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SUBROUTINE <a name="ZLAR1V.1"></a><a href="zlar1v.f.html#ZLAR1V.1">ZLAR1V</a>( N, B1, BN, LAMBDA, D, L, LD, LLD,
$ PIVMIN, GAPTOL, Z, WANTNC, NEGCNT, ZTZ, MINGMA,
$ R, ISUPPZ, NRMINV, RESID, RQCORR, WORK )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment"> Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment"> November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Scalar Arguments ..
</span> LOGICAL WANTNC
INTEGER B1, BN, N, NEGCNT, R
DOUBLE PRECISION GAPTOL, LAMBDA, MINGMA, NRMINV, PIVMIN, RESID,
$ RQCORR, ZTZ
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Array Arguments ..
</span> INTEGER ISUPPZ( * )
DOUBLE PRECISION D( * ), L( * ), LD( * ), LLD( * ),
$ WORK( * )
COMPLEX*16 Z( * )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Purpose
</span><span class="comment">*</span><span class="comment"> =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> <a name="ZLAR1V.25"></a><a href="zlar1v.f.html#ZLAR1V.1">ZLAR1V</a> computes the (scaled) r-th column of the inverse of
</span><span class="comment">*</span><span class="comment"> the sumbmatrix in rows B1 through BN of the tridiagonal matrix
</span><span class="comment">*</span><span class="comment"> L D L^T - sigma I. When sigma is close to an eigenvalue, the
</span><span class="comment">*</span><span class="comment"> computed vector is an accurate eigenvector. Usually, r corresponds
</span><span class="comment">*</span><span class="comment"> to the index where the eigenvector is largest in magnitude.
</span><span class="comment">*</span><span class="comment"> The following steps accomplish this computation :
</span><span class="comment">*</span><span class="comment"> (a) Stationary qd transform, L D L^T - sigma I = L(+) D(+) L(+)^T,
</span><span class="comment">*</span><span class="comment"> (b) Progressive qd transform, L D L^T - sigma I = U(-) D(-) U(-)^T,
</span><span class="comment">*</span><span class="comment"> (c) Computation of the diagonal elements of the inverse of
</span><span class="comment">*</span><span class="comment"> L D L^T - sigma I by combining the above transforms, and choosing
</span><span class="comment">*</span><span class="comment"> r as the index where the diagonal of the inverse is (one of the)
</span><span class="comment">*</span><span class="comment"> largest in magnitude.
</span><span class="comment">*</span><span class="comment"> (d) Computation of the (scaled) r-th column of the inverse using the
</span><span class="comment">*</span><span class="comment"> twisted factorization obtained by combining the top part of the
</span><span class="comment">*</span><span class="comment"> the stationary and the bottom part of the progressive transform.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Arguments
</span><span class="comment">*</span><span class="comment"> =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> N (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The order of the matrix L D L^T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> B1 (input) INTEGER
</span><span class="comment">*</span><span class="comment"> First index of the submatrix of L D L^T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> BN (input) INTEGER
</span><span class="comment">*</span><span class="comment"> Last index of the submatrix of L D L^T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LAMBDA (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> The shift. In order to compute an accurate eigenvector,
</span><span class="comment">*</span><span class="comment"> LAMBDA should be a good approximation to an eigenvalue
</span><span class="comment">*</span><span class="comment"> of L D L^T.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> L (input) DOUBLE PRECISION array, dimension (N-1)
</span><span class="comment">*</span><span class="comment"> The (n-1) subdiagonal elements of the unit bidiagonal matrix
</span><span class="comment">*</span><span class="comment"> L, in elements 1 to N-1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> D (input) DOUBLE PRECISION array, dimension (N)
</span><span class="comment">*</span><span class="comment"> The n diagonal elements of the diagonal matrix D.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LD (input) DOUBLE PRECISION array, dimension (N-1)
</span><span class="comment">*</span><span class="comment"> The n-1 elements L(i)*D(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LLD (input) DOUBLE PRECISION array, dimension (N-1)
</span><span class="comment">*</span><span class="comment"> The n-1 elements L(i)*L(i)*D(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> PIVMIN (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> The minimum pivot in the Sturm sequence.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> GAPTOL (input) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> Tolerance that indicates when eigenvector entries are negligible
</span><span class="comment">*</span><span class="comment"> w.r.t. their contribution to the residual.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Z (input/output) COMPLEX*16 array, dimension (N)
</span><span class="comment">*</span><span class="comment"> On input, all entries of Z must be set to 0.
