When the roots are searched in the interval, the algebraic equation is
expanded by the orthogonal polynomials associated with the interval. The
roots are solved in the neighbor of the interval as the eigenvalues of
the generalized companion matrix which corresponds to the orthogonal
expansion. Similarly, the smooth nonlinear equation is approximated by
the truncated orthogonal expansion in the interval, and the approximated
roots of the nonlinear equation in the interval are solved by the
above generalized companion method as the roots of the expansion. The
condition of the set of orthogonal polynomials in the interval is much
better than that of monomials (xc)^k (which is also the basis for
Taylor expansion). Therefore, the calculated approximated roots are
expected to give smaller residuals with the restricted precision of the
floating point numbers. Some examples would be shown which solved
equations approximately by the applications of the orthogonal polynomial
expansion.
