chpevx man page on Scientific

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

NAME
       CHPEVX - computes selected eigenvalues and, optionally, eigenvectors of
       a complex Hermitian matrix A in packed storage

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

	   CHARACTER	  JOBZ, RANGE, UPLO

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

	   REAL		  ABSTOL, VL, VU

	   INTEGER	  IFAIL( * ), IWORK( * )

	   REAL		  RWORK( * ), W( * )

	   COMPLEX	  AP( * ), WORK( * ), Z( LDZ, * )

PURPOSE
       CHPEVX computes selected eigenvalues and, optionally, eigenvectors of a
       complex Hermitian matrix A in packed storage.  Eigenvalues/vectors  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.

       AP      (input/output) COMPLEX array, dimension (N*(N+1)/2)
	       On entry, the upper or lower triangle of the  Hermitian	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,	 AP  is	 over‐
	       written by values generated during the reduction to tridiagonal
	       form.  If UPLO = 'U', the diagonal and first  superdiagonal  of
	       the  tridiagonal	 matrix T overwrite the corresponding elements
	       of A, and if UPLO = 'L', the diagonal and first subdiagonal  of
	       T overwrite the corresponding elements of A.

       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  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 AP 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 eigen‐
	       vectors did not converge, try setting ABSTOL to	2*SLAMCH('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) REAL array, dimension (N)
	       If INFO = 0, the selected eigenvalues in ascending order.

       Z       (output) COMPLEX 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) COMPLEX array, dimension (2*N)

       RWORK   (workspace) REAL 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 driver routine (version 3November 2008			     CHPEVX(1)
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