#---------------------------------------------------------------------
# Minimizes a multivariable objective function
# using steepest descent directions. Step length is
# determined using the R function optimize()
#
# Usage: steepest(obj, gradient, hessian , x0, eps)
#
# Required Arguments:
#    obj(x): returns a value of the objective function evaluated 
#            at a vector x
#    gradient(x): returns the derivative vector of the objective 
#            function evaluated at a vector x
#    hessian(x): returns the hessian matrix of the objective function
#            evaluated at a vector x
#    x0: initial vector
#    eps: tolerance
#-----------------------------------------------------------------------
steepest=function(obj, gradient, hessian , x0, maxiter=200, eps=.001)
{
	xi = x0
	cat(" Starting Values: ", x0, fill = T)
	i=0
	repeat {
		i = i + 1
		cat("\n Current Function Value: ", obj(xi), fill = T)
		gi = gradient(xi)
                gi.star=gi/sqrt(sum(gi^2))	
                fi=function(a,x,g,fun) fun(x - a*g)
                alpha=optimize(fi,c(0,1),lower=0,upper=1,maximum=F,tol=.0001,xi,gi.star,obj)[[1]][1]     
	        if(i>100)break
		xinew = xi - alpha*gi.star
		if( sqrt((sum((xinew - xi)^2))) < eps)break
		xi=xinew
		cat("\n Estimates at Iteration ",i , ":", xi, fill = T)
		} 
        if(i>maxiter) cat ("\n ****** No. of Iterations",maxiter," Exceeded: Terminated",fill = T) 
           else
	cat("\n Convergence Criterion of ",eps, " met ", fill = T)
	cat("\n Final Estimates Are: ",fill = T)
        cat("\n       ", signif(xinew,8), fill = T)
	cat("\n Final Function Value: ",fill = T)
        cat("\n       ",  signif(obj(xinew),12), fill = T)
	cat("\n Value of Gradient at Convergence:",fill = T)
	cat("\n       ", signif(gradient(xinew),8) , fill = T)	
	cat("\n Inverse of Negative Hessian",fill = T)
	cat(signif(solve(-1 *hessian(xinew)),8),fill = T)
}