1SLAIC1(1)           LAPACK auxiliary routine (version 3.2)           SLAIC1(1)
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NAME

6       SLAIC1  -  applies  one step of incremental condition estimation in its
7       simplest version
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SYNOPSIS

10       SUBROUTINE SLAIC1( JOB, J, X, SEST, W, GAMMA, SESTPR, S, C )
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12           INTEGER        J, JOB
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14           REAL           C, GAMMA, S, SEST, SESTPR
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16           REAL           W( J ), X( J )
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PURPOSE

19       SLAIC1 applies one step of incremental condition estimation in its sim‐
20       plest version: Let x, twonorm(x) = 1, be an approximate singular vector
21       of an j-by-j lower triangular matrix L, such that
22                twonorm(L*x) = sest
23       Then SLAIC1 computes sestpr, s, c such that
24       the vector
25                       [ s*x ]
26                xhat = [  c  ]
27       is an approximate singular vector of
28                       [ L     0  ]
29                Lhat = [ w' gamma ]
30       in the sense that
31                twonorm(Lhat*xhat) = sestpr.
32       Depending on JOB, an estimate for  the  largest  or  smallest  singular
33       value is computed.
34       Note that [s c]' and sestpr**2 is an eigenpair of the system
35           diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]
36                                                 [ gamma ]
37       where  alpha =  x'*w.
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ARGUMENTS

40       JOB     (input) INTEGER
41               = 1: an estimate for the largest singular value is computed.
42               = 2: an estimate for the smallest singular value is computed.
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44       J       (input) INTEGER
45               Length of X and W
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47       X       (input) REAL array, dimension (J)
48               The j-vector x.
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50       SEST    (input) REAL
51               Estimated singular value of j by j matrix L
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53       W       (input) REAL array, dimension (J)
54               The j-vector w.
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56       GAMMA   (input) REAL
57               The diagonal element gamma.
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59       SESTPR  (output) REAL
60               Estimated singular value of (j+1) by (j+1) matrix Lhat.
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62       S       (output) REAL
63               Sine needed in forming xhat.
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65       C       (output) REAL
66               Cosine needed in forming xhat.
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70 LAPACK auxiliary routine (versionNo3v.e2m)ber 2008                       SLAIC1(1)
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