Application of filter diagonalization method to numerical solution of algebraic equations

Abstract

By the use of symbolic computation, a problem given by a set of multivariate algebraic relations is often reduced to a univariate algebraic equation which is quite high degree. And, if the roots are required in numbers we generally have to solve the higher degree algebraic equation by some iterative method. In this paper, an application of the filter diagonalization method (FDM) [5] is studied to solve the higher degree univariate algebraic equation of numerical coefficients when only a small portion of roots are required which are near the specified location in the complex plane or near the specified interval. Recently, FDM has been developing as the technique to solve a small portion of eigenpairs of a matrix selectively depending on their eigenvalues.By the companion method, roots of an algebraic equation of higher degree N are solved as eigenvalues of companion matrix A after the balancing is made. Usually all eigenvalues can be solved by the method of shifted QR iteration. The amount of computation of the ordinal shifted QR iteration which does not use the special non-zero structure of the Frobenius companion matrix of degree N is O(N3). In the paper [1], it was shown that the amount of computation to solve all eigenvalues of size N Frobenius companion matrix by the special QR iteration which uses the structure is O(N2).In this paper, we assume that not all but only a small portion of roots are required. To reduce the elapsed time, the inverse iteration (Rayleigh-quotient iteration) will be used in parallel. In the inverse iteration, the linear equation of shifted matrix A-ρI is solved. (For the case of a univariate algebraic equation of N-th degree, A is the Frobenius companion (or its balanced matrix). By the use of the special sparse structure of the shifted matrix in LU-decomposition or QR-decomposition, the complexity of the solution of the linear equation is O(N) in both arithmetics and space.) By the use of a well-tuned filter which is a linear combination of resolvents, FDM gives well approximated eigenpairs whose eigenvalues are near the specified location in the complex plane. From the approximated eigenpairs as initial values, inverse-iteration method quickly improves eigenpairs in a few iterations.