clarrv man page on Scientific

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CLARRV(1)	    LAPACK auxiliary routine (version 3.2)	     CLARRV(1)

NAME
       CLARRV  -  computes  the eigenvectors of the tridiagonal matrix T = L D
       L^T given L, D and APPROXIMATIONS to the eigenvalues of L D L^T

SYNOPSIS
       SUBROUTINE CLARRV( N, VL, VU, D, L, PIVMIN, ISPLIT, M, DOL,  DOU,  MIN‐
			  RGP,	RTOL1,	RTOL2,	W, WERR, WGAP, IBLOCK, INDEXW,
			  GERS, Z, LDZ, ISUPPZ, WORK, IWORK, INFO )

	   INTEGER	  DOL, DOU, INFO, LDZ, M, N

	   REAL		  MINRGP, PIVMIN, RTOL1, RTOL2, VL, VU

	   INTEGER	  IBLOCK( * ), INDEXW( * ), ISPLIT( * ), ISUPPZ( *  ),
			  IWORK( * )

	   REAL		  D(  * ), GERS( * ), L( * ), W( * ), WERR( * ), WGAP(
			  * ), WORK( * )

	   COMPLEX	  Z( LDZ, * )

PURPOSE
       CLARRV computes the eigenvectors of the tridiagonal matrix T = L D  L^T
       given L, D and APPROXIMATIONS to the eigenvalues of L D L^T.  The input
       eigenvalues should have been computed by SLARRE.

ARGUMENTS
       N       (input) INTEGER
	       The order of the matrix.	 N >= 0.

       VL      (input) REAL
	       VU      (input) REAL Lower and upper  bounds  of	 the  interval
	       that  contains the desired eigenvalues. VL < VU. Needed to com‐
	       pute gaps on the left or right end of the extremal  eigenvalues
	       in the desired RANGE.

       D       (input/output) REAL	       array, dimension (N)
	       On entry, the N diagonal elements of the diagonal matrix D.  On
	       exit, D may be overwritten.

       L       (input/output) REAL	       array, dimension (N)
	       On entry, the (N-1) subdiagonal elements of the unit bidiagonal
	       matrix  L  are  in elements 1 to N-1 of L (if the matrix is not
	       splitted.) At the end of each block is stored the corresponding
	       shift as given by SLARRE.  On exit, L is overwritten.

       PIVMIN  (in) DOUBLE PRECISION
	       The minimum pivot allowed in the Sturm sequence.

       ISPLIT  (input) INTEGER array, dimension (N)
	       The  splitting  points,	at which T breaks up into blocks.  The
	       first block consists of rows/columns 1 to ISPLIT( 1 ), the sec‐
	       ond of rows/columns ISPLIT( 1 )+1 through ISPLIT( 2 ), etc.

       M       (input) INTEGER
	       The total number of input eigenvalues.  0 <= M <= N.

       DOL     (input) INTEGER
	       DOU	(input)	 INTEGER  If  the  user	 wants to compute only
	       selected eigenvectors from all the eigenvalues supplied, he can
	       specify	an  index  range  DOL:DOU.  Or else the setting DOL=1,
	       DOU=M should be applied.	 Note that DOL and DOU	refer  to  the
	       order  in  which	 the eigenvalues are stored in W.  If the user
	       wants to compute only selected  eigenpairs,  then  the  columns
	       DOL-1  to DOU+1 of the eigenvector space Z contain the computed
	       eigenvectors. All other columns of Z are set to zero.

       MINRGP  (input) REAL

       RTOL1   (input) REAL
	       RTOL2	(input)	 REAL  Parameters   for	  bisection.	RIGHT-
	       LEFT.LT.MAX( RTOL1*GAP, RTOL2*MAX(|LEFT|,|RIGHT|) )

       W       (input/output) REAL	       array, dimension (N)
	       The  first  M elements of W contain the APPROXIMATE eigenvalues
	       for which eigenvectors are to  be  computed.   The  eigenvalues
	       should  be grouped by split-off block and ordered from smallest
	       to largest within the block ( The output array W from SLARRE is
	       expected	 here  ).  Furthermore,	 they  are with respect to the
	       shift of the corresponding root representation for their block.
	       On exit, W holds the eigenvalues of the UNshifted matrix.

