NAME

SSBEVX - compute selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A

SYNOPSIS

SUBROUTINE SSBEVX(
JOBZ, RANGE, UPLO, N, KD, AB, LDAB, Q, LDQ, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )

CHARACTER JOBZ, RANGE, UPLO

INTEGER IL, INFO, IU, KD, LDAB, LDQ, LDZ, M, N

REAL ABSTOL, VL, VU

INTEGER IFAIL( ∗ ), IWORK( ∗ )

REAL AB( LDAB, ∗ ), Q( LDQ, ∗ ), W( ∗ ), WORK( ∗ ), Z( LDZ, ∗ )

PURPOSE

SSBEVX computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A.  Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for the desired eigenvalues. 
 

ARGUMENTS

JOBZ    (input) CHARACTER∗1
= ’N’:  Compute eigenvalues only;
= ’V’:  Compute eigenvalues and eigenvectors.

RANGE   (input) CHARACTER∗1
= ’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) CHARACTER∗1
= ’U’:  Upper triangle of A is stored;
= ’L’:  Lower triangle of A is stored.

N       (input) INTEGER
The order of the matrix A.  N >= 0.

KD      (input) INTEGER
The number of superdiagonals of the matrix A if UPLO = ’U’, or the number of subdiagonals if UPLO = ’L’.  KD >= 0.

AB      (input/output) REAL array, dimension (LDAB, N)
On entry, the upper or lower triangle of the symmetric band matrix A, stored in the first KD+1 rows of the array.  The j-th column of A is stored in the j-th column of the array AB as follows: if UPLO = ’U’, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = ’L’, AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
 
On exit, AB is overwritten by values generated during the reduction to tridiagonal form.  If UPLO = ’U’, the first superdiagonal and the diagonal of the tridiagonal matrix T are returned in rows KD and KD+1 of AB, and if UPLO = ’L’, the diagonal and first subdiagonal of T are returned in the first two rows of AB.

LDAB    (input) INTEGER
The leading dimension of the array AB.  LDAB >= KD + 1.

Q       (output) REAL array, dimension (LDQ, N)
If JOBZ = ’V’, the N-by-N orthogonal matrix used in the reduction to tridiagonal form. If JOBZ = ’N’, the array Q is not referenced.

LDQ     (input) INTEGER
The leading dimension of the array Q.  If JOBZ = ’V’, then LDQ >= max(1,N).

VL      (input) REAL
VU      (input) REAL 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) INTEGER
IU      (input) INTEGER 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) 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 + 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 AB 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’).
 
See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3.

M       (output) INTEGER
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)
The first M elements contain the selected eigenvalues in ascending order.

Z       (output) REAL array, dimension (LDZ, max(1,M))
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). 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. If JOBZ = ’N’, then Z is not referenced. 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) INTEGER
The leading dimension of the array Z.  LDZ >= 1, and if JOBZ = ’V’, LDZ >= max(1,N).

WORK    (workspace) REAL array, dimension (7∗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) 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.

  —  LAPACK version 2.0  —  08 October 1994