dspgvx man page on Scientific

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DSPGVX(1)	      LAPACK driver routine (version 3.2)	     DSPGVX(1)

NAME
       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

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

	   CHARACTER	  JOBZ, RANGE, UPLO

	   INTEGER	  IL, INFO, ITYPE, IU, LDZ, M, N

	   DOUBLE	  PRECISION ABSTOL, VL, VU

	   INTEGER	  IFAIL( * ), IWORK( * )

	   DOUBLE	  PRECISION AP( * ), BP( * ), W( * ), WORK(  *	),  Z(
			  LDZ, * )

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

       N       (input) INTEGER
	       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 factorization B = U**T*U or B =
	       L*L**T, in the same storage format as B.

       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 pre‐
	       cision.	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	eigen‐
	       vectors did not converge, try setting ABSTOL to 2*DLAMCH('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) DOUBLE PRECISION array, dimension (N)
	       On normal exit, the first M elements contain the	 selected  ei‐
	       genvalues in ascending 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
	       eigenvectors  of the matrix A corresponding to the selected ei‐
	       genvalues, with the i-th column of Z  holding  the  eigenvector
	       associated  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)  col‐
	       umns  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 (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) INTEGER
	       = 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 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 driver routine (version 3November 2008			     DSPGVX(1)
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