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Interval solutions of linear interval equations. (English) Zbl 0716.65047
We use the notations of the preceding review. Let a nonsingular interval matrix $$A^ I$$ be given by the $$n\times n$$ matrices $$A^-=(a^- _{ij})$$ and $$A^+=(a^+_{ij})$$, and similarly the interval vector $$b^ I=[b^-,b^+]$$ by $$b^-=(b^-_ i)$$, $$b^+=(b^+_ i)\in {\mathbb{R}}^ n$$. The problem of finding an interval n-vector solution $$x^ I=[x^-,x^+]$$ of $(*)\;\sum^{n}_{j=1}[a^- _{ij},a^+_{ij}]\cdot [x^-_ j,x^+_ j]=[b^-_ i,b^+_ i],\;i=1,...,n,$ is considered under the additional assumption that there exist $$A',A''\in A^ I$$ and $$x',x''\in x^ I$$ such that $$A'x'=b^-$$ and $$A''x''=b^+$$ hold. The solvability of this problem is related to solving the equations $$A_ cx-\Delta (| x_ j|)=b^-$$ and $$A_ cx+\Delta (| x_ j|)=b^+$$. The unique solutions of these two equations are utilized to obtain an algorithm to find a solution of the above type for (*) or alternatively to confirm that no such solution exists.
Reviewer: M.Kracht

##### MSC:
 65G30 Interval and finite arithmetic 65F30 Other matrix algorithms (MSC2010)
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##### References:
 [1] W. Barth E. Nuding: Optimale Lösung von Intervallgleichungssystemen. Computing 12 (1974), 117-125. · Zbl 0275.65008 [2] H. Beeck: Zur Problematik der Hüllenbestimmung von Intervallgleichungssystemen. Interval Mathematics (K. Nickel. Lecture Notes, Springer 1975, 150-159. · Zbl 0303.65025 [3] E. Hansen: On Linear Algebraic Equations with Interval Coefficients. Topics in Interval Analysis (E. Hansen. Clarendon Press, Oxford 1969. [4] R. E. Moore: Interval Analysis. Prentice-Hall, Englewood Cliffs 1966. · Zbl 0176.13301 [5] K. Nickel: Die Auflösbarkeit linearer Kreisscheiben- und Intervall-Gleichungssysteme. Freiburger Intervall-Berichte 81/3, 11 - 46. · Zbl 0483.65019 [6] W. Oettli W. Prager: Compatibility of Approximate Solution of Linear Equations with Given Error Bounds for Coefficients and Right-Hand Sides. Numerische Mathematik 6 (1964), 405-409. · Zbl 0133.08603 [7] H. Ratschek W. Sauer: Linear Interval Equations. Computing 28 (1982), 105-115. · Zbl 0468.65025 [8] J. Rohn: Some Results on Interval Linear Systems. Freiburger Intervall-Berichte 85/4, 93-116. · Zbl 0684.65046 [9] J. Rohn: A Note on Solving Equations of Tupe $$A^1 x^1 = b^1$$. Freiburger Intervall-Berichte 86/4, 29-31.
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