1CLAIC1(1)           LAPACK auxiliary routine (version 3.1)           CLAIC1(1)
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NAME

6       CLAIC1  -  one step of incremental condition estimation in its simplest
7       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:
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24       Let  x,  twonorm(x) = 1, be an approximate singular vector of an j-by-j
25       lower triangular matrix L, such that
26                twonorm(L*x) = sest
27       Then CLAIC1 computes sestpr, s, c such that
28       the vector
29                       [ s*x ]
30                xhat = [  c  ]
31       is an approximate singular vector of
32                       [ L     0  ]
33                Lhat = [ w' gamma ]
34       in the sense that
35                twonorm(Lhat*xhat) = sestpr.
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37       Depending on JOB, an estimate for  the  largest  or  smallest  singular
38       value is computed.
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40       Note that [s c]' and sestpr**2 is an eigenpair of the system
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42           diag(sest*sest, 0) + [alpha  gamma] * [ conjg(alpha) ]
43                                                 [ conjg(gamma) ]
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45       where  alpha =  conjg(x)'*w.
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ARGUMENTS

49       JOB     (input) INTEGER
50               = 1: an estimate for the largest singular value is computed.
51               = 2: an estimate for the smallest singular value is computed.
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53       J       (input) INTEGER
54               Length of X and W
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56       X       (input) COMPLEX array, dimension (J)
57               The j-vector x.
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59       SEST    (input) REAL
60               Estimated singular value of j by j matrix L
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62       W       (input) COMPLEX array, dimension (J)
63               The j-vector w.
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65       GAMMA   (input) COMPLEX
66               The diagonal element gamma.
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68       SESTPR  (output) REAL
69               Estimated singular value of (j+1) by (j+1) matrix Lhat.
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71       S       (output) COMPLEX
72               Sine needed in forming xhat.
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74       C       (output) COMPLEX
75               Cosine needed in forming xhat.
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79 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       CLAIC1(1)
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