求解偏微分方程當採用配置法配上非局部函數當基底，例如正交多項式和徑向基函數時得出指數型收斂行為。然而，它產生一個滿矩陣且condition過大。此論文中使用配置法配局部基底函數，即再生核心函數，結果為代數收斂行為。此方法類似有限元素法。 本論文主旨為擾動性和穩定性分析，並估計離散方程的condition number。Condition number是用來估計解的相對誤差上界，它在數值穩定上扮演一個關鍵的角色。另外，論文中介紹新condition number公式，被稱作effective condition number。此新估計式中線性系統中的矩陣及右端向量兩者都被考量在condition number的估計上，它們提供一個比傳統condition number更好的測量條件。數值結果也證實了數學分析。 Solving partial differential equations with strong form collocation and nonlocal approximation functions such as orthogonal polynomials and radial basis functions exhibits exponential convergence rate; however, it yields a full matrix and suffers from ill conditioning. In this work, the local approximation functions, reproducing kernel functions, are used as basis functions. This approach offers algebra convergence rate, but the method is stable like the finite element. We provide the perturbation and stability analysis of this approach, and the estimation of condition number of the discrete equation is derived. Condition number is used to measure the solution errors resulting from rounding errors, and it plays a critical role in numerical stability. In addition, the new formulas of condition number, called effective condition numbers, are given. Both matrix and right hand side vector of a linear system are taken into consideration in the estimation of condition number, they offer a better measure of conditioning than traditional condition numbers. Numerical results are also presented to validate the mathematical analysis.
