Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
2.21 Notes and references
J. Albrecht, Monotone Iterationsfolgen und ihre Verwendung zur Lösung linearer Gleichungssysteme, Numerische Mathematik, 3 (1961), pp. 345–358.
G. Alefeld, V. Kreinovich, and G. Mayer, The shape of the symmetric solution set, in Applications of Interval Computations, R. B. Kearfott and V. Kreinovich, eds., Dordrecht, 1996, Kluwer, pp. 61–79.
G. Alefeld, V. Kreinovich, and G. Mayer, On the shape of the symmetric, persymmetric, and skew-symmetric solution set, SIAM Journal on Matrix Analysis and Applications, 18 (1997), pp. 693–705.
G. Alefeld, V. Kreinovich, and G. Mayer, The shape of the solution set for systems of interval linear equations with dependent coefficients, Mathematische Nachrichten, 192 (1998), pp. 23–36.
G. Alefeld and G. Mayer, On the symmetric and unsymmetric solution set of interval systems, SIAM Journal on Matrix Analysis and Applications, 16 (1995), pp. 1223–1240.
M. Baumann, A regularity criterion for interval matrices, in Collection of Scientific Papers Honouring Prof. Dr. K. Nickel on Occasion of his 60th Birthday, Part I, J. Garloff et al., eds., Freiburg, 1984, Albert-Ludwigs-Universität, pp. 45–50.
H. Beeck, Zur Problematik der Hüllenbestimmung von Intervallgleichungssystemen, in Interval Mathematics, K. Nickel, ed., Lecture Notes in Computer Science 29, Springer-Verlag, Berlin, 1975, pp. 150–159.
C. Bliek, Computer Methods for Design Automation, PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, July 1992.
G. E. Coxson, Computing exact bounds on elements of an inverse interval matrix is NP-hard, Reliable Computing, 5 (1999), pp. 137–142.
A. Deif, Sensitivity Analysis in Linear Systems, Springer-Verlag, Berlin, 1986.
J. Farkas, Theorie der einfachen Ungleichungen, Journal für die Reine und Angewandte Mathematik, 124 (1902), pp. 1–27.
J. Garloff, Totally nonnegative interval matrices, in Interval Mathematics 1980, K. Nickel, ed., Academic Press, New York, 1980, pp. 317–327.
W. Gerlach, Zur Lösung linearer Ungleichungssysteme bei Störung der rechten Seite und der Koeffizientenmatrix, Mathematische Operationsforschung und Statistik, Series Optimization, 12 (1981), pp. 41–43.
G. H. Golub and C. F. van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, 1996.
F. Gray, Pulse code communication. United States Patent Number 2,632,058. March 17, 1953.
E. R. Hansen, Bounding the solution of interval linear equations, SIAM Journal on Numerical Analysis, 29 (1992), pp. 1493–1503.
G. Heindl, Some inclusion results based on a generalized version of the Oettli-Prager theorem, Zeitschrift für Angewandte Mathematik und Mechanik, Supplement 3, 76 (1996), pp. 263–266.
J. Herzberger and D. Bethke, On two algorithms for bounding the inverse of an interval matrix, Interval Computations, 1 (1991), pp. 44–53.
N. J. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Philadelphia, 1996.
K.-U. Jahn, Eine Theorie der Gleichungssysteme mit Intervallkoeffizienten, Zeitschrift für Angewandte Mathematik und Mechanik, 54 (1974), pp. 405–412.
C. Jansson, Calculation of exact bounds for the solution set of linear interval systems, Linear Algebra and Its Applications, 251 (1997), pp. 321–340.
B. Kelling, Methods of solution of linear tolerance problems with interval arithmetic, in Computer Arithmetic and Enclosure Methods, L. Atanassova and J. Herzberger, eds., North-Holland, Amsterdam, 1992, pp. 269–277.
B. Kelling, Geometrische Untersuchungen zur eingeschränkten Lösungsmenge linearer Intervallgleichungssysteme, Zeitschrift für Angewandte Mathematik und Mechanik, 74 (1994), pp. 625–628.
