The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze

Invariant
polynomials
The entries a1 ( s), a2 ( s), , ar ( s)
are monic polynomials such that a1 ( s)
divides
ai+1 ( s)
for i 1, 2, , r 1.
The Smith form is particularly useful for theoretical
considerations as it reveals many important properties of the matrix. Its practical
use, however is limited because it is quite sensitive to small parameter perturbations.
The computation of the Smith form becomes numerically troublesome as soon as the
matrix size and degree become larger. The Polynomial Toolbox offers a choice of three
different algorithms to achieve the Smith form, all programmed in macro smith.
For larger examples, a manual change of tolerance may be necessary. To compute the
Smith form of a simple matrix
A=[1+s, 0, s+s^2; 0, s+2, 2*s+s^2]
A =
simply call
smith(A)
ans =
1 + s 0 s + s^2
0 2 + s 2s + s^2
1 0 0
0 2 + 3s + s^2 0
The polynomials a1 ( s), a2 ( s), , ar ( s)
that appear in the Smith form are uniquely
determined and are called the invariant polynomials of A( s).
They may be retrieved
by typing
diag(smith(A))
ans =
1
2 + 3s + s^2
Polynomial matrix equations
Diophantine
equations
The simplest type of linear scalar polynomial equation — called Diophantine
equation after the Alexandrian mathematician Diophantos (A.D. 275) — is
a( s) x( s) b( s) y( s) c( s)
The polynomials a( s),
b( s)
and c( s)
are given while the polynomials x( s)
and y( s)
are
unknown. The equation is solvable if and only if the greatest common divisor of a( s)
and b( s)
divides c( s)
. This implies that with a( s)
and b( s)
coprime the equation is
solvable for any right hand side polynomial, including c( s)
1.
The Diophantine equation possesses infinitely many solutions whenever it is
solvable. If x( s),
y( s)
is any (particular) solution then the general solution of the
Diophantine equation is
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