Prof. Kevin G. TeBeest - Newton's Method

Dr. Kevin G. TeBeest
Kettering University

NOTE: This is NOT the Maple code for Newton's method.
This is merely DEMONSTRATING what Newton's method is doing.

After starting Maple, you must first load Maple's "student" library of commands:
This is the case ONLY if you want to write this demo in a Maple session.

> with(student) :

Define the function f( x ) . As an example, we'll use function

> f := x −> evalf( cos( ln(x) ) ) ;
The evalf command automatically converts the results to floating point (decimal) form any time function f is evaluated.

Determine the derivative f '( x ) and store it AS A FUNCTION in fp (using the unapply command):

> fp := unapply( diff( f(x) , x ) , x ) ;

Plot function to determine roughly where its zero is. We'll plot it on the x interval [0,10] and restict the y range to [−1,1]:

> plot( f(x), x = 0.0 .. 10.0, −1.0 .. 1.0, thickness = 4 ) ;

We see that ONE zero of function f lies near x = 5. Let's suppose we want to approximate that zero of f.
Also, remember to STAY AWAY from the critical value near x = 1.5.

Although x = 5 would be an excellent starting value, to demonstrate Newton's method we'll use a starting value of x = 9.5

> x0 := 9.5 ;

FOR ILLUSTRATION PURPOSES ONLY, here we will use Maple's "showtangent" command to automatically plot function f (x) and its tangent line at x₀ = 9.5 :
To obtain a clean plot, we'll restrict the x interval (horizontal range) to [0,10] and the y interval (vertical range) to [−1,1].

> showtangent( f(x), x = x0, x = 0.0 .. 10.0, −1.0 .. 1.0, thickness = 4 ) ;

Now use Newton's method to calculate x₁, the first approximation to z :

> x1 := x0 − f(x0) / fp(x0) ;

FOR ILLUSTRATION PURPOSES ONLY, here we will use Maple's "showtangent" command to automatically plot function f (x) and its tangent line at x₁.
To obtain a clean plot, we'll restrict the x interval (horizontal range) to [0,10] and the y interval (vertical range) to [−1,1].

> showtangent( f(x), x = x1, x = 0.0 .. 10.0, −1.0 .. 1.0, thickness = 4 ) ;

Now use Newton's method to calculate x₂, the second approximation to z :

> x2 := x1 − f(x1) / fp(x1) ;

FOR ILLUSTRATION PURPOSES ONLY, here we will use Maple's "showtangent" command to automatically plot function f (x) and its tangent line at x₂.
To obtain a clean plot, we'll restrict the x interval (horizontal range) to [0,10] and the y interval (vertical range) to [−1,1], or we may "zoom in" by changing these ranges.

> showtangent( f(x), x = x2, x = 0.0 .. 10.0, −1.0 .. 1.0, thickness = 4 ) ;

Now use Newton's method to calculate x₃, the third approximation to z :

> x3 := x2 − f(x2) / fp(x2) ;

FOR ILLUSTRATION PURPOSES ONLY, here we will use Maple's "showtangent" command to automatically plot function f (x) and its tangent line at x₃.
To obtain a clean plot, we'll restrict the x interval to [0,10] and the y interval to [−1,1], or we may "zoom in" by changing these ranges.

> showtangent( f(x), x = x3, x = 0.0 .. 10.0, −1.0 .. 1.0, thickness = 4 ) ;

We may use Maple's "fsolve" command to approximate the zero automatically. We'll search on the interval [2,8].

> correct_zero := fsolve( f(x), x = 2.0 .. 8.0 ) ;

Return to Section 1.3 assignment

Return to course web site

Dr. Kevin G. TeBeest
Applied Mathematics
Kettering University