.TH ZLAIC1 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) " .SH NAME ZLAIC1 - one step of incremental condition estimation in its simplest version .SH SYNOPSIS .TP 19 SUBROUTINE ZLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C ) .TP 19 .ti +4 INTEGER J, JOB .TP 19 .ti +4 DOUBLE PRECISION SEST, SESTPR .TP 19 .ti +4 COMPLEX*16 C, GAMMA, S .TP 19 .ti +4 COMPLEX*16 W( J ), X( J ) .SH PURPOSE ZLAIC1 applies one step of incremental condition estimation in its simplest version: Let x, twonorm(x) = 1, be an approximate singular vector of an j-by-j lower triangular matrix L, such that .br twonorm(L*x) = sest .br Then ZLAIC1 computes sestpr, s, c such that .br the vector .br [ s*x ] .br xhat = [ c ] .br is an approximate singular vector of .br [ L 0 ] .br Lhat = [ w\(aq gamma ] .br in the sense that .br twonorm(Lhat*xhat) = sestpr. .br Depending on JOB, an estimate for the largest or smallest singular value is computed. .br Note that [s c]\(aq and sestpr**2 is an eigenpair of the system diag(sest*sest, 0) + [alpha gamma] * [ conjg(alpha) ] [ conjg(gamma) ] where alpha = conjg(x)\(aq*w. .br .SH ARGUMENTS .TP 8 JOB (input) INTEGER = 1: an estimate for the largest singular value is computed. .br = 2: an estimate for the smallest singular value is computed. .TP 8 J (input) INTEGER Length of X and W .TP 8 X (input) COMPLEX*16 array, dimension (J) The j-vector x. .TP 8 SEST (input) DOUBLE PRECISION Estimated singular value of j by j matrix L .TP 8 W (input) COMPLEX*16 array, dimension (J) The j-vector w. .TP 8 GAMMA (input) COMPLEX*16 The diagonal element gamma. .TP 8 SESTPR (output) DOUBLE PRECISION Estimated singular value of (j+1) by (j+1) matrix Lhat. .TP 8 S (output) COMPLEX*16 Sine needed in forming xhat. .TP 8 C (output) COMPLEX*16 Cosine needed in forming xhat.