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§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

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

Cui Jun-zhi computing Center, Academia Sinica

Abstract:

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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