Interval Analysis

5Linear systems of equations

5.1Motivation

Linear systems of equations are basic in mathematical modeling and in numerical analysis. Even if a model is nonlinear or nondiscrete, linearization and discretization finally can lead to a linear system. The reason why one tries to end up with such systems lies in their simplicity and their ‘easy’ numerical treatment. A typical example is Newton’s method for the computation of a zero x^∗ of a nonlinear (sufficiently smooth) function f : D ⊆ ℝⁿ → ℝⁿ. It results from a Taylor expansion of f(x^∗) at some vector x^k ≠ x^∗ cut off behind the linear term. Thus the nonlinear system f(x^∗) = 0 transforms the linear approximation 0 ≈ f(x^k) + f′(x^k)(x^∗ − x^k). Changing this approximation to an equality by replacing ...