# Solving Underdetermined Systems with Interval Methods

Solving Underdetermined Systems with Interval Methods In this paper, we use the concepts of {1} - and {2} -inverse of a matrix to construct such a Krawczyk-like operator as $$\overline K (X) = x - Yf(x) + (YA - YF'(X))(X - x),{\text{ }}x \in X,$$ where Y ∈ Rnxm, and A ∈ Rmxn satisfies YAY = Y. We also construct a generalized Krawczyk-Moore algorithm using this operator. Our main result is that, under the conditions $$\overline K (X) \subseteq X{\text{ and }}r(\overline K (X)) < r(X),$$ there is a unique solution of Yf(x) = 0 on X $$\cap$$ {x + R(Y)} , and the generalized Krawczyk-Moore algorithm converges to this solution at least linearly. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliable Computing Springer Journals

# Solving Underdetermined Systems with Interval Methods

, Volume 5 (1) – Oct 21, 2004
11 pages

/lp/springer_journal/solving-underdetermined-systems-with-interval-methods-0QoBTTqfcv
Publisher
Springer Journals
Subject
Mathematics; Numeric Computing; Approximations and Expansions; Computational Mathematics and Numerical Analysis; Mathematical Modeling and Industrial Mathematics
ISSN
1385-3139
eISSN
1573-1340
D.O.I.
10.1023/A:1026489507711
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we use the concepts of {1} - and {2} -inverse of a matrix to construct such a Krawczyk-like operator as $$\overline K (X) = x - Yf(x) + (YA - YF'(X))(X - x),{\text{ }}x \in X,$$ where Y ∈ Rnxm, and A ∈ Rmxn satisfies YAY = Y. We also construct a generalized Krawczyk-Moore algorithm using this operator. Our main result is that, under the conditions $$\overline K (X) \subseteq X{\text{ and }}r(\overline K (X)) < r(X),$$ there is a unique solution of Yf(x) = 0 on X $$\cap$$ {x + R(Y)} , and the generalized Krawczyk-Moore algorithm converges to this solution at least linearly.

### Journal

Reliable ComputingSpringer Journals

Published: Oct 21, 2004

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