B. Kelling and D. Oelschlägel, Zur Lösung von linearen Toleranzproblemen, Wissenschaftliche Zeitschrift TH Leuna-Merseburg, 33 (1991), pp. 121–131.
V. Kreinovich, A. Lakeyev, J. Rohn, and P. Kahl, Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer Academic Publishers, Dordrecht, 1998. Chapter 21 and 22
A. V. Lakeyev and S. I. Noskov, A description of the set of solutions of a linear equation with intervally defined operator and right-hand side (in Russian), Doklady of the Russian Academy of Sciences, 330 (1993), pp. 430–433.
A. V. Lakeyev and S. I. Noskov, On the set of solutions of a linear equation with intervally defined operator and right-hand side (in Russian), Siberian Mathematical Journal, 35 (1994), pp. 1074–1084.
G. Mayer and J. Rohn, On the applicability of the interval Gaussian algorithm, Reliable Computing, 4 (1998), pp. 205–222.
R. E. Moore, Interval Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1966.
R. E. Moore, Methods and Applications of Interval Analysis, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1979.
J. Nedoma, Sign-stable solutions of column-vague linear equation systems, Reliable Computing, 3 (1997), pp. 173–180.
J. Nedoma, Inaccurate linear equation systems with a restricted-rank error matrix, Linear and Multilinear Algebra, 44 (1998), pp. 29–44.
J. Nedoma, On solving vague systems of linear equations with patternshaped columns, Linear Algebra and Its Applications, 324 (2001), pp. 107–118.
J. Nedoma, Positively regular vague matrices, Linear Algebra and Its Applications, 326 (2001), pp. 85–100.
A. Neumaier, Linear interval equations, in Interval Mathematics 1985, K. Nickel, ed., Lecture Notes in Computer Science 212, Springer-Verlag, Berlin, 1986, pp. 109–120.
A. Neumaier, Tolerance analysis with interval arithmetic, Freiburger Intervall-Berichte 86/9, Albert-Ludwigs-Universität, Freiburg, 1986.
A. Neumaier, Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.
A. Neumaier, A simple derivation of the Hansen-Bliek-Rohn-Ning-Kearfott enclosure for linear interval equations, Reliable Computing, 5 (1999), pp. 131–136.
K. Nickel, Die Überschätzung des Wertebereichs einer Funktion in der Intervallrechnung mit Anwendungen auf lineare Gleichungssysteme, Computing, 18 (1977), pp. 15–36.
S. Ning and R. B. Kearfott, A comparison of some methods for solving linear interval equations, SIAM Journal on Numerical Analysis, 34 (1997), pp. 1289–1305.
E. Nuding, Ein einfacher Beweis der Sätze von Oettli-Prager und J. Rohn, Freiburger Intervall-Berichte 86/9, Albert-Ludwigs-Universität, Freiburg, 1986.
E. Nuding and J. Wilhelm, Über Gleichungen und über Lösungen, Zeitschrift für Angewandte Mathematik und Mechanik, 52 (1972), pp. T188–T190.
W. Oettli, On the solution set of a linear system with inaccurate coefficients, SIAM Journal on Numerical Analysis, 2 (1965), pp. 115–118.
W. Oettli and W. Prager, Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides, Numerische Mathematik, 6 (1964), pp. 405–409.
S. Poljak and J. Rohn, Radius of nonsingularity, Research Report, KAM Series 88-117, Faculty of Mathematics and Physics, Charles University, Prague, December 1988.
S. Poljak and J. Rohn, Checking robust nonsingularity is NP-hard, Mathematics of Control, Signals, and Systems, 6 (1993), pp. 1–9.
H. Ratschek and W. Sauer, Linear interval equations, Computing, 28 (1982), pp. 105–115.
G. Rex, Zum Regularitätsnachweis von Matrizen, Zeitschrift fur Angewandte Mathematik und Mechanik, 75 (1995), pp. S549–S550.
G. Rex and J. Rohn, A note on checking regularity of interval matrices, Linear and Multilinear Algebra, 39 (1995), pp. 259–262.
