Existence verification for singular and nonsmooth zeros of real nonlinear systems
Existence verification for singular and nonsmooth zeros of real nonlinear systems
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DOI：
10.1090/s0025-5718-02-01427-8
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发表时间：
2003-04
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影响因子：
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通讯作者：
Jianwei Dian;R. Baker Kearfott
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文献类型：
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作者：
Jianwei Dian;R. Baker Kearfott
传统的计算不动点定理，如康托洛维奇定理（通过有向多项式严格化）、克劳奇克方法或区间牛顿方法，使用计算机的浮点硬件计算来数学地证明在给定的n空间区域内非线性方程组的解的存在性和唯一性。这样的计算要求系统的雅可比矩阵在解的邻域内是非奇异的。然而，在以前的工作中，我们展示了如何在数学上验证奇异解的存在性，在一个小区域的复杂的n-空间包含一个近似的真实的解决方案。我们通过验证一个小区域的拓扑度是非零的，验证了这种奇异解的存在性;非零拓扑度意味着该区域内部存在一个解。本文证明了当复空间中的实际拓扑度为奇数且Jacobi矩阵的秩亏为1时，在真实的空间中，包含奇异解的小区域的拓扑度可被证明为正或负1.用于在真实的空间中验证的算法明显更简单和更有效。我们证明了这种效率与数值experiments.Since我们的验证过程中只使用的边界框，包含解决方案的表面上的值，该方法也可以适用于系统是非光滑的解决方案的情况下。
Traditional computational fixed point theorems, such as the Kantorovich theorem (made rigorous with directed roundings), Krawczyk's method, or interval Newton methods use a computer's floating-point hardware computations to mathematically prove existence and uniqueness of a solution to a nonlinear system of equations within a given region of n-space. Such computations require the Jacobi matrix of the system to be nonsingular in a neighborhood of a solution. However, in previous work we showed how we could mathematically verify existence of singular solutions in a small region of complex n-space containing an approximate real solution. We verified existence of such singular solutions by verifying that the topological degree of a small region is nonzero; a nonzero topological degree implies existence of a solution in the interior of the region. Here, we show that, when the actual topological degree in complex space is odd and the rank defect of the Jacobi matrix is one, the topological degree of a small region containing the singular solution can be verified to be plus or minus one in real space. The algorithm for verification in real space is significantly simpler and more efficient. We demonstrate this efficiency with numerical experiments.Since our verification procedure uses only values on the surfaces of a bounding box that contains the solution, the method can also be applied to cases where the system is nonsmooth at the solution.