roots

Polynomial roots

Description

example

r = roots(p) returns the roots of the polynomial represented by p as a column vector. Input p is a vector containing n+1 polynomial coefficients, starting with the coefficient of xn. A coefficient of 0 indicates an intermediate power that is not present in the equation. For example, p = [3 2 -2] represents the polynomial $3{x}^{2}+2x-2$.

The roots function solves polynomial equations of the form ${p}_{1}{x}^{n}+...+{p}_{n}x+{p}_{n+1}=0$. Polynomial equations contain a single variable with nonnegative exponents.

Examples

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Solve the equation $3{x}^{2}-2x-4=0$.

Create a vector to represent the polynomial, then find the roots.

p = [3 -2 -4];
r = roots(p)
r = 2×1

1.5352
-0.8685

Solve the equation ${x}^{4}-1=0$.

Create a vector to represent the polynomial, then find the roots.

p = [1 0 0 0 -1];
r = roots(p)
r = 4×1 complex

-1.0000 + 0.0000i
0.0000 + 1.0000i
0.0000 - 1.0000i
1.0000 + 0.0000i

Input Arguments

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Polynomial coefficients, specified as a vector. For example, the vector [1 0 1] represents the polynomial ${x}^{2}+1$, and the vector [3.13 -2.21 5.99] represents the polynomial $3.13{x}^{2}-2.21x+5.99$.

Data Types: single | double
Complex Number Support: Yes

Tips

• Use the poly function to obtain a polynomial from its roots: p = poly(r). The poly function is the inverse of the roots function.

• Use the fzero function to find the roots of nonlinear equations. While the roots function works only with polynomials, the fzero function is more broadly applicable to different types of equations.

Algorithms

The roots function considers p to be a vector with n+1 elements representing the nth degree characteristic polynomial of an n-by-n matrix, A. The roots of the polynomial are calculated by computing the eigenvalues of the companion matrix, A.

A = diag(ones(n-1,1),-1);
A(1,:) = -p(2:n+1)./p(1);
r = eig(A)

The results produced are the exact eigenvalues of a matrix within roundoff error of the companion matrix, A. However, this does not mean that they are the exact roots of a polynomial whose coefficients are within roundoff error of those in p.