Matthias Hocks

Nonlinear Equations in One Variable

One of the most important tasks in scientific computing is the problem of finding zeros (or roots) of nonlinear functions. In classical numerical analysis, root-finding methods for nonlinear functions begin with an approximation and apply an iterative method (such as Newton’s or Halley’s methods), which hopefully improves the approximation. It is a myth that no numerical algorithm is able to compute all zeros of a nonlinear equation with guaranteed error bounds, or even more, that no method is able to give concrete information about the existence and uniqueness of solutions of such a problem.

Zeros of Complex Polynomials

We consider the complex polynomial p: C → C defined by

The Features of C-XSC

p(z)=\sum\limits_{i=0}^n{{p_i}{z^i}},{p_i}\in\mathbb{C}{\text{, }}i=0, . . . , n, {p_n}\ne 0.

Linear Systems of Equations

(1) (9.1) The Fundamental Theorem of algebra asserts that this polynomial has n zeros counted by multiplicity. Finding these roots is a non trivial problem in numerical mathematics. Most algorithms deliver only approximations of the exact zeros without any or with only weak statements concerning the accuracy.

The Features of PASCAL-XSC

The evaluation of arithmetic expressions using floating-point arithmetic may lead to unpredictable results due to an accumulation of roundoff errors. As an example, the evaluation of x + 1 — x for x > 1020 using the standard floating-point format on almost every digital computer yields the wrong result 0. Since the evaluation of arithmetic expressions is a basic task in digital computing, we should have a method to evaluate a given expression to an arbitrary accuracy. We will develop such a method for real expressions composed of the operations +, —, •, /, and ↑,where ↑ denotes exponentiation by an integer. The method is based on the principle of iterative refinement as discussed in Section 3.8.

Numerical Toolbox for Verified Computing I

In this chapter, we present a brief review of the C-XSC library, a C++ class library for eXtended Scientific Computation. For a complete library reference and examples, refer to [49].

C++ Toolbox for Verified Computing I

In Chapter 6, we considered the problem of finding zeros (or roots) of nonlinear functions of a single variable. Now, we consider its generalization, the problem of finding the solution vectors of a system of nonlinear equations. We give a method for finding all solutions of a nonlinear system of equations f(x) = 0 for a continuously differentiable function f: ℝn→ ℝn in a given interval vector (box). Our method computes close bounds on the solution vectors, and it delivers information about existence and uniqueness of the computed solutions. The method we present is a variant of the interval Gauss-Seidel method based on the method of Hansen and Sengupta [3], [32], and a modification of Ratz [79]. Our method makes use of the extended interval operations defined in Section 3.3.

C++ Toolbox for Verified Computing I: Basic Numerical Problems Theory, Algorithms, and Programs

Finding the solution of a linear system of equations is one of the basic problems in numerical algebra. We will develop a verification algorithm for square systems with full matrix based on a Newton-like method for an equivalent fixed-point problem.