#  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$f(xcur)
  fs
  qfun$gr(xcur)
 #                  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)
 # done
   rm(list=ls())
   q()