Polynomial
Roots  'Zero finding' in Matlab
To find polynomial roots
(aka 'zero finding'
process), Matlab has a specific command, namely 'roots'.
A
polynomial is an expression of finite length built from variables and
constants, using only the operations of addition, subtraction,
multiplication, and nonnegative integer exponents.
If you type on the command window:
>>
help roots
the Matlab online help
will display:
ROOTS
Find polynomial roots.
ROOTS(C) computes the roots of the polynomial whose
coefficients are the elements of the
vector C. If C has N+1
components, the polynomial is C(1)*X^N +
... + C(N)*X +
C(N+1).
This means that the coefficients of the polynomial are placed in a
vector C and the builtin function returns the corresponding roots or zeros. It is simple
to use, but care is needed when entering the coefficients of the
polynomial.
For example, find the roots of the equation
f(x) =
3x^{5} − 2x^{4} + x^{3}
+ 2x^{2}
− 1
You can use the simple Matlab code:
c = [3
2 1 2 0 1];
roots(c)
And Matlab will deliver the following five zeros (xvalues that produce
a function f(x) = 0):
ans =
0.5911 + 0.9284i
0.5911  0.9284i
0.6234
0.5694 + 0.3423i
0.5694  0.3423i
There’s a couple of things to note from this example:
The polynomial’s
coefficients are listed starting with the one
corresponding to the largest
power. It is important that zerocoefficients are included
in the sequence where necessary. A polynomial of order p has p + 1 coefficients,
this is, a quadratic has three coefficients and a polynomial of degree p will have p roots.
As long as the coefficients of the polynomial are real, the roots will
be real or
occur in complex
conjugate pairs.
Builtin Command fzero
The Matlab command 'fzero'
is powerful. It actually chooses which internal method should be used
to solve a given problem, and can be used for nonpolynomical functions.
Let us try the form of the command
options
= optimset('disp','iter');
fzero('func',
3, options)
The command uses many different forms but here we are looking for a
root of the function defined in 'func.m'
near the value x
= 3 and the options are set to 'display
iterations'.
For example, determine the zero of the function f(x) =
3x − 2x^{2} sin(x), closest to x = 2.
A fast plot of the function (for exploration
purposes) results in:
So we need to set up the code:
function
mf = myfunction(x)
mf = 3*x
– 2*x.ˆ2.*sin(x);
and later we may use the inline command fzero like this (no
input vector is needed)
>>
fzero('myfunction', 2)
The Matlab answer is:
ans = 1.5034
More
operations with polynomials
From
'Polynomial Roots' to home
From
'Polynomial Roots' to 'Matlab Cookbook II'

