• 论文 •

### 高阶稀疏局部非线性方程组的一种拟牛顿方法

1. 中国科学院计算中心
• 出版日期:1982-04-14 发布日期:1982-04-14

### ONE QUASI-NEWTON METHOD ON LARGE SPARSE AND LOCALLY NONLINEAR EQUATIONS

1. Cui Jun-zhi computing Center, Academia Sinica
• Online:1982-04-14 Published:1982-04-14
§1.引言 当用有限元法或有限差分法分析非线性偏微分方程问题时,必然会导致求解非线性方程组的问题,即求 F(x)=0 (1.1)的解.其中,x=(x_1,x_2,…,Xx_n)~T∈D,D?R~n;F:D→R~n是一个非线性映射.因此,有效地求解非线性方程组(1.1),是分析相应的非线性问题的关键. 不管这些非线性问题是来自流体力学、固体力学,还是其他的物理范畴,它们所对应
First, an exact definition of sparsity and local nonlinearity for large nonlinear equations isgiven. Let ?~((1)) = {λ_(ij)~((1))} and ?~((2)) = {λ_(ij)~((2))}, where λ_(ij)~((1)) = 1 if x_j appears in f_i(x), 0 otherwise, λ_(ij)~((1)) = 1 if x_j appears as nonlinear term in f_i(x), 0 otherwise.Furthermore, E~((2)) = {(i, j)|λ_(ij)~((2)) = 1 } is defined. The quasi-newton scheme is well known x~((l+1)) = x~((l)) + αp, K~((l))p = -- F(x~((1))), l = 0, 1, 2,…,where K~((l+1)) = K~((l)) + M~((l)), M~((l))αp = r, r = F(x~((l+1)) M (α -- 1)F(x~((l))).We suppose that M~((l)) has the following form Then α_(ii)αp_i + α_(ij)αp_j = ξ_(ij)r_i, α_(ij)αp_i + α_(ij)αp_j = ξ_(ij)r_j.Minimizing under constraints ∑ j ξ_(ij) = 1, we can obtain {ξ_(ij)}, and theh A_(ij). The practical example shows that the above method has good efficiency, especially, for∑λ_(ij)~((2))<<∑λ_(ij)~((1)) << n~2.
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 [1] J. E. Dennis, J. More, Quasi-Newton methods motivation and theory, SIAM Review, 19, 1977, 46-89． [2] Mordecoi Avriel, Nonlinear Programming. Analysis and methods. [3] L. K. Schubert, Modification of a quasi-Newton method for nonlinear equations with a sparse Jacobian, Math. Comp., 24, 1970, 27-30． [4] Bim Lam, On the convergence of a quasi-Newton methods for sparse nonlinear systems, Math. Comp., 32, 1978, 447-451． [5] Toint Ph. L., Some numerical results using a sparse matrix updating formula in unconstrained optimization, Math. Comp., 32, 1978, 839-851． [6] 赵瑞安,大型稀疏非线性方程组解法述评,1979． [7] R.S.瓦格著,蒋尔雄,游兆永,张玉德译,矩阵迭代分析,上海科学技术出版社,1966． [8] D. Greenspan, M. Yohe, On the approximate solution of △u-F(u), Comm. ACM, 6, 1963, 564-568．
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