对称矩阵分解因子的直接换元修正法

DIRECT EXCHANGE ENTRY UPDATES OF SYMMETRIC MATRIX FACTORIZATIONS

§1.引言 考虑非线性方程组 F(x)=0, (1)其中F:Ω?R^n→R^n使F′(x)对称.本文给出求解(1)的一种分解修正法,这种方法始于Jacobian F′(x)的初始对称三角分解,然后利用换元技巧直接修正上三角分解因子,进而前代与回代求迭代点.本文分析了分解修正法的运算量,证明了这个算法不用重新启动仍具有局部超线性收敛性和大范围收敛性.此外,这个算法自然保持分解因子的稀疏传递性和修正矩阵的对称传递性,特别当Jacobian正定时,还具有正定传递性.由此本文完成了[1]和[2]无法完成的工作.本算法特别适于大规模带状方程组和最优化问题,数值例子也表明了这一点.

In this paper, a new method for solving nonlinear simultaneous equations is proposed.This method employs an initial symmetric factorization of the Jacobian matrix, and then up-dates the upper triangular factors directly at each step. Iterations are generated using forwardand backward substitutions employing the updates factorizations. It is shown that this me-thod takes the principal advantage of the quasi-Newton method, and is globally convergent. Inaddition, it has the property of hereditary symmetry and sparsity. It can be applied to the lar-ge scale problems and the optimization problems. Numerical results show that the method iseffective.