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dspgvx(3P)		    Sun Performance Library		    dspgvx(3P)

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

SYNOPSIS
       SUBROUTINE DSPGVX(ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL,
	     IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)

       CHARACTER * 1 JOBZ, RANGE, UPLO
       INTEGER ITYPE, N, IL, IU, M, LDZ, INFO
       INTEGER IWORK(*), IFAIL(*)
       DOUBLE PRECISION VL, VU, ABSTOL
       DOUBLE PRECISION AP(*), BP(*), W(*), Z(LDZ,*), WORK(*)

       SUBROUTINE DSPGVX_64(ITYPE, JOBZ, RANGE, UPLO, N, AP, BP, VL, VU, IL,
	     IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)

       CHARACTER * 1 JOBZ, RANGE, UPLO
       INTEGER*8 ITYPE, N, IL, IU, M, LDZ, INFO
       INTEGER*8 IWORK(*), IFAIL(*)
       DOUBLE PRECISION VL, VU, ABSTOL
       DOUBLE PRECISION AP(*), BP(*), W(*), Z(LDZ,*), WORK(*)

   F95 INTERFACE
       SUBROUTINE SPGVX(ITYPE, JOBZ, RANGE, UPLO, [N], AP, BP, VL, VU, IL,
	      IU, ABSTOL, M, W, Z, [LDZ], [WORK], [IWORK], IFAIL, [INFO])

       CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
       INTEGER :: ITYPE, N, IL, IU, M, LDZ, INFO
       INTEGER, DIMENSION(:) :: IWORK, IFAIL
       REAL(8) :: VL, VU, ABSTOL
       REAL(8), DIMENSION(:) :: AP, BP, W, WORK
       REAL(8), DIMENSION(:,:) :: Z

       SUBROUTINE SPGVX_64(ITYPE, JOBZ, RANGE, UPLO, [N], AP, BP, VL, VU,
	      IL, IU, ABSTOL, M, W, Z, [LDZ], [WORK], [IWORK], IFAIL, [INFO])

       CHARACTER(LEN=1) :: JOBZ, RANGE, UPLO
       INTEGER(8) :: ITYPE, N, IL, IU, M, LDZ, INFO
       INTEGER(8), DIMENSION(:) :: IWORK, IFAIL
       REAL(8) :: VL, VU, ABSTOL
       REAL(8), DIMENSION(:) :: AP, BP, W, WORK
       REAL(8), DIMENSION(:,:) :: Z

   C INTERFACE
       #include <sunperf.h>

       void  dspgvx(int itype, char jobz, char range, char uplo, int n, double
		 *ap, double *bp, double vl, double vu, int il, int iu, double
		 abstol,  int  *m,  double *w, double *z, int ldz, int *ifail,
		 int *info);

       void dspgvx_64(long itype, char jobz, char range, char  uplo,  long  n,
		 double	 *ap,  double *bp, double vl, double vu, long il, long
		 iu, double abstol, long *m, double *w, double *z,  long  ldz,
		 long *ifail, long *info);

PURPOSE
       dspgvx computes selected eigenvalues, and optionally, eigenvectors of a
       real  generalized  symmetric-definite   eigenproblem,   of   the	  form
       A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A and B
       are assumed to be symmetric, stored in packed storage, and  B  is  also
       positive	 definite.   Eigenvalues  and  eigenvectors can be selected by
       specifying either a range of values or  a  range	 of  indices  for  the
       desired eigenvalues.

ARGUMENTS
       ITYPE (input)
		 Specifies the problem type to be solved:
		 = 1:  A*x = (lambda)*B*x
		 = 2:  A*B*x = (lambda)*x
		 = 3:  B*A*x = (lambda)*x

       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 triangle of A and B are stored;
		 = 'L':	 Lower triangle of A and B are stored.

       N (input) The order of the matrix pencil (A,B).	N >= 0.

       AP (input/output)
		 Double	 precision  array, dimension (N*(N+1)/2) On entry, the
		 upper or lower triangle of the	 symmetric  matrix  A,	packed
		 columnwise in a linear array.	The j-th column of A is stored
		 in the array AP as follows: if UPLO = 'U', AP(i +  (j-1)*j/2)
		 =  A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2)
		 = A(i,j) for j<=i<=n.

		 On exit, the contents of AP are destroyed.

       BP (input/output)
		 Double precision array, dimension (N*(N+1)/2) On  entry,  the
		 upper	or  lower  triangle  of the symmetric matrix B, packed
		 columnwise in a linear array.	The j-th column of B is stored
		 in  the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2)
		 = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i +  (j-1)*(2*n-j)/2)
		 = B(i,j) for j<=i<=n.

		 On  exit, the triangular factor U or L from the Cholesky fac‐
		 torization B = U**T*U or B = L*L**T, in the same storage for‐
		 mat as B.

       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)
		 Double	 precision  array,  dimension  (N) On normal exit, the
		 first M elements contain the selected eigenvalues in  ascend‐
		 ing order.

       Z (output)
		 Double	 precision  array, dimension (LDZ, max(1,M)) If JOBZ =
		 'N', then Z is not referenced.	 If JOBZ = 'V', then if INFO =
		 0, the first M columns of Z contain the orthonormal eigenvec‐
		 tors of the matrix A corresponding to the selected  eigenval‐
		 ues,  with the i-th column of Z holding the eigenvector asso‐
		 ciated with W(i).  The eigenvectors are  normalized  as  fol‐
		 lows:	if  ITYPE  =  1	 or  2,	 Z**T*B*Z  =  I; if ITYPE = 3,
		 Z**T*inv(B)*Z = 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.   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)
		 The  leading dimension of the array Z.	 LDZ >= 1, and if JOBZ
		 = 'V', LDZ >= max(1,N).

       WORK (workspace)
		 Double precision array, dimension(8*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)
		 = 0:  successful exit
		 < 0:  if INFO = -i, the i-th argument had an illegal value
		 > 0:  DPPTRF or DSPEVX returned an error code:
		 <= N:	if INFO = i, DSPEVX failed to converge; i eigenvectors
		 failed to converge.  Their indices are stored in array IFAIL.
		 >  N:	  if  INFO  = N + i, for 1 <= i <= N, then the leading
		 minor of order i of B is not positive definite.  The  factor‐
		 ization  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			    dspgvx(3P)
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