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Simple bounds for zeros of systems of equations

Monotone Iterations And Computational Error Bounds

Part of the Lecture Notes in Mathematics book series (LNM,volume 953)

Abstract

Let \(F:D( \subseteq \mathbb{R}^n ) \to \mathbb{R}^n\)be a continuous function, and suppose that for some XO ∈ D and some nonsingular matrix A the vector δO:=A−1 F(xO) is "small". Assuming the existence of a nonnegative vector c ∈ ℝn such that

$$\left| {F(x) - F(x_0 ) - A(x - x_0 )} \right| \leqslant \left\| {\delta _0 } \right\|c$$

for all x in a suitable neighbourhood S of xO, a simple condition is given which guarantees that S contains a zero \(\hat x\)of F. The resulting bounds are shown to be quite accurate. They contain as a special case the bounds obtainable from a theorem of Kantorovic.

It is discussed how to compute the required vector c, and how to make efficient use of sparsity. The practical use of the bounds is demonstrated by an extensive example, the finite difference equations obtained by discretizing the minimal surface equation.

Keywords

  • Newton Iteration
  • Interval Arithmetic
  • Finite Difference Equation
  • Unique Zero
  • Polynomial Operator

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 1982 Springer-Verlag

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Neumaier, A. (1982). Simple bounds for zeros of systems of equations. In: Ansorge, R., Meis, T., Törnig, W. (eds) Iterative Solution of Nonlinear Systems of Equations. Lecture Notes in Mathematics, vol 953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069376

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  • DOI: https://doi.org/10.1007/BFb0069376

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  • Print ISBN: 978-3-540-11602-8

  • Online ISBN: 978-3-540-39379-5

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