Division of generalized polynomials using the comrade matrix
A polynomial in the generalized form is an expression \(a(x)=p_ n(x)+a_ 1p_{n-1}(x)+...+a_ np_ 0(x),\) where \(a_ i's\) are real numbers and \(\{p_ i(x)\}\) is an orthogonal polynomial basis defined by \(p_ 0(x)=1,\) \(p_ 1(x)=u_ 1x+v_ 1,\) \(p_ i(x)=(u_ ix+v_ i)p_{i-1}(x)-w_ ip_{i-2}(x),\) \(i=2,3,...\), with \(u_ i>0\), \(w_ i\geq 0\). The author shows that the greatest common divisor d(x) of polynomials a(x) and f(x) in the generalized form, the quotient a(x)/d(x), and the quotient and remainder on division of a(x) by f(x) can be determined simultaneously, without any conversion into standard power form. This is done by applying elementary row operations to the matrix f(A), where A is the comrade matrix associated with a(x). The notation and terminology follow that of the author in ibid. 41, 277-298 (1981; Zbl 0479.15012).
scientific article; zbMATH DE number 3879072
scientific article