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首页> 外文学位 >THE ITERATIVE SOLUTION OF A NONLINEAR EQUATION USING THE INTEGRAL-EQUATION FORMULATIONS OF AN IMBEDDING.
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THE ITERATIVE SOLUTION OF A NONLINEAR EQUATION USING THE INTEGRAL-EQUATION FORMULATIONS OF AN IMBEDDING.

机译：使用嵌入的积分方程式的非线性方程的迭代解。

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摘要

The thesis demonstrates how iterative algorithms for the numerical solution of a nonlinear equation can be deduced from the integral-equation representations of a homotopic imbedding.;Instead of numerically solving the differential expression for the end point, as is the case with continuation methods, we first find approximate analytical solutions to the corresponding Volterra integral equations. The approximations are obtained by standard techniques such as Picard's method and the local linearization procedure of Newton. Their end points are then used to define inductively iterative algorithms without memory for the numerical solution of the given nonlinear equation. We also consider approximating the integral operator by means of quadrature rules. Similar iterative algorithms are defined after analytically or numerically approximating the integrand.;Various one-point algorithms, including those of Newton and Olver, have been derived by the analytical approach. With the aid of quadrature rules, multipoint algorithms have been obtained, some of which have supercubic rates of convergence.;As with continuation methods, we begin by imbedding the nonlinear equation in a continuous one-parameter family of problems by means of a homotopy. A root of the original equation is then the end point of the path of the homotopy's zeros which connects the root to an initial estimate. The path is formulated as the solution of a first-order initial value problem.
机译：论文论证了如何从同构嵌入的积分方程表示中推导出非线性方程数值解的迭代算法。代替数值方法求解端点的微分表达式，这与连续方法一样。首先找到相应的Volterra积分方程的近似解析解。通过标准技术（例如Picard方法和牛顿局部线性化过程）获得近似值。然后将它们的端点用于定义归纳迭代算法，而无需为给定非线性方程的数值解提供存储。我们还考虑通过正交规则逼近积分算子。在对被积体进行解析或数值逼近后，定义了类似的迭代算法。通过分析方法，得出了包括牛顿算法和奥尔弗算法在内的各种单点算法。借助正交规则，获得了多点算法，其中一些算法具有超三次收敛速度。与连续方法一样，我们首先通过同伦将非线性方程式嵌入到连续的一参数系列问题中。然后，原始方程的根是同位点零点路径的终点，该点将根连接到初始估计值。路径被公式化为一阶初值问题的解决方案。

著录项

  • 作者

    BEAUDOIN, YVES.;

  • 作者单位

    The University of Western Ontario (Canada).;

  • 授予单位 The University of Western Ontario (Canada).;
  • 学科 Mathematics.
  • 学位 Ph.D.
  • 年度 1984
  • 页码 1 p.
  • 总页数 1
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

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