polyvalm

Matrix polynomial evaluation

Syntax

Y = polyvalm(p,X)

Description

Y = polyvalm(p,X) returns the evaluation of polynomial p in a matrix sense. This evaluation is the same as substituting matrix X in the polynomial, p.

Examples

Matrix Evaluation of Characteristic Polynomial

Find the characteristic polynomial of a Pascal Matrix of order 4.

X =  pascal(4)

X = 4×4

     1     1     1     1
     1     2     3     4
     1     3     6    10
     1     4    10    20

p = poly(X)

p = 1×5

    1.0000  -29.0000   72.0000  -29.0000    1.0000

The characteristic polynomial is

$$p(x)=x^4-29x^3+72x^2-29x+1$$

Pascal matrices have the property that the vector of coefficients of the characteristic polynomial is the same forward and backward (palindromic).

Substitute the matrix, X, into the characteristic equation, p. The result is very close to being a zero matrix. This example is an instance of the Cayley-Hamilton theorem, where a matrix satisfies its own characteristic equation.

Y = polyvalm(p,X)

Y = 4×4
10-10 ×

   -0.0014   -0.0064   -0.0105   -0.0242
   -0.0049   -0.0220   -0.0362   -0.0801
   -0.0116   -0.0514   -0.0827   -0.1819
   -0.0230   -0.0976   -0.1567   -0.3424

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

X — Input matrix

square matrix

Input matrix, specified as a square matrix.

Data Types: single | double
Complex Number Support: Yes

Output Arguments

Y — Output polynomial coefficients

row vector

Output polynomial coefficients, returned as a row vector.

Extended Capabilities

C/C++ Code Generation
Generate C and C++ code using 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 polyvalm 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).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a