12.9: Finding Roots

A root of a function is an x-value where the function equals zero. Roots are also called zeros or x-intercepts. Root-finding is common in engineering because many problems can be written as equations of the form f(x) = 0.

Roots of Polynomials with roots

If the function is a polynomial represented by a coefficient vector, MATLAB can find its roots using the roots function. The basic format is:

r = roots(polynomialCoefficients);

Example \(\PageIndex{1}\)

finding the roots of \(f(x) = 4*x^3 - 2*x^2 - 8*x + 3\).

r = roots([4 -2 -8 3])

Solution

r =

-1.3660
1.5000
0.3660

Roots of Nonlinear Functions with fzero

Not every function is a polynomial. For nonlinear functions such as sine, cosine, exponential, or Bessel functions, MATLAB provides fzero.

The basic format is:

xRoot = fzero(functionHandle, startingGuess);

The second input can be a single starting guess or a two-value interval. If it is an interval, the function should change sign over that interval.

Example \(\PageIndex{2}\)

Finding the root of cos(2x) near x = 0.1.

myFun = @(x, c) cos(c*x);

c = 2;

xRoot = fzero(@(x) myFun(x, c), 0.1)

Solution

xRoot = 0.7854

Example \(\PageIndex{3}\)

Finding the root of the Bessel function of the first kind between the [2 3] interval.

anonFct = @(x) besselj(0, x);

xBracket = [2 3];

xRoot = fzero(anonFct, xBracket)

Solution

xRoot = 2.4048

Note

Common Mistake

fzero finds one root near the starting guess or inside the interval. If a function has multiple roots, you may need to try different starting guesses or intervals.