poly

Polynomial with specified roots or characteristic polynomial

Syntax

p = poly(r)

p = poly(A)

Description

p = poly(r), where r is a vector, returns the coefficients of the polynomial whose roots are the elements of r.

p = poly(A), where A is an n-by-n matrix, returns the n+1 coefficients of the characteristic polynomial of the matrix, det(λI – A).

Examples

Characteristic Polynomial from Eigenvalues

Calculate the eigenvalues of a matrix, A.

A = [1 8 -10; -4 2 4; -5 2 8]

A = 3×3

     1     8   -10
    -4     2     4
    -5     2     8

e = eig(A)

e = 3×1 complex

  11.6219 + 0.0000i
  -0.3110 + 2.6704i
  -0.3110 - 2.6704i

Since the eigenvalues in e are the roots of the characteristic polynomial of A, use poly to determine the characteristic polynomial from the values in e.

p = poly(e)

p = 1×4

    1.0000  -11.0000    0.0000  -84.0000

Characteristic Polynomial of Matrix

Use poly to calculate the characteristic polynomial of a matrix, A.

A = [1 2 3; 4 5 6; 7 8 0]

A = 3×3

     1     2     3
     4     5     6
     7     8     0

p = poly(A)

p = 1×4

    1.0000   -6.0000  -72.0000  -27.0000

Calculate the roots of p using roots. The roots of the characteristic polynomial are the eigenvalues of matrix A.

r = roots(p)

r = 3×1

   12.1229
   -5.7345
   -0.3884

Input Arguments

r — Polynomial roots

vector

Polynomial roots, specified as a vector.

Example: poly([2 -3])

Example: poly([2 -2 3 -3])

Example: poly(roots(k))

Example: poly(eig(A))

Data Types: single | double
Complex Number Support: Yes

A — Input matrix

matrix

Input matrix.

Example: poly([0 -1; 1 0])

Data Types: single | double
Complex Number Support: Yes

Output Arguments

p — Polynomial coefficients

row vector

Polynomial coefficients, returned as a row vector.

• If the input is a square n-by-n matrix, A, then p contains the coefficients for the characteristic polynomial of A.

• If the input is a vector of roots, r, then p contains the coefficients for the polynomial whose roots are in r.

In each case, the n+1 coefficients in p describe the polynomial

$$p_1{x}^n+p_2{x}^{n-1}+\mathrm{...}+p_{n}x+p_{n+1}.$$

Tips

• For vectors, r = roots(p) and p = poly(r) are inverse functions of each other, up to roundoff error, ordering, and scaling.

Algorithms

The algorithms employed for poly and roots illustrate an interesting aspect of the modern approach to eigenvalue computation. poly(A) generates the characteristic polynomial of A, and roots(poly(A)) finds the roots of that polynomial, which are the eigenvalues of A. But both poly and roots use eig, which is based on similarity transformations. The classical approach, which characterizes eigenvalues as roots of the characteristic polynomial, is actually reversed.

If A is an n-by-n matrix, poly(A) produces the coefficients p(1) through p(n+1), with p(1) = 1, in

$$\det \left(\lambda I-A\right)=p_1{\lambda }^n+\dots +p_n\lambda +p_{n+1}.$$

The algorithm is

z = eig(A);
p = zeros(n+1,1); 
p(1) = 1;
for j = 1:n
    p(2:j+1) = p(2:j+1)-z(j)*p(1:j);
end

This recursion is derived by expanding the product,

$$(\lambda -{\lambda }_1)(\lambda -{\lambda }_2)\dots (\lambda -{\lambda }_n).$$

It is possible to prove that poly(A) produces the coefficients in the characteristic polynomial of a matrix within roundoff error of A. This is true even if the eigenvalues of A are badly conditioned. The traditional algorithms for obtaining the characteristic polynomial do not use the eigenvalues, and do not have such satisfactory numerical properties.

Extended Capabilities

C/C++ Code Generation

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

Usage notes and limitations:

• Code generation does not discard nonfinite input values.

• See Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

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 poly function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Version History

Introduced before R2006a