Computing Sylvester resultant matrix in hermite polynomials

Abstract

There has been increasing interest in the theory of polynomials in different fields of science and engineering. Recent work has shown that enhanced numerical solution can be obtained via expressing polynomials in the orthogonal basis such as the Chebyshev, Legendre or Hermite basis. In some problems, such expression requires transforming resultant matrix between the monomial and the orthogonal or generalized basis. This dissertation concentrates on the possibility of constructing and implementing the Sylvester matrix in the Hermite basis as a computational tool in its orthogonal form. The transformation of the Sylvester resultant matrix between the monomial basis and the orthogonal basis is first studied. The multiplication formulas for some Hermite basis polynomials needed in the computation of the resultant matrix are first derived. Then the computation of the Sylvester resultant matrix in the Hermite basis and the representation of Hermite polynomials with Sylvester type determinants are carried out. The outcomes of this study proved that the Sylvester matrix resultant can be constructed and computed in the Hermite basis. In this form, the matrix can further be applied for working with polynomials in the Hermite basis. Thus. ill-conditioned conversion of polynomials from the orthogonal basis to the monomial basis can be avoided when the input polynomials are represented in the orthogonal basis, in particular the Hermite basis.