Question

Use R and need the R code for every question.

3. Write a program to use Newton's Method to find the minimum of f(x1,x2)--log(1-ㄨㄧ-x2)-log(x1)-log(T2) Use starting value (0.1,0.1) (a) give the value of z1, 2 where the minimum occurs (b) give the minimum function value, f(xi, 2) (c) give the value of the tolerance level you use (d) give the number of iterations to convergence

Answer 1

write the equation as an R function:-

func3 <- function(x) {
  -1 * log(1-x1-x2)-log(x1)-log(x2)
}

Plot the function to visualize where the roots may exist:-

curve(func3, col = 'blue', lty = 2, lwd = 2, xlim=c(-5,5), ylim=c(-5,5), ylab='f(x)')
abline(h=0)
abline(v=0)

Now a graph will be made.Select the interval in which roots are present.Suppose the interval is 0.5 and 1.5.Then

uniroot(func3, c(.5, 1.5))

newton.raphson(func3, .5, 1.5)

The numDeriv package is used to compute the derivative f′(x)f′(x), though this could also be done by taking the limit with a sufficiently small h.

newton.raphson <- function(f, a, b, tol = 1e-5, n = 1000) {
  require(numDeriv) # Package for computing f'(x)
  
  x0 <- a # Set start value to supplied lower bound
  k <- n # Initialize for iteration results
  
  # Check the upper and lower bounds to see if approximations result in 0
  fa <- f(a)
  if (fa == 0.0) {
    return(a)
  }
  
  fb <- f(b)
  if (fb == 0.0) {
    return(b)
  }

  for (i in 1:n) {
    dx <- genD(func = f, x = x0)$D[1] # First-order derivative f'(x0)
    x1 <- x0 - (f(x0) / dx) # Calculate next value x1
    k[i] <- x1 # Store x1
    # Once the difference between x0 and x1 becomes sufficiently small, output the results.
    if (abs(x1 - x0) < tol) {
      root.approx <- tail(k, n=1)
      res <- list('root approximation' = root.approx, 'iterations' = k)
      return(res)
    }
    # If Newton-Raphson has not yet reached convergence set x1 as x0 and continue
    x0 <- x1
  }
  print('Too many iterations in method')
}

Description: at a minimum of a function, the derivative is zero.

Newton-Rhapson is a root-finding algorithm, and hence is well-suited to computing zeros of functions.

Edit: We need to find the minimum. The first step of this is to find points where ∂f/∂x1 and ∂f/∂x2 are both zero.

Now you have two conditions, so you want to compute:

g(x1,x2)=def⎛⎝⎜⎜∂f∂x1∂f∂x2⎞⎠⎟⎟=(00).g(x1,x2)=def(∂f∂x1∂f∂x2)=(00).

This gives you two equations with two unknowns. Now perform Newton-Raphson on gg.