sylm

Arguments

a, b

Two polynomials with real or complex coefficients.

S

matrix of real or complex numbers

Description

sylm(a,b) gives the Sylvester matrix associated to polynomials a and b, i.e. the matrix S such that:

coeff( a*x + b*y )' = S * [coeff(x)';coeff(y)'].

Dimension of S is equal to degree(a)+degree(b).

If a and b are coprime polynomials then rank(sylm(a,b))=degree(a)+degree(b)) and the instructions

u = sylm(a,b) \ eye(na+nb,1)
x = poly(u(1:nb),'z','coeff')
y = poly(u(nb+1:na+nb),'z','coeff')

compute Bezout factors x and y of minimal degree such that a*x+b*y = 1

Examples

x = poly(0,"x");
y = poly ([1, 2, 3], "x", "coeff")
sylm(x, y)

--> x = poly(0, "x");
--> y = poly([1, 2, 3], "x","coeff")
 y  =
  1 +2x +3x²

--> sylm(x, y)
 ans  =
   0.   0.   1.
   1.   0.   2.
   0.   1.   3.

See Also

• bezout — GCD of two polynomials or two integers, by the Bezout method
• diophant — Solves the diophantine (Bezout) equation p1*x1 + p2*x2 = b