Newton's Method Tutorial
In this tutorial we will explore Newton's method for finding the roots
of equations, as explained in Chapter 14.
Simulation
We will be using a Newton's method simulator throughout this
tutorial. You can start it by clicking on the following button.
Finding Roots
This tutorial explores a numerical method for finding the root of an
equation: Newton's method. Newton's method is discussed in Chapter 14
as a way to solve equations in one unknown that cannot be solved
symbolically.
For example, suppose that we would like to solve the simple equation
2
x = 5
To solve this equation using Newton's method, we first manipulate it
algebraically so that one side is zero.
2
x - 5 = 0
Finding a solution to this equation is then equivalent to finding a
root of the function
2
f(x) = x - 5
This function is plotted in the simulation window.
We next make a guess for the root. In the simulation window, the
guess is -5. The point
(guess, f(guess))
is displayed with a pink dot. The coordinates of the dot are
displayed at the bottom of the simulation window.
The yellow line is tangent (to the curve whose root we are seeking) at
the pink dot. Newton's method relies on the observation that this
tangent line will often cross the x-axis at a point closer to a root
than is the guess.
To see Newton's method in action, click on the button labeled "Step".
The pink dot will slide down the tangent line until it reaches the
x-axis, and it will then move vertically until it reaches the curve.
A new tangent line will be displayed. The new x-coordinate of the
pink dot is the new guess to the root of the function. For many
functions and for many initial guesses, repeating this process a few
times will yield an excellent approximation to a root.
If you click on Step a few more times, the pink dot will move closer
to the point where the curve crosses the x-axis. (You can zoom in by
using the mouse to drag a rectangle around the region that you'd like
to enlarge. There is also a "Zoom" menu in the menu bar.) At any
point of the simulation, the x-coordinate of the guess will be an
approximation to the root, and the y-coordinate of the guess will be
the value of the function at that guess.
Exercises
- You can place the guess by clicking the mouse where you would
like the guess to go. Experiment with placing the guess and observing
convergence.
- If you place the guess at -5, how many steps are required until
the approximate root is good to three decimal places? (The root near
-2 is -2.236067978 to ten digits.) How this compare to the behavior
of the bisection method when the positive guess is at -5 and the
negative guess is at 1? (You can switch to the bisection method by
using the Method menu.)
- Use the "Function" menu to display the curve for cos(x). Notice
that four different roots are displayed. What guesses lead to
which roots? (Be sure to watch the coordinates of the guess. You may
need to zoom out to see where the guess is.)
- It can be difficult to predict exactly how Newton's method will
behave. Use the "Function" menu to display the curve for cos(10x) +
4x. Watch what happens if you start from the guess that is displayed
by default.
- The function x^2 + 1 has no root. Experiment with how Newton's
method behaves with it.
- Choose the function sin(5x) + x^2 - 3x and place the guess so
between 1.2 and 2.0 so that the tangent line crosses the x-axis
between 1.0 and 2.0. Step through Newton's method several times.
What do you notice? What does this say about finding the roots of
this function with Newton's method?
- Choose the function ln(x^2 - (4/5)x + 1) and zoom out by 5.
There are 2 points on the function which, if you choose your first
guess between these two points (excluding the function minimum and
it's very close proximity), Newton's method will converge and if you
choose your first guess outside these points Newton's method will
diverge. What are these two points?
- Experiment with some of the other functions to get a feel for how
Newton's method works and for how many steps it takes for it to come
up with a good approximation.
Last modified 19Nov96.