       WERR    (input/output) REAL	       array, dimension (N)
	       The  first  M elements contain the semiwidth of the uncertainty
	       interval of the corresponding eigenvalue in W

       WGAP    (input/output) REAL	       array, dimension (N)
	       The separation from the right neighbor eigenvalue in W.

       IBLOCK  (input) INTEGER array, dimension (N)
	       The indices of the blocks  (submatrices)	 associated  with  the
	       corresponding  eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i)
	       belongs to the first block from the top, =2 if W(i) belongs  to
	       the second block, etc.

       INDEXW  (input) INTEGER array, dimension (N)
	       The  indices  of the eigenvalues within each block (submatrix);
	       for example, INDEXW(i)= 10 and IBLOCK(i)=2 imply that the  i-th
	       eigenvalue W(i) is the 10-th eigenvalue in the second block.

       GERS    (input) REAL		array, dimension (2*N)
	       The  N  Gerschgorin intervals (the i-th Gerschgorin interval is
	       (GERS(2*i-1), GERS(2*i)). The Gerschgorin intervals  should  be
	       computed from the original UNshifted matrix.

       Z       (output) COMPLEX		 array, dimension (LDZ, max(1,M) )
	       If  INFO	 = 0, the first M columns of Z contain the orthonormal
	       eigenvectors of the matrix T corresponding to the input	eigen‐
	       values, with the i-th column of Z holding the eigenvector asso‐
	       ciated with W(i).  Note: the user must  ensure  that  at	 least
	       max(1,M) columns are supplied in the array Z.

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

       ISUPPZ  (output) INTEGER array, dimension ( 2*max(1,M) )
	       The support of the eigenvectors in Z, i.e., the	indices	 indi‐
	       cating  the  nonzero  elements  in  Z.  The I-th eigenvector is
	       nonzero only in elements ISUPPZ( 2*I-1 ) through ISUPPZ( 2*I ).

       WORK    (workspace) REAL		    array, dimension (12*N)

       IWORK   (workspace) INTEGER array, dimension (7*N)

       INFO    (output) INTEGER
	       = 0:  successful exit > 0:  A problem occured in CLARRV.
	       < 0:  One of the called subroutines signaled an internal	 prob‐
	       lem.  Needs inspection of the corresponding parameter IINFO for
	       further information.

       =-1:  Problem in SLARRB when refining a child's eigenvalues.
	     =-2:  Problem in SLARRF when computing the RRR of a child.	  When
	     a child is inside a tight cluster, it can be difficult to find an
	     RRR. A partial remedy from the user's point of view  is  to  make
	     the  parameter  MINRGP  smaller  and  recompile.  However, as the
	     orthogonality of the computed vectors is proportional  to	1/MIN‐
	     RGP,  the user should be aware that he might be trading in preci‐
	     sion when he decreases MINRGP.   =-3:   Problem  in  SLARRB  when
	     refining  a  single  eigenvalue after the Rayleigh correction was
	     rejected.	= 5:  The Rayleigh Quotient Iteration failed  to  con‐
	     verge to full accuracy in MAXITR steps.

FURTHER DETAILS
       Based on contributions by
	  Beresford Parlett, University of California, Berkeley, USA
	  Jim Demmel, University of California, Berkeley, USA
	  Inderjit Dhillon, University of Texas, Austin, USA
	  Osni Marques, LBNL/NERSC, USA
	  Christof Voemel, University of California, Berkeley, USA

 LAPACK auxiliary routine (versioNovember 2008			     CLARRV(1)
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