
DSBEVX(l) ) DSBEVX(l)
NAME
DSBEVX  compute selected eigenvalues and, optionally, eigenvectors of a real symmetric
band matrix A
SYNOPSIS
SUBROUTINE DSBEVX( 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
DOUBLE PRECISION ABSTOL, VL, VU
INTEGER IFAIL( * ), IWORK( * )
DOUBLE PRECISION AB( LDAB, * ), Q( LDQ, * ), W( * ), WORK( * ), Z( LDZ, * )
PURPOSE
DSBEVX 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 halfopen interval (VL,VU] will be found; = 'I': the
ILth through IUth 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 sub
diagonals if UPLO = 'L'. KD >= 0.
AB (input/output) DOUBLE PRECISION 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 jth column of A is stored in the jth col
umn of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for
max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,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) DOUBLE PRECISION array, dimension (LDQ, N)
If JOBZ = 'V', the NbyN 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) DOUBLE PRECISION
VU (input) DOUBLE PRECISION 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) DOUBLE PRECISION
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 1norm 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*DLAMCH('S'), not zero. If this routine returns with INFO>0,
indicating that some eigenvectors did not converge, try setting ABSTOL to
2*DLAMCH('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 = IUIL+1.
W (output) DOUBLE PRECISION array, dimension (N)
The first M elements contain the selected eigenvalues in ascending order.
Z (output) DOUBLE PRECISION 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
ith 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) DOUBLE PRECISION 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 ith argument had an illegal value.
> 0: if INFO = i, then i eigenvectors failed to converge. Their indices are
stored in array IFAIL.
LAPACK version 3.0 15 June 2000 DSBEVX(l) 
