The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze

Bézout
equations
Matrix
polynomial
equations
To investigate the case of different minimal degree solutions, consider a right hand
side of higher degree
c = 15+15*s^4;
As before, the call
[x1,y1] = axbyc(a,b,c)
x1 =
y1 =
8 - 13s + 15s^2
7 + 12s + 2s^2
results in the solution of minimal overall degree (in this case deg x1 deg y1
2 ).
A slightly different command
[x2,y2] = axbyc(a,b,c,'minx')
x2 =
y2 =
10.0000
5 - 5s - 15s^2 - 15s^3
returns another solution with the minimal degree of the first unknown. Finally,
typing
[x2,y2] = axbyc(a,b,c,'miny')
x2 =
y2 =
10 - 15s + 15s^2
5 + 10s
produces the solution of minimal degree in the second unknown.
Should the equation be unsolvable, the function returns NaNs..
[x,y] = axbyc(s,s,1)
x =
y =
NaN
NaN
A Diophantine equation with 1 on its right hand side is called a Bézout equation. It
may look like
a( s) x( s) b( s) y( s)
1
with a( s),
b( s)
given and x( s)
, y( s)
unknown.
In the matrix case, the polynomial equation becomes a polynomial matrix equation.
The basic matrix polynomial (or polynomial matrix) equations are
and
A( s) X ( s) B( s)
X ( s) A( s) B( s)
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