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Unformatted text preview: returns a symbolic answer. In this case, we are looking for a numerical (decimal) approximation to the
solution for f (x) = 0. So we will use Maple’s evalf command.
[> evalf( Newton(1) );
Maple tells us that the next approximation for c is x2 = 1.059167373. Let’s see if f (1.059167373) is close to zero?
Indeed, x2 = 1.059167373 is an even better approximation for f (x) = 0 than was x1 = 1. Let’s see if we can do better by
doing another iteration of Newton’s Method. This time we must calculate x3 = x2 − f (x2 )
f (x2 ) But we already know that x2 = 1.059167373 and we have Newton, so we don’t have to type this formula in again.
[> evalf( Newton(1.059167373) );
Maple tells us that the next approximation for c is x3 = 1.058006881. Let’s see if f (1.058006881) is even closer to zero?
Maple tells us that when x3 = 1.058006881 is used as an approximation for c, we get f (1.058006881) = −0.832 × 10−6 !!!
This means that f (1.058006881) = 0.000000832 which is very close to 0! We can stop now as we have just shown that
x3 = 1.058006881 is a very good approximation for one solution of f (x) = 0.
In real-world applications, the functions that are used are much more complicated and Newton’s Method is a technique
that scientists can use to quickly get solutions that have acceptable errors.
As we did with the Bisection Method, we can also use Maple’s programming facilities to write a simple routine that contains
a loop to calculate the Newton Method iterates.
Let’s repeat the previous example using a Maple loop for Newton’s Method. Try entering the following lines in Maple (you
do not need to type the comments). Note: As before, Maple will give you the comment Warning, premature end of input
until you type in all of the lines.
[> f := x -> ln(4 - x∧2) - x; deﬁne the function [> x := 1; initial guess to estimate c [> k := 10; number of times you want to iterate Newton’s Method [> for n from 1 to k do do k iterations of the Newton’s Method [> x := evalf( x - f(x)/D(f)(x) ); calculate the next iterate [> end do;
Notice that after the third iteration of Newton’s Method, all of the values are the SAME! This means that three iterations
of Newton’s Method gives an es...
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