roots

Polynomial roots

Syntax

r = roots(p)

Description

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

The roots function solves polynomial equations of the form $$p_1{x}^n+\mathrm{...}+p_{n}x+p_{n+1}=0$$. Polynomial equations contain a single variable with nonnegative exponents.

Examples

Roots of Quadratic Polynomial

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

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

Roots of Quartic Polynomial

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

p — Polynomial coefficients

vector

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 $$3.13x^2-2.21x+5.99$$.

For more information, see Create and Evaluate Polynomials.

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.

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• In the generated code, the output of the roots function is always complex.

• The order of the roots in output vector r might differ between the generated code and MATLAB^®.

• The roots returned by the generated code might not be exactly equal to those returned by MATLAB.

• For input argument p:

  • You must specify the input vector as a fixed-size or variable-length vector at code generation time. Either the first or the second dimension of the vector can be variable size. All other dimensions must have a fixed size of 1.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

The roots function supports GPU array input with these usage notes and limitations:

• The output r is always complex even if all the imaginary parts are zero.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a