Enclosure for the Solution Set of Parametric Linear Systems with Non-affine Dependencies

The problem of solving linear systems whose coefficients are nonlinear functions of parameters varying within prescribed intervals is investigated. A new method for outer interval solution of such system is proposed. In order to reduce memory usage, nonlinear dependencies between parameters are handled using revised affine arithmetic. Some numerical experiments which aim to show the properties of the proposed method are reported.

[1]  Evgenija D. Popova.  Generalization of a Parametric Fixed‐Point Iteration , 2004 .

[2]  Frédéric Messine,et al.  Extentions of Affine Arithmetic: Application to Unconstrained Global Optimization , 2002, J. Univers. Comput. Sci..

[3]  Evgenija D. Popova.  Solving Linear Systems Whose Input Data Are Rational Functions of Interval Parameters , 2006, Numerical Methods and Applications.

[4]  Iwona Skalna,et al.  Direct Method for Solving Parametric Interval Linear Systems with Non-affine Dependencies , 2009, PPAM.

[5]  E. R. Hansen,et al.  A Generalized Interval Arithmetic , 1975, Interval Mathematics.

[6]  Lubomir V. Kolev.  Solving Linear Systems Whose Elements Are Nonlinear Functions of Intervals , 2004, Numerical Algorithms.

[7]  H. El-Owny.  Parametric Linear System of Equations, Whose Elements are Nonlinear Functions , 2006, 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006).

[9]  Z. Kulpa,et al.  Analysis of linear mechanical structures with uncertainties by means of interval methods , 1998 .

[10]  Iwona Skalna.  Evolutionary Optimization Method for Approximating the Solution Set Hull of Parametric Linear Systems , 2006, Numerical Methods and Applications.

[11]  Boi Faltings,et al.  Combining multiple inclusion representations in numerical constraint propagation , 2004, 16th IEEE International Conference on Tools with Artificial Intelligence.

[12]  Milan Hladík,et al.  Enclosures for the solution set of parametric interval linear systems , 2012, Int. J. Appl. Math. Comput. Sci..

[13]  Lubomir V. Kolev,et al.  An Improved Interval Linearization for Solving Nonlinear Problems , 2004, Numerical Algorithms.

[14]  S. Rump.  Verification methods for dense and sparse systems of equations , 1994 .

[15]  E. Popova.  On the Solution of Parametrised Linear Systems , 2001 .

[16]  J. Stolfi,et al.  An Introduction to Affine Arithmetic , 2003 .

[17]  Jarosław Knabel,et al.  COMPUTER ASSISTED MECHANICS AND ENGINEERING SCIENCES , 2009 .

[18]  G. Marchuk,et al.  Numerical methods and applications , 1995 .

[19]  Lubomir V. Kolev.  Automatic Computation of a Linear Interval Enclosure , 2001, Reliab. Comput..

[20]  Ramil R. Akhmerov.  Interval-Affine Gaussian Algorithm for Constrained Systems , 2005, Reliab. Comput..

[21]  Iwona Skalna,et al.  A Method for Outer Interval Solution of Systems of Linear Equations Depending Linearly on Interval Parameters , 2006, Reliab. Comput..

[22]  A. Neumaier.  Interval methods for systems of equations , 1990 .