R version 2.9.0 (2009-04-17)
Copyright (C) 2009 The R Foundation for Statistical Computing
ISBN 3-900051-07-0

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>  #  chex84.r
>  # Example 8.4 -- zig-zag behavior of steepest descent search
>  #
>  ##########################
>  # create function to be minimized (& gradient)
>   mkqfun <- function(A,b,c) {
+     list( f = function(x){ t(x)%*%A%*%x+t(b)%*%x+c},   # function
+           gr = function(x) as.vector( 2*A%*%x + b ) ) }         # gradient
>  #
>  # create functions
>   qfun <- mkqfun(matrix(c(1,0,0,.5),2,2),c(0,0),0)
>  #
>  # start here and then loop
>   xcur <- c(1,2)
>   x1s <- 1
>   x2s <- 1
>   fs <- qfun$f(xcur)
>   fs
     [,1]
[1,]    3
>   qfun$gr(xcur)
[1] 2 2
>  #                  start loop
>   for (i in(1:10) ) {
+     grad <- qfun$gr(xcur)     # gradient at current
+     lfun <- function(u) qfun$f(xcur-u*grad)       # function on line
+     ym1 <- lfun(-1)
+     y0 <- lfun(0)
+     yp1 <- lfun(+1)
+     ustar <- - (yp1-ym1)/(2*(yp1-2*y0+ym1))       # best point on line 
+     xcur <- xcur - ustar*grad                     # new point
+     x1s <- c(x1s,xcur[1])
+     x2s <- c(x2s,xcur[2])
+     fs <- c(fs,qfun$f(xcur))  }                   # new f & end loop
>   cbind(x1s,x2s,fs)
                x1s          x2s           fs
 [1,]  1.000000e+00 1.000000e+00 3.000000e+00
 [2,] -3.333333e-01 6.666667e-01 3.333333e-01
 [3,]  1.111111e-01 2.222222e-01 3.703704e-02
 [4,] -3.703704e-02 7.407407e-02 4.115226e-03
 [5,]  1.234568e-02 2.469136e-02 4.572474e-04
 [6,] -4.115226e-03 8.230453e-03 5.080526e-05
 [7,]  1.371742e-03 2.743484e-03 5.645029e-06
 [8,] -4.572474e-04 9.144947e-04 6.272255e-07
 [9,]  1.524158e-04 3.048316e-04 6.969172e-08
[10,] -5.080526e-05 1.016105e-04 7.743524e-09
[11,]  1.693509e-05 3.387018e-05 8.603916e-10
>  # done
>    rm(list=ls())
>    q()