Maple Usage

The CalcP package (which must be called using the with(CalcP); command) contains an implementation of the Newton-Raphson method called Newton. This Newton command takes three arguments. The first is the function that is set equal to zero, next is the initial guess x₀, and third is the number of repetitions of Newton-Raphson.

For instance, to see the first 15 approximations to a root of x⁵-6x⁴+x³+2x²-5x+7=0 when x₀ is taken as -9, the syntax would be as follows.

  > with(CalcP);

  > f := x -> x^5-6*x^4+x^3+2*x^2-5*x+7;

  > Newton(f,x=-9,15);

This will display x₀ and f(x₀) and x_n and f(x_n) for n equal 1 to 15. By looking at the f(x_n), one can see if f(x_n) is tending to 0 as n increases.

If an expression is used as the first argument of Newton as in

  > Newton(x^7-5*x^3-3*x+7,x=2,10);

then each line of output will be of the form

x_i = . . . . . . , Expr(x_i) = . . . . . . .

.

In the CalcP package there is a routine called NewtonPlot which can help you visualize the action of the Newton-Raphson algorithm. You may find it useful in dealing with some of the exercises below. Basically, it plots the sequence of approximations to a root generated by Newton-Raphson. The syntax is as follows where a is the initial guess for a root.

  > NewtonPlot(f,x=a);