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RedHat 9 (Linux i386) - man page for ssbgvx (redhat section l)

SSBGVX(l)					)					SSBGVX(l)

NAME
       SSBGVX  - compute selected eigenvalues, and optionally, eigenvectors of a real generalized
       symmetric-definite banded eigenproblem, of the form A*x=(lambda)*B*x

SYNOPSIS
       SUBROUTINE SSBGVX( JOBZ, RANGE, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, Q, LDQ, VL,  VU,  IL,
			  IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )

	   CHARACTER	  JOBZ, RANGE, UPLO

	   INTEGER	  IL, INFO, IU, KA, KB, LDAB, LDBB, LDQ, LDZ, M, N

	   REAL 	  ABSTOL, VL, VU

	   INTEGER	  IFAIL( * ), IWORK( * )

	   REAL 	  AB( LDAB, * ), BB( LDBB, * ), Q( LDQ, * ), W( * ), WORK( * ), Z( LDZ, *
			  )

PURPOSE
       SSBGVX computes selected eigenvalues, and optionally, eigenvectors of a	real  generalized
       symmetric-definite  banded  eigenproblem,  of  the form A*x=(lambda)*B*x. Here A and B are
       assumed to be symmetric and banded, and B is  also  positive  definite.	 Eigenvalues  and
       eigenvectors  can be selected by specifying either all eigenvalues, 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 triangles of A and B are stored;
	       = 'L':  Lower triangles of A and B are stored.

       N       (input) INTEGER
	       The order of the matrices A and B.  N >= 0.

       KA      (input) INTEGER
	       The  number of superdiagonals of the matrix A if UPLO = 'U', or the number of sub-
	       diagonals if UPLO = 'L'.  KA >= 0.

       KB      (input) INTEGER
	       The number of superdiagonals of the matrix B if UPLO = 'U', or the number of  sub-
	       diagonals if UPLO = 'L'.  KB >= 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 ka+1 rows of the array.  The j-th column of A is stored in the j-th col-
	       umn  of	the  array  AB	as  follows:  if  UPLO = 'U', AB(ka+1+i-j,j) = A(i,j) for
	       max(1,j-ka)<=i<=j; if UPLO = 'L', AB(1+i-j,j)	= A(i,j) for j<=i<=min(n,j+ka).

	       On exit, the contents of AB are destroyed.

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

       BB      (input/output) REAL array, dimension (LDBB, N)
	       On entry, the upper or lower triangle of the symmetric band matrix  B,  stored  in
	       the first kb+1 rows of the array.  The j-th column of B is stored in the j-th col-
	       umn of the array BB as follows:	if  UPLO  =  'U',  BB(ka+1+i-j,j)  =  B(i,j)  for
	       max(1,j-kb)<=i<=j; if UPLO = 'L', BB(1+i-j,j)	= B(i,j) for j<=i<=min(n,j+kb).

	       On  exit,  the  factor  S  from	the  split  Cholesky factorization B = S**T*S, as
	       returned by SPBSTF.

       LDBB    (input) INTEGER
	       The leading dimension of the array BB.  LDBB >= KB+1.

       Q       (output) REAL array, dimension (LDQ, N)
	       If JOBZ = 'V', the n-by-n matrix used in the reduction of A*x  =  (lambda)*B*x  to
	       standard  form, i.e. C*x = (lambda)*x, and consequently C to tridiagonal form.  If
	       JOBZ = 'N', the array Q is not referenced.

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

       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)
	       If INFO = 0, the eigenvalues in ascending order.

       Z       (output) REAL array, dimension (LDZ, N)
	       If JOBZ = 'V', then if INFO = 0, Z contains the matrix Z of eigenvectors, with the
	       i-th  column  of Z holding the eigenvector associated with W(i).  The eigenvectors
	       are normalized so Z**T*B*Z = I.	If JOBZ = 'N', then Z is not referenced.

       LDZ     (input) INTEGER
	       The leading dimension of the array Z.  LDZ >=  1,  and  if  JOBZ  =  'V',  LDZ  >=
	       max(1,N).

       WORK    (workspace/output) REAL array, dimension (7N)

       IWORK   (workspace/output) INTEGER array, dimension (5N)

       IFAIL   (input) INTEGER array, dimension (M)
	       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 eigenvalues 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
	       <=  N:  if  INFO  =  i, then i eigenvectors failed to converge.	Their indices are
	       stored in IFAIL.  > N : SPBSTF returned an error code; i.e., 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.

FURTHER DETAILS
       Based on contributions by
	  Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

LAPACK version 3.0			   15 June 2000 				SSBGVX(l)


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