1DLAIC1(1)           LAPACK auxiliary routine (version 3.1)           DLAIC1(1)
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NAME

6       DLAIC1  -  one step of incremental condition estimation in its simplest
7       version
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SYNOPSIS

10       SUBROUTINE DLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
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12           INTEGER        J, JOB
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14           DOUBLE         PRECISION C, GAMMA, S, SEST, SESTPR
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16           DOUBLE         PRECISION W( J ), X( J )
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PURPOSE

19       DLAIC1 applies one step of incremental condition estimation in its sim‐
20       plest version:
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22       Let  x,  twonorm(x) = 1, be an approximate singular vector of an j-by-j
23       lower triangular matrix L, such that
24                twonorm(L*x) = sest
25       Then DLAIC1 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.
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35       Depending on JOB, an estimate for  the  largest  or  smallest  singular
36       value is computed.
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38       Note that [s c]' and sestpr**2 is an eigenpair of the system
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40           diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
41                                                 [ gamma ]
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43       where  alpha =  x'*w.
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ARGUMENTS

47       JOB     (input) INTEGER
48               = 1: an estimate for the largest singular value is computed.
49               = 2: an estimate for the smallest singular value is computed.
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51       J       (input) INTEGER
52               Length of X and W
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54       X       (input) DOUBLE PRECISION array, dimension (J)
55               The j-vector x.
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57       SEST    (input) DOUBLE PRECISION
58               Estimated singular value of j by j matrix L
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60       W       (input) DOUBLE PRECISION array, dimension (J)
61               The j-vector w.
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63       GAMMA   (input) DOUBLE PRECISION
64               The diagonal element gamma.
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66       SESTPR  (output) DOUBLE PRECISION
67               Estimated singular value of (j+1) by (j+1) matrix Lhat.
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69       S       (output) DOUBLE PRECISION
70               Sine needed in forming xhat.
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72       C       (output) DOUBLE PRECISION
73               Cosine needed in forming xhat.
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77 LAPACK auxiliary routine (versionNo3v.e1m)ber 2006                       DLAIC1(1)
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