一类非线性矩阵方程对称解的双迭代算法

利用逆矩阵的Neumann级数形式,将在Schur插值问题中遇到的含未知矩阵二次项之逆的非线性矩阵方程转化为高次多项式矩阵方程,然后采用牛顿算法求高次多项式矩阵方程的对称解,并采用修正共轭梯度法求由牛顿算法每一步迭代计算导出的线性矩阵方程的对称解或者对称最小二乘解,建立求非线性矩阵方程的对称解的双迭代算法.双迭代算法仅要求非线性矩阵方程有对称解,不要求它的对称解唯一,也不对它的系数矩阵做附加限定.数值算例表明,双迭代算法是有效的.

DOUBLE ITERATIVE ALGORITHM FOR SYMMETRIC SOLUTION OF A NONLINEAR MATRIX EQUATION

By using Neumann series of inverse matrix, nonlinear matrix equation with the inverse matrix of quadratic unknown matrix polynomial in the Schur's interpolation problem can be transformed into the high-order polynomial matrix equation. Then Newton's method is applied to find symmetric solution of the high-order polynomial matrix equation, and the modified conjugate gradient method is used to solve symmetric solution or symmetric least-square solution of linear matrix equation derived from each iterative step of Newton's method. In this way, a double iterative algorithm is established to find symmetric solution of nonlinear matrix equation. Nonlinear matrix equation is only required to have symmetric solution by double iterative algorithm, and the solution may not be unique. Besides, there are not additional limits to the coefficient matrix of the nonlinear matrix equation. Numerical experiments confirm that the double iterative algorithm is effective.