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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 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) 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 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 super‐
	       diagonal 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 N-by-N orthogonal matrix used in the reduc‐
	       tion 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 approxi‐
	       mate 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 reduc‐
	       ing 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	eigen‐
	       vectors 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 = IU-IL+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 con‐
	       tain 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 approxi‐
	       mation 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 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 3.0		 15 June 2000			     DSBEVX(l)
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