Create and Evaluate Polynomials

This example shows how to represent a polynomial as a vector in MATLAB® and evaluate the polynomial at points of interest.

Representing Polynomials

MATLAB® represents polynomials as row vectors containing coefficients ordered by descending powers. For example, the three-element vector

p = [p2 p1 p0];

represents the polynomial

$$p(x)=p_2{x}^2+p_{1}x+p_0.$$

Create a vector to represent the quadratic polynomial $$p(x)=x^2-4x+4$$.

p = [1 -4 4];

Intermediate terms of the polynomial that have a coefficient of 0 must also be entered into the vector, since the 0 acts as a placeholder for that particular power of x.

Create a vector to represent the polynomial $$p(x)=4x^5-3x^2+2x+33$$.

p = [4 0 0 -3 2 33];

Evaluating Polynomials

After entering the polynomial into MATLAB® as a vector, use the polyval function to evaluate the polynomial at a specific value.

Use polyval to evaluate $$p(2)$$.

polyval(p,2)

ans = 
153

Alternatively, you can evaluate a polynomial in a matrix sense using polyvalm. The polynomial expression in one variable, $$p(x)=4x^5-3x^2+2x+33$$, becomes the matrix expression

$$p(X)=4X^5-3X^2+2X+33I,$$

where X is a square matrix and I is the identity matrix.

Create a square matrix, X, and evaluate p at X.

X = [2 4 5; -1 0 3; 7 1 5];
Y = polyvalm(p,X)

Y = 3×3

      154392       78561      193065
       49001       24104       59692
      215378      111419      269614