Enclosures for the solution set of parametric interval linear systems

Abstract We investigate parametric interval linear systems of equations. The main result is a generalization of the Bauer–Skeel and the Hansen–Bliek–Rohn bounds for this case, comparing and refinement of both. We show that the latter bounds are not provable better, and that they are also sometimes too pessimistic. The presented form of both methods is suitable for combining them into one to get a more efficient algorithm. Some numerical experiments are carried out to illustrate performances of the methods.

[1]  M. Padberg.  Linear Optimization and Extensions , 1995 .

[2]  V. Kreinovich,et al.  On the Shape of the Symmetric, Persymmetric, and Skew-Symmetric Solution Set , 1997, SIAM J. Matrix Anal. Appl..

[3]  R. B. Kearfott,et al.  A Comparison of some Methods for Solving Linear Interval Equations , 1997 .

[4]  J. Rohn.  Systems of linear interval equations , 1989 .

[5]  Robert Rohn,et al.  Institute of Computer Science , 2012 .

[6]  Christian Jansson,et al.  Interval linear systems with symmetric matrices, skew-symmetric matrices and dependencies in the right hand side , 1991, Computing.

[7]  Evgenija D. Popova,et al.  Visualizing parametric solution sets , 2008 .

[8]  Lubomir V. Kolev,et al.  A Method for Outer Interval Solution of Linear Parametric Systems , 2004, Reliab. Comput..

[9]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[10]  J. Rohn,et al.  Cheap and Tight Bounds : The Recent Result , 2013 .

[11]  Götz Alefeld,et al.  The Cholesky method for interval data , 1993 .

[12]  Helmut Beeck,et al.  Zur Problematik der Hüllenbestimmung von Intervallgleichungssystemen , 1975, Interval Mathematics.

[13]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[14]  Arnold Neumaier,et al.  Linear Systems with Large Uncertainties, with Applications to Truss Structures , 2007, Reliab. Comput..

[15]  Evgenija D. Popova,et al.  Non-commercial Research and Educational Use including without Limitation Use in Instruction at Your Institution, Sending It to Specific Colleagues That You Know, and Providing a Copy to Your Institution's Administrator. All Other Uses, Reproduction and Distribution, including without Limitation Comm , 2022 .

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

[17]  Götz Alefeld,et al.  New Criteria for the Feasibility of the Cholesky Method with Interval Data , 2008, SIAM J. Matrix Anal. Appl..

[18]  G. W. Stewart,et al.  Matrix algorithms , 1998 .

[19]  Evgenija D. Popova,et al.  Quality of the Solution Sets of Parameter‐Dependent Interval Linear Systems , 2002 .

[20]  Milan Hladík.  Description of Symmetric and Skew-Symmetric Solution Set , 2008, SIAM J. Matrix Anal. Appl..

[21]  Lubomir V. Kolev.  Improvement of a Direct Method for Outer Solution of Linear Parametric Systems , 2006, Reliab. Comput..

[22]  Siegfried M. Rump,et al.  Solving Algebraic Problems with High Accuracy , 1983, IMACS World Congress.

[23]  Evgenija D. Popova,et al.  Parametric Interval Linear Solver , 2004, Numerical Algorithms.

[24]  M. Fiedler,et al.  Linear Optimization Problems with Inexact Data , 2006 .

[25]  Mikolaj Buslowicz.  Robust stability of positive continuous-time linear systems with delays , 2010, Int. J. Appl. Math. Comput. Sci..

[26]  E.D. Popova.  Computer-Assisted Proofs in Solving Linear Parametric Problems , 2006, 12th GAMM - IMACS International Symposium on Scientific Computing, Computer Arithmetic and Validated Numerics (SCAN 2006).

[27]  Jiri Rohn,et al.  Sufficient Conditions for Regularity and Singularity of Interval Matrices , 1999, SIAM J. Matrix Anal. Appl..

[28]  Iwona Skalna.  On Checking the Monotonicity of Parametric Interval Solution of Linear Structural Systems , 2007, PPAM.

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

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

[31]  Siegfried M. Rump,et al.  Verification methods: rigorous results using floating-point arithmetic , 2010, Acta Numerica.

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

[33]  Minjie Zhang,et al.  WEBCOMPUTING SERVICE FRAMEWORK , 2006 .

[34]  G. W. Stewart,et al.  Matrix Algorithms: Volume 1, Basic Decompositions , 1998 .

[35]  Arnold Neumaier,et al.  A Simple Derivation of the Hansen-Bliek-Rohn-Ning-Kearfott Enclosure for Linear Interval Equations , 1999, Reliab. Comput..

[36]  V. Kreinovich,et al.  On the solution sets of particular classes of linear interval systems , 2003 .

[37]  E. Popova.  EXPLICIT CHARACTERIZATION OF A CLASS OF PARAMETRIC SOLUTION SETS , 2009 .

[38]  Jean-Pierre Merlet.  Interval Analysis for Certified Numerical Solution of Problems in Robotics , 2009, Int. J. Appl. Math. Comput. Sci..

[39]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[40]  Jürgen Garloff.  Pivot Tightening for the Interval Cholesky Method , 2010 .