ZLAR1V(3S)ZLAR1V(3S)NAMEZLAR1V - compute 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
SYNOPSIS
SUBROUTINE ZLAR1V( N, B1, BN, SIGMA, D, L, LD, LLD, GERSCH, Z, ZTZ,
MINGMA, R, ISUPPZ, WORK )
INTEGER B1, BN, N, R
DOUBLE PRECISION MINGMA, SIGMA, ZTZ
INTEGER ISUPPZ( * )
DOUBLE PRECISION D( * ), GERSCH( * ), L( * ), LD( * ), LLD( *
), WORK( * )
COMPLEX*16 Z( * )
IMPLEMENTATION
These routines are part of the SCSL Scientific Library and can be loaded
using either the -lscs or the -lscs_mp option. The -lscs_mp option
directs the linker to use the multi-processor version of the library.
When linking to SCSL with -lscs or -lscs_mp, the default integer size is
4 bytes (32 bits). Another version of SCSL is available in which integers
are 8 bytes (64 bits). This version allows the user access to larger
memory sizes and helps when porting legacy Cray codes. It can be loaded
by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
only one of the two versions; 4-byte integer and 8-byte integer library
calls cannot be mixed.
PURPOSEZLAR1V 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. 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.
ARGUMENTS
N (input) INTEGER
The order of the matrix L D L^T.
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ZLAR1V(3S)ZLAR1V(3S)
B1 (input) INTEGER
First index of the submatrix of L D L^T.
BN (input) INTEGER
Last index of the submatrix of L D L^T.
SIGMA (input) DOUBLE PRECISION
The shift. Initially, when R = 0, SIGMA should be a good
approximation to an eigenvalue of L D L^T.
L (input) DOUBLE PRECISION array, dimension (N-1)
The (n-1) subdiagonal elements of the unit bidiagonal matrix L,
in elements 1 to N-1.
D (input) DOUBLE PRECISION array, dimension (N)
The n diagonal elements of the diagonal matrix D.
LD (input) DOUBLE PRECISION array, dimension (N-1)
The n-1 elements L(i)*D(i).
LLD (input) DOUBLE PRECISION array, dimension (N-1)
The n-1 elements L(i)*L(i)*D(i).
GERSCH (input) DOUBLE PRECISION array, dimension (2*N)
The n Gerschgorin intervals. These are used to restrict the
initial search for R, when R is input as 0.
Z (output) COMPLEX*16 array, dimension (N)
The (scaled) r-th column of the inverse. Z(R) is returned to be
1.
ZTZ (output) DOUBLE PRECISION
The square of the norm of Z.
MINGMA (output) DOUBLE PRECISION
The reciprocal of the largest (in magnitude) diagonal element of
the inverse of L D L^T - sigma I.
R (input/output) INTEGER
Initially, R should be input to be 0 and is then output as the
index where the diagonal element of the inverse is largest in
magnitude. In later iterations, this same value of R should be
input.
ISUPPZ (output) 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 ).
WORK (workspace) DOUBLE PRECISION array, dimension (4*N)
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ZLAR1V(3S)ZLAR1V(3S)FURTHER DETAILS
Based on contributions by
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of
California at Berkeley, USA
SEE ALSOINTRO_LAPACK(3S), INTRO_SCSL(3S)
This man page is available only online.
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