Numerical Solution of Matrix Polynomial Equations by Newton's Method

Let P(X) = _v=1ⁿ A_vX^v with A_v, X C^m?m (v = 1, ..., n) be a matrixpolynomial. We present a Newton method to solve the equationP(X) = B, and we prove that the algorithm converges quadraticallynear simple solvents. We need the inverse of the Fr?chet-derivativeP' of P. This leads to linear equations for the correctionsH of type In the second part, we turn to the case of scalar coefficients, i.e. A_v = _vI, with_v C (v = 1, ..., n). The derivative P' and the usual algebraicderivative P' are compared and we show that the use of P' leadsto 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 existencetheorem is proved by using a modified Newton method.