Page 80 - Numerical Analysis Using MATLAB and Excel

Summary

2.4 Summary

• Newton’s (or Newton−Raphson) method can be used to approximate the roots of any linear or
non−linear equation of any degree. It uses the formula

fx ) (
n
x n + 1 = x – --------------- )
n
f ' x (
n
To apply Newton’s method, we must begin with a reasonable approximation of the root value.
In all cases, this can best be done by plotting fx() versus . x
• We can use a spreadsheet to approximate the real roots of linear and non−linear equations but
to approximate all roots (real and complex conjugates) it is advisable to use MATLAB.
• The MATLAB the while end loop evaluates a group of statements an indefinite number of
times and thus can be effectively used for root approximation.

• For approximating real roots we can use Excel’s Goal Seek feature. We use Goal Seek when
we know the desired result of a single formula, but we do not know the input value which sat-
isfies that result. Thus, if we have the function y = f x() , we can use Goal Seek to set the
dependent variable to the desired value (goal) and from it, find the value of the indepen-
y
dent variable which satisfies that goal.
x
• For repetitive tasks, such as finding the roots of polynomials, it is prudent to construct a tem-
plate (model spreadsheet) with the appropriate formulas and then enter the coefficients of the
polynomial to find its real roots.
• The Bisection (or interval halving) method is an algorithm for locating the real roots of a
function. The objective is to find two values of x, say x 1 and x 2 , so that fx ) ( 1 and fx ) ( 2 have

opposite signs, that is, either fx ( 1 ) > 0 and fx ) ( 2 < 0 , or fx ) ( 1 < 0 and fx ) ( 2 > 0 . If any of these
two conditions is satisfied, we can compute the midpoint x of the interval x ≤ 1 x ≤ x 2 with
m
x + x
1
x m = ----------------- 2
2
• We can use the Bisection Method with MATLAB to approximate one of the roots by specify-
ing a number of iterations using a for end or by specifying a tolerance using a while end loop
program.

• We can use an Excel spreadsheet to construct a template that approximates a real root of a
function with the bisection method. This requires repeated use of the IF function which has
the =IF(logical_test,value_if_true,value_if_false)

Numerical Analysis Using MATLAB® and Excel®, Third Edition 2−27
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