CONVERGENCE OF NEWTON'S METHOD FOR SOLVING A CLASS OF QUADRATIC MATRIX EQUATIONS

Abstract

We consider the most generalized quadratic matrix equation, Q(X) = <TEX>$A_7XA_6XA_5+A_4XA_3+A_2XA_1+A_0=0$</TEX>, where X is m <TEX>${\times}$</TEX> n, <TEX>$A_7$</TEX>, <TEX>$A_4$</TEX> and <TEX>$A_2$</TEX> are p <TEX>${\times}$</TEX> m, <TEX>$A_6$</TEX> is n <TEX>${\times}$</TEX> m, <TEX>$A_5$</TEX>, <TEX>$A_3$</TEX> and <TEX>$A_l$</TEX> are n <TEX>${\times}$</TEX> q and <TEX>$A_0$</TEX> is p <TEX>${\times}$</TEX> q matrices with complex elements. The convergence of Newton's method for solving some different types of quadratic matrix equations are considered and we show that the elementwise minimal positive solvents can be found by Newton's method with the zero starting matrices. We finally give numerical results.