Geometrically Isolated Nonisolated Solutions and Their Approximation
Geometrically Isolated Nonisolated Solutions and Their Approximation
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几何孤立的非孤立解及其逼近
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发表时间：
1981
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通讯作者：
H. Keller
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作者：
H. Keller
F（x）=0 $的解x = x^0 $称为“孤立的”，如果Frechet导数F '（x^0）$是非奇异的。如果对于某个$\rho > 0$，没有其他解在$\| {x-x ^0} \| \leqq \rho $中，则称之为“几何孤立的”。孤立解总是几何孤立的。得到了非孤立解也是几何孤立解的充分条件。然后我们研究的应用程序的近似方法，在一般形式$F_h（x_h）= 0$，近似几何孤立的非孤立的解决方案。在强相容性条件下，结果是有些负面的近似可能有偶数（包括零）或奇数根接近x^0 $，取决于“多重性”cf x^0 $作为根。如果精度为O（h^p）$，重数为N，则近似的误差为O（h^{p /N}）$。简要讨论了这些结果与极限点和分歧点的关系。
A solution $x = x^0 $ of $F(x)=0 $ is said to be “isolated” if the Frechet derivative $F'(x^0 )$ is nonsingular. It is said to be “geometrically isolated” if no other solution is in $\| {x - x^0 } \| \leqq \rho $ for some $\rho > 0$. Isolated solutions are always geometrically isolated. Sufficient conditions are obtained to insure that a nonisolated solution is also geometrically isolated. We then study the application of approximation methods, in the general form $F_h (x_h ) = 0$, to approximate nonisolated solutions which are geometrically isolated. Under strong consistency conditions the results are somewhat negative—the approximations may have an even number (including zero) or an odd number of roots near $x^0$, depending upon the “multiplicity” cf $ x^0 $ as a root. If the accuracy is $O(h^p )$ and the multiplicity is N, then the approximations have error $O(h^{p /N})$. The relation of these results to limit and bifurcation points is discussed briefly.