</span><span class="comment">*</span><span class="comment"> On output, Z contains the (scaled) r-th column of the
</span><span class="comment">*</span><span class="comment"> inverse. The scaling is such that Z(R) equals 1.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WANTNC (input) LOGICAL
</span><span class="comment">*</span><span class="comment"> Specifies whether NEGCNT has to be computed.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NEGCNT (output) INTEGER
</span><span class="comment">*</span><span class="comment"> If WANTNC is .TRUE. then NEGCNT = the number of pivots < pivmin
</span><span class="comment">*</span><span class="comment"> in the matrix factorization L D L^T, and NEGCNT = -1 otherwise.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ZTZ (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> The square of the 2-norm of Z.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> MINGMA (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> The reciprocal of the largest (in magnitude) diagonal
</span><span class="comment">*</span><span class="comment"> element of the inverse of L D L^T - sigma I.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> R (input/output) INTEGER
</span><span class="comment">*</span><span class="comment"> The twist index for the twisted factorization used to
</span><span class="comment">*</span><span class="comment"> compute Z.
</span><span class="comment">*</span><span class="comment"> On input, 0 <= R <= N. If R is input as 0, R is set to
</span><span class="comment">*</span><span class="comment"> the index where (L D L^T - sigma I)^{-1} is largest
</span><span class="comment">*</span><span class="comment"> in magnitude. If 1 <= R <= N, R is unchanged.
</span><span class="comment">*</span><span class="comment"> On output, R contains the twist index used to compute Z.
</span><span class="comment">*</span><span class="comment"> Ideally, R designates the position of the maximum entry in the
</span><span class="comment">*</span><span class="comment"> eigenvector.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> ISUPPZ (output) INTEGER array, dimension (2)
</span><span class="comment">*</span><span class="comment"> The support of the vector in Z, i.e., the vector Z is
</span><span class="comment">*</span><span class="comment"> nonzero only in elements ISUPPZ(1) through ISUPPZ( 2 ).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> NRMINV (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> NRMINV = 1/SQRT( ZTZ )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RESID (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> The residual of the FP vector.
</span><span class="comment">*</span><span class="comment"> RESID = ABS( MINGMA )/SQRT( ZTZ )
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> RQCORR (output) DOUBLE PRECISION
</span><span class="comment">*</span><span class="comment"> The Rayleigh Quotient correction to LAMBDA.
</span><span class="comment">*</span><span class="comment"> RQCORR = MINGMA*TMP
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Further Details
</span><span class="comment">*</span><span class="comment"> ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> Based on contributions by
</span><span class="comment">*</span><span class="comment"> Beresford Parlett, University of California, Berkeley, USA
</span><span class="comment">*</span><span class="comment"> Jim Demmel, University of California, Berkeley, USA
</span><span class="comment">*</span><span class="comment"> Inderjit Dhillon, University of Texas, Austin, USA
</span><span class="comment">*</span><span class="comment"> Osni Marques, LBNL/NERSC, USA
</span><span class="comment">*</span><span class="comment"> Christof Voemel, University of California, Berkeley, USA
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> .. Parameters ..
</span> DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
COMPLEX*16 CONE
PARAMETER ( CONE = ( 1.0D0, 0.0D0 ) )
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Local Scalars ..
</span> LOGICAL SAWNAN1, SAWNAN2
INTEGER I, INDLPL, INDP, INDS, INDUMN, NEG1, NEG2, R1,
$ R2
DOUBLE PRECISION DMINUS, DPLUS, EPS, S, TMP
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. External Functions ..
</span> LOGICAL <a name="DISNAN.150"></a><a href="disnan.f.html#DISNAN.1">DISNAN</a>
DOUBLE PRECISION <a name="DLAMCH.151"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>
EXTERNAL <a name="DISNAN.152"></a><a href="disnan.f.html#DISNAN.1">DISNAN</a>, <a name="DLAMCH.152"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Intrinsic Functions ..
</span> INTRINSIC ABS, DBLE
<span class="comment">*</span><span class="comment"> ..
</span><span class="comment">*</span><span class="comment"> .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span> EPS = <a name="DLAMCH.159"></a><a href="dlamch.f.html#DLAMCH.1">DLAMCH</a>( <span class="string">'Precision'</span> )
IF( R.EQ.0 ) THEN
R1 = B1
R2 = BN
ELSE
R1 = R
R2 = R
END IF
<span class="comment">*</span><span class="comment"> Storage for LPLUS
</span> INDLPL = 0
<span class="comment">*</span><span class="comment"> Storage for UMINUS
</span> INDUMN = N
INDS = 2*N + 1
INDP = 3*N + 1
IF( B1.EQ.1 ) THEN
WORK( INDS ) = ZERO
ELSE
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