Numerical Solution of Matrix Polynomial Equations by Newton's Method

Abstract

Let P(X) = Σv=1nAvXv with Av, X ɛ Cm×m (v = 1, …, n) be a matrix polynomial. We present a Newton method to solve the equation P(X) = B, and we prove that the algorithm converges quadratically near simple solvents. We need the inverse of the Fréchet-derivative P′ of P. This leads to linear equations for the corrections H of type ∑v=1nA˜vHB¯v=C In the second part, we turn to the case of scalar coefficients, i.e. Av = αvI, with αv ε C (v = 1, …, n). The derivative P′ and the usual algebraic derivative P′ are compared and we show that the use of P′ leads to difficulties. In particular, those algorithms based on P′ are not self-correcting, while our proposed method is self-correcting. Numerical examples are included. In the Appendix, an existence theorem is proved by using a modified Newton method.