# A Method for Outer Interval Solution of Systems of Linear Equations Depending Linearly on Interval Parameters

- 104 Downloads
- 25 Citations

## Abstract

Consider the systems of linear interval equations whose coefficients are linear functions of interval parameters. Such systems, called parametrized systems of linear interval equations, are encountered in many practical problems, e.g in electrical engineering and structure mechanics. A direct method for computing a tight enclosure for the solution set is proposed in this paper. It is proved that for systems with real matrix and interval right-hand vector the method generates the hull of the solution set. For such systems an explicit formula for the hull is also given. Finally some numerical examples are provided to demonstrate the usefulness of the method in structure mechanics.

## Keywords

Mathematical Modeling Linear Function Linear Equation Computational Mathematic Parametrized System## Preview

Unable to display preview. Download preview PDF.

## References

- 1.Jansson, C.: Interval Linear Systems with Symmetric Matrices, Skew-Symmetric Matrices and Dependencies in the Right Hand Side,
*Computing***46**(3) (1991), pp. 265–274.CrossRefzbMATHMathSciNetGoogle Scholar - 2.Kolev, L. V.: A Method for Outer Interval Solution of Linear Parametric Systems,
*Reliable Computing***10**(3) (2004), pp. 227–239.CrossRefzbMATHMathSciNetGoogle Scholar - 3.Kolev, L. V.:
*Interval Methods for Circuit Analysis*, Word Scientific Ltd., Singapore, New Jersey, London, 1993.Google Scholar - 4.Kolev, L. V.: Outer Solution of Linear Systems Whose Elements Are Affine Functions of Interval Parameters,
*Reliable Computing***6**(12) (2002), pp. 493–501.zbMATHMathSciNetGoogle Scholar - 5.Kolev, L. V.: Worst-Case Tolerance Analysis of Linear DC and AC Electric Circuits,
*IEEE Trans, on Circuits and Systems, I, Fund. Theory and Appl*.**49**(12) (2002), pp. 1693–1702.MathSciNetGoogle Scholar - 6.Kulpa, Z., Pownuk, A., and Skalna, I.: Analysis of Linear Mechanical Structures with Uncertain ties by Means of Interval Methods,
*Computer Assisted Mechanics and Engineering Sciences***5**(4) (1998), pp. 443–477.MathSciNetGoogle Scholar - 7.Neumaier, A.:
*Interval Methods for Systems of Equations*, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 1990.Google Scholar - 8.Popova, E. D.: On the Solution of Parametrised Linear Systems,
*Scientific Computing, Validated Numerics, Interval Methods*(2001), pp. 127–138.Google Scholar - 9.Rohn, J.: On the Interval Hull of the Solution Set of an Interval Linear System,
*Freiburger Intervall-Ber*. 81/5, Universität Freiburg, Freiburg, 1981, pp. 47–57.Google Scholar - 10.Rohn, J.: Systems of Linear Interval Equations,
*Linear Algebra and Its Applications***126**(1989), pp. 39–78.CrossRefzbMATHMathSciNetGoogle Scholar - 11.Rump, S. M.: Verification Methods for Dense and Sparse Systems of Equations, in: Herzberger, J. (ed.),
*Topics in Validated Computations: Proceedings of IMACS-GAMM International Workshop on Validated Computation, Oldenburg, Germany, 30 August–3 September 1993, Studies in Com putational Mathematics***5**, Elsevier, Amsterdam, 1994, pp. 63–135.Google Scholar - 12.Skalna, I.: Methods for Solving Systems of Linear Equations of Structure Mechanics with Interval Parameters,
*Computer Assisted Mechanics and Engineering Sciences***10**(3) (2003), pp. 281–293.zbMATHGoogle Scholar