CLAR1V (3) - Linux Manuals

NAME

clar1v.f -

SYNOPSIS


Functions/Subroutines


subroutine clar1v (N, B1, BN, LAMBDA, D, L, LD, LLD, PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA, R, ISUPPZ, NRMINV, RESID, RQCORR, WORK)
CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

Function/Subroutine Documentation

subroutine clar1v (integerN, integerB1, integerBN, realLAMBDA, real, dimension( * )D, real, dimension( * )L, real, dimension( * )LD, real, dimension( * )LLD, realPIVMIN, realGAPTOL, complex, dimension( * )Z, logicalWANTNC, integerNEGCNT, realZTZ, realMINGMA, integerR, integer, dimension( * )ISUPPZ, realNRMINV, realRESID, realRQCORR, real, dimension( * )WORK)

CLAR1V computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the tridiagonal matrix LDLT - λI.

Purpose: 

 CLAR1V computes the (scaled) r-th column of the inverse of
 the sumbmatrix in rows B1 through BN of the tridiagonal matrix
 L D L**T - sigma I. When sigma is close to an eigenvalue, the
 computed vector is an accurate eigenvector. Usually, r corresponds
 to the index where the eigenvector is largest in magnitude.
 The following steps accomplish this computation :
 (a) Stationary qd transform,  L D L**T - sigma I = L(+) D(+) L(+)**T,
 (b) Progressive qd transform, L D L**T - sigma I = U(-) D(-) U(-)**T,
 (c) Computation of the diagonal elements of the inverse of
     L D L**T - sigma I by combining the above transforms, and choosing
     r as the index where the diagonal of the inverse is (one of the)
     largest in magnitude.
 (d) Computation of the (scaled) r-th column of the inverse using the
     twisted factorization obtained by combining the top part of the
     the stationary and the bottom part of the progressive transform.


 

Parameters:

N 

          N is INTEGER
           The order of the matrix L D L**T.


B1 

          B1 is INTEGER
           First index of the submatrix of L D L**T.


BN 

          BN is INTEGER
           Last index of the submatrix of L D L**T.


LAMBDA 

          LAMBDA is REAL
           The shift. In order to compute an accurate eigenvector,
           LAMBDA should be a good approximation to an eigenvalue
           of L D L**T.


L 

          L is REAL array, dimension (N-1)
           The (n-1) subdiagonal elements of the unit bidiagonal matrix
           L, in elements 1 to N-1.


D 

          D is REAL array, dimension (N)
           The n diagonal elements of the diagonal matrix D.


LD 

          LD is REAL array, dimension (N-1)
           The n-1 elements L(i)*D(i).


LLD 

          LLD is REAL array, dimension (N-1)
           The n-1 elements L(i)*L(i)*D(i).


PIVMIN 

          PIVMIN is REAL
           The minimum pivot in the Sturm sequence.


GAPTOL 

          GAPTOL is REAL
           Tolerance that indicates when eigenvector entries are negligible
           w.r.t. their contribution to the residual.


Z 

          Z is COMPLEX array, dimension (N)
           On input, all entries of Z must be set to 0.
           On output, Z contains the (scaled) r-th column of the
           inverse. The scaling is such that Z(R) equals 1.


WANTNC 

          WANTNC is LOGICAL
           Specifies whether NEGCNT has to be computed.


NEGCNT 

          NEGCNT is INTEGER
           If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
           in the  matrix factorization L D L**T, and NEGCNT = -1 otherwise.


ZTZ 

          ZTZ is REAL
           The square of the 2-norm of Z.


MINGMA 

          MINGMA is REAL
           The reciprocal of the largest (in magnitude) diagonal
           element of the inverse of L D L**T - sigma I.


R 

          R is INTEGER
           The twist index for the twisted factorization used to
           compute Z.
           On input, 0 <= R <= N. If R is input as 0, R is set to
           the index where (L D L**T - sigma I)^{-1} is largest
           in magnitude. If 1 <= R <= N, R is unchanged.
           On output, R contains the twist index used to compute Z.
           Ideally, R designates the position of the maximum entry in the
           eigenvector.


ISUPPZ 

          ISUPPZ is INTEGER array, dimension (2)
           The support of the vector in Z, i.e., the vector Z is
           nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).


NRMINV 

          NRMINV is REAL
           NRMINV = 1/SQRT( ZTZ )


RESID 

          RESID is REAL
           The residual of the FP vector.
           RESID = ABS( MINGMA )/SQRT( ZTZ )


RQCORR 

          RQCORR is REAL
           The Rayleigh Quotient correction to LAMBDA.
           RQCORR = MINGMA*TMP


WORK 

          WORK is REAL array, dimension (4*N)


 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

September 2012

Contributors:

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 

Definition at line 229 of file clar1v.f.

Author

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