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dsbgvx(3P)		    Sun Performance Library		    dsbgvx(3P)

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

SYNOPSIS
       SUBROUTINE DSBGVX(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 * 1 JOBZ, RANGE, UPLO
       INTEGER N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
       INTEGER IWORK(*), IFAIL(*)
       DOUBLE PRECISION VL, VU, ABSTOL
       DOUBLE  PRECISION  AB(LDAB,*),  BB(LDBB,*),  Q(LDQ,*),  W(*), Z(LDZ,*),
       WORK(*)

       SUBROUTINE DSBGVX_64(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 * 1 JOBZ, RANGE, UPLO
       INTEGER*8 N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
       INTEGER*8 IWORK(*), IFAIL(*)
       DOUBLE PRECISION VL, VU, ABSTOL
       DOUBLE PRECISION	 AB(LDAB,*),  BB(LDBB,*),  Q(LDQ,*),  W(*),  Z(LDZ,*),
       WORK(*)

   F95 INTERFACE
       SUBROUTINE SBGVX(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(LEN=1) :: JOBZ, RANGE, UPLO
       INTEGER :: N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
       INTEGER, DIMENSION(:) :: IWORK, IFAIL
       REAL(8) :: VL, VU, ABSTOL
       REAL(8), DIMENSION(:) :: W, WORK
       REAL(8), DIMENSION(:,:) :: AB, BB, Q, Z

       SUBROUTINE SBGVX_64(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(LEN=1) :: JOBZ, RANGE, UPLO
       INTEGER(8) :: N, KA, KB, LDAB, LDBB, LDQ, IL, IU, M, LDZ, INFO
       INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
       REAL(8) :: VL, VU, ABSTOL
       REAL(8), DIMENSION(:) :: W, WORK
       REAL(8), DIMENSION(:,:) :: AB, BB, Q, Z

   C INTERFACE
       #include <sunperf.h>

       void  dsbgvx(char  jobz,	 char range, char uplo, int n, int ka, int kb,
		 double *ab, int ldab, double *bb, int ldbb,  double  *q,  int
		 ldq, double vl, double vu, int il, int iu, double abstol, int
		 *m, double *w, double *z, int ldz, int *ifail, int *info);

       void dsbgvx_64(char jobz, char range, char uplo, long n, long ka,  long
		 kb,  double *ab, long ldab, double *bb, long ldbb, double *q,
		 long ldq, double vl, double vu,  long	il,  long  iu,	double
		 abstol, long *m, double *w, double *z, long ldz, long *ifail,
		 long *info);

PURPOSE
       dsbgvx 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)
		 = 'N':	 Compute eigenvalues only;
		 = 'V':	 Compute eigenvalues and eigenvectors.

       RANGE (input)
		 = '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)
		 = 'U':	 Upper triangles of A and B are stored;
		 = 'L':	 Lower triangles of A and B are stored.

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

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

       KB (input)
		 The number of superdiagonals of the matrix B if UPLO  =  'U',
		 or the number of subdiagonals if UPLO = 'L'.  KB >= 0.

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

       BB (input/output)
		 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 column 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 DPBSTF.

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

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

       VL (input)
		 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'.

       VU (input)
		 See the description of VL.

       IL (input)
		 If RANGE='I', the indices (in ascending order) of the	small‐
		 est 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'.

       IU (input)
		 See the description of IL.

       ABSTOL (input)
		 The absolute error tolerance for the eigenvalues.  An approx‐
		 imate eigenvalue is accepted as converged when it  is	deter‐
		 mined 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*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').

       M (output)
		 The  total  number  of	 eigenvalues  found.  0 <= M <= N.  If
		 RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.

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

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

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

       WORK (workspace)
		 dimension(7*N)

       IWORK (workspace/output)
		 dimension(5*N)

       IFAIL (input)
		 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 eigenvalues that failed to	converge.   If
		 JOBZ = 'N', then IFAIL is not referenced.

       INFO (output)
		 = 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 : DPBSTF 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

				  6 Mar 2009			    dsbgvx(3P)
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