vpoly, vroots (Mu Analysis and Synthesis Toolbox)

Find coefficients and roots of a characteristic polynomial from a CONSTANT or VARYING matrix

Syntax

matout = vpoly(matin) 
vecout = vpoly(vecin) 
vecout = vroots(vecin)

Description

vpoly forms an n + 1 element VARYING row vector whose elements form the coefficients of the characteristic polynomial, det(sI - matin(i)), if matin is an
n x n VARYING matrix. The coefficients are ordered in descending powers of s. If the input is a column vector vecin containing the roots of a polynomial, vpoly(vecin) returns a VARYING row vector whose elements are the coefficients of the corresponding characteristic polynomial.

vroots returns as a VARYING column vector vecout whose elements are the roots of the polynomial at each independent variable, if vecin is a VARYING row vector containing the coefficients of a polynomial. vpoly and vroots are identical to the MATLAB poly and roots commands, but also work on VARYING matrices.

Examples

Given a 3 x 3 VARYING matrix, find the characteristic polynomial and its roots. Compare this to finding the eigenvalues of the input matrix via veig.

see(matin)
3 rows      3 columns
iv = 0.1
1   2   3
4   5   6
7   8   9
iv = 0.4
10   11   12
13   14   15
16   17   18
matout = vpoly(matin);
see(matout)
1 row      4 columns
iv = 0.1
l.0000e+00    -1.5000e+01    -1.8000e+01    -1.4483e-14
iv = 0.4
l.0000e+00    -4.2000e+01    -1.8000e+01     1.2818e-14
vecout = vroots(matout);
see(vecout)
3 rows       1 column
iv = 0.1
 1.6117e+01
-1.1168e+00
-8.0463e-16

iv = 0.4
 4.2424e+01
-4.2429e-01
7.1212e-16
evals = veig(matin);
see(evals)
3 rows        1 column

iv = 0.1
 1.6117e+01
-1.1168e+00
-8.0463e-16
iv = 0.4
 4.2424e+01
-4.2429e-01
 7.1212e-16

Algorithm

vpoly and vroots call the MATLAB poly and roots commands.