poly - Polynomial with specified roots or characteristic polynomial - MATLAB (original) (raw)

Polynomial with specified roots or characteristic polynomial

Syntax

Description

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

example

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

example

Examples

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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 = 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 = 1×4

1.0000  -11.0000   -0.0000  -84.0000

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 = 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 = 3×1

12.1229 -5.7345 -0.3884

Input Arguments

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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

Input matrix.

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

Data Types: single | double
Complex Number Support: Yes

Output Arguments

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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

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

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,

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

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Usage notes and limitations:

The input argument must not contain Inf orNaN values.
If the input to poly is complex, the output will be complex as well, even in cases where the polynomial coefficient has only zero-valued imaginary parts
If the input argument is a variable-size matrix at code generation time, this argument cannot be a vector at run time. To generate code for vector inputs, specify a fixed-length vector at code generation time.

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