newtonf.html

Newton-Raphson Method

This work sheet features two approaches. The first one introduces loops in Maple and shows how they can be used with Newton's method to find solutions to equations. The second approach uses an animation to demonstrate Newton's method visually.

EXAMPLE 1 (numerical approach)

Assume you want to solve 11 - x^2 = 0. Use the Enter Expression button on MenuMaple and select the function option . Enter f as the name of the function, x as the independent variable, and 11 - x^2 as the expression.

> f:=x->11-x^2;

From the calculus menu of MenuMaple find the derivative of f with respect to x and call the result df

> df:=D(f);

We will use an initial value of 1 for xn in Newton's method.

> xn:=1;

The following loop will perform 7 iterations of Newton's method. Each time the new value of xn is based on the old value of xn by means of xn - f(xn)/f '(xn). To insure that decimal values are obtained evalf is used. To get the loop to execute place the cursor any where inside the red commands and press the enter key.

> for i from 1 to 7 do
> xn:=evalf(xn-f(xn)/df(xn));
> od;

Change the first value of xn to -1 and re-execute the loop. What do you get now?

> xn:=-1;

> for i from 1 to 7 do
> xn:=evalf(xn-f(xn)/df(xn));
> od;

This next command loads the package math191 which contains a custom Maple command that will create an animation (of the first four steps) of Newton's method.

> with(math191):
> newtonraphson(f(x),x=1);

EXAMPLE 2

Make changes in example 1 so that you can find the solution to
2*x - 5 - sin(x) = 0. You'll need to modify the expression used in f, find its derivative, perhaps change the starting value of xn, and then re-execute the loop.

> f:=x->2*x-5-sin(x);

> df:=D(f);

> xn:=0;

> for i from 1 to 7 do
> xn:=evalf(xn-f(xn)/df(xn));
> od;

> newtonraphson(f(x),x=0);

EXAMPLE 3

Make changes in example 1 so that you can try to find the solution to
x^(1/3)=0 by Newton's method. In order to be sure of getting real values use surd(x,3) in place of x^(1/3). What happens?

> f:=x->surd(x,3);