SLAIC1 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


          twonorm(L*x) = sest


 Then SLAIC1 computes sestpr, s, c such that


 the vector


                 [ s*x ]


          xhat = [  c  ]


 is an approximate singular vector of


                 [ L      0  ]


          Lhat = [ w**T gamma ]


 in the sense that


          twonorm(Lhat*xhat) = sestpr.


 Depending on JOB, an estimate for the largest or smallest singular


 value is computed.


 Note that [s c]**T and sestpr**2 is an eigenpair of the system


     diag(sest*sest, 0) + [alpha  gamma] * [ alpha ]


                                           [ gamma ]


 where  alpha =  x**T*w.

JOB

          JOB is INTEGER


          = 1: an estimate for the largest singular value is computed.


          = 2: an estimate for the smallest singular value is computed.

J

          J is INTEGER


          Length of X and W

X

          X is REAL array, dimension (J)


          The j-vector x.

SEST

          SEST is REAL


          Estimated singular value of j by j matrix L

W

          W is REAL array, dimension (J)


          The j-vector w.

GAMMA

          GAMMA is REAL


          The diagonal element gamma.

SESTPR

          SESTPR is REAL


          Estimated singular value of (j+1) by (j+1) matrix Lhat.

S

          S is REAL


          Sine needed in forming xhat.

C

          C is REAL


          Cosine needed in forming xhat.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016