Page 74 - Numerical Analysis Using MATLAB and Excel

The Bisection Method for Root Approximation

We will illustrate the Bisection Method with examples using both MATLAB and Excel.

Example 2.7
Use the Bisection Method with MATLAB to approximate one of the roots of

5
3
y = f x() = 3x – 2x + 6x – 8 (2.19)
by
a. by specifying 16 iterations, and using a for end loop MATLAB program

b. by specifying 0.00001 tolerance for fx() , and using a while end loop MATLAB program
Solution:

This is the same polynomial as in Example 2.4.
a. The for end loop allows a group of functions to be repeated a fixed and predetermined num-
ber of times. The syntax is:

for x = array
commands...
end
Before we write the program script, we must define a function assigned to the given polyno-
mial and save it as a function m−file. We will define this function as funcbisect01 and will save
it as funcbisect01.m.

function y= funcbisect01(x);
y = 3 .* x .^ 5 − 2 .* x .^ 3 + 6 .* x − 8;
% We must not forget to type the semicolon at the end of the line above;
% otherwise our script will fill the screen with values of y
,
,
,
On the script below, the statement for k = 1:16 says for k = 1 k = 2 … k = 16 , evaluate all
commands down to the end command. After the k = 16 iteration, the loop ends and any
commands after the end are computed and displayed as commanded.
Let us also review the meaning of the fprintf('%9.6f %13.6f \n', xm,fm) line. Here, %9.6f and
%13.6f are referred to as format specifiers or format scripts; the first specifies that the value of
xm must be expressed in decimal format also called fixed point format, with a total of 9 digits, 6
of which will be to the right of the decimal point. Likewise, fm must be expressed in decimal
format with a total of 13 digits, 6 of which will be to the right of the decimal point. Some other
specifiers are %e for scientific format, %s for string format, and %d for integer format. For
more information, we can type help fprintf. The special format \n specifies a linefeed, that is, it
prints everything specified up to that point and starts a new line. We will discuss other special
formats as they appear in subsequent examples.

Numerical Analysis Using MATLAB® and Excel®, Third Edition 2−21
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