the program is
terminated with an error message.