1CLAIC1(1)           LAPACK auxiliary routine (version 3.2)           CLAIC1(1)
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NAME

6       CLAIC1  -  applies  one step of incremental condition estimation in its
7       simplest version
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SYNOPSIS

10       SUBROUTINE CLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
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12           INTEGER        J, JOB
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14           REAL           SEST, SESTPR
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16           COMPLEX        C, GAMMA, S
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18           COMPLEX        W( J ), X( J )
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PURPOSE

21       CLAIC1 applies one step of incremental condition estimation in its sim‐
22       plest version: Let x, twonorm(x) = 1, be an approximate singular vector
23       of an j-by-j lower triangular matrix L, such that
24                twonorm(L*x) = sest
25       Then CLAIC1 computes sestpr, s, c such that
26       the vector
27                       [ s*x ]
28                xhat = [  c  ]
29       is an approximate singular vector of
30                       [ L     0  ]
31                Lhat = [ w' gamma ]
32       in the sense that
33                twonorm(Lhat*xhat) = sestpr.
34       Depending on JOB, an estimate for  the  largest  or  smallest  singular
35       value is computed.
36       Note that [s c]' and sestpr**2 is an eigenpair of the system
37           diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
38                                                 [ conjg(gamma) ] where  alpha
39       =  conjg(x)'*w.
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ARGUMENTS

42       JOB     (input) INTEGER
43               = 1: an estimate for the largest singular value is computed.
44               = 2: an estimate for the smallest singular value is computed.
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46       J       (input) INTEGER
47               Length of X and W
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49       X       (input) COMPLEX array, dimension (J)
50               The j-vector x.
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52       SEST    (input) REAL
53               Estimated singular value of j by j matrix L
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55       W       (input) COMPLEX array, dimension (J)
56               The j-vector w.
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58       GAMMA   (input) COMPLEX
59               The diagonal element gamma.
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61       SESTPR  (output) REAL
62               Estimated singular value of (j+1) by (j+1) matrix Lhat.
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64       S       (output) COMPLEX
65               Sine needed in forming xhat.
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67       C       (output) COMPLEX
68               Cosine needed in forming xhat.
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72 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008                       CLAIC1(1)
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