G. Rex and J. Rohn, Sufficient conditions for regularity and singularity of interval matrices, SIAM Journal on Matrix Analysis and Applications, 20 (1999), pp. 437–445.
F. N. Ris, Interval Analysis and Applications to Linear Algebra, PhD thesis, Oxford University, Oxford, 1972.
J. Rohn, A perturbation theorem for linear equations. To appear.
J. Rohn, Soustavy lineárních rovnic s intervalově zadanými koeficienty, Ekonomicko-matematický obzor, 12 (1976), pp. 311–315.
J. Rohn, Input-output planning with inexact data, Freiburger Intervall-Berichte 78/9, Albert-Ludwigs-Universität, Freiburg, 1978.
J. Rohn, Interval linear systems with prescribed column sums, Linear Algebra and Its Applications, 39 (1981), pp. 143–148.
J. Rohn, Strong solvability of interval linear programming problems, Computing, 26 (1981), pp. 79–82.
J. Rohn, Proofs to “Solving interval linear systems”, Freiburger Intervall-Berichte 84/7, Albert-Ludwigs-Universität, Freiburg, 1984.
J. Rohn, Solving interval linear systems, Freiburger Intervall-Berichte 84/7, Albert-Ludwigs-Universität, Freiburg, 1984.
J. Rohn, Miscellaneous results on linear interval systems, Freiburger Intervall-Berichte 85/9, Albert-Ludwigs-Universität, Freiburg, 1985. Theorem 1.2
J. Rohn, Inner solutions of linear interval systems, in Interval Mathematics 1985, K. Nickel, ed., Lecture Notes in Computer Science 212, Springer-Verlag, Berlin, 1986, pp. 157–158.
J. Rohn, Systems of linear interval equations, Linear Algebra and Its Applications, 126 (1989), pp. 39–78.
J. Rohn, Characterization of a linear program in standard form by a family of linear programs with inequality constraints, Ekonomicko-matematický obzor, 26 (1990), pp. 71–73.
J. Rohn, Interval solutions of linear interval equations, Aplikace matematiky, 35 (1990), pp. 220–224.
J. Rohn, An existence theorem for systems of linear equations, Linear and Multilinear Algebra, 29 (1991), pp. 141–144.
J. Rohn, Cheap and tight bounds: The recent result by E. Hansen can be made more efficient, Interval Computations, 4 (1993), pp. 13–21.
J. Rohn, NP-hardness results for some linear and quadratic problems, Technical Report 619, Institute of Computer Science, Academy of Sciences of the Czech Republic, Prague, January 1995.
J. Rohn, Linear programming with inexact data is NP-hard, Zeitschrift für Angewandte Mathematik und Mechanik, Supplement 3, 78 (1998), pp. S1051–S1052.
J. Rohn, Computing the norm ‖A‖∞,1 is NP-hard, Linear and Multilinear Algebra, 47 (2000), pp. 195–204.
J. Rohn, Solvability of systems of linear interval equations, SIAM Journal on Matrix Analysis and Applications, 25 (2003), pp. 237–245.
J. Rohn and V. Kreinovich, Computing exact componentwise bounds on solutions of linear systems with interval data is NP-hard, SIAM Journal on Matrix Analysis and Applications, 16 (1995), pp. 415–420.
J. Rohn and J. Kreslová, Linear interval inequalities, Linear and Multilinear Algebra, 38 (1994), pp. 79–82.
S. M. Rump, Solving algebraic problems with high accuracy, in A New Approach to Scientific Computation, U. Kulisch and W. Miranker, eds., Academic Press, New York, 1983, pp. 51–120.
S. M. Rump, On the solution of interval linear systems, Computing, 47 (1992), pp. 337–353.
S. M. Rump, Verification methods for dense and sparse systems of equations, in Topics in Validated Computations, J. Herzberger, ed., North-Holland, Amsterdam, 1994, pp. 63–135.
S. P. Shary, O nekotorykh metodakh resheniya lineinoi zadachi o dopuskakh, Preprint 6, Siberian Branch of the Soviet Academy of Sciences, Krasnoyarsk, 1989.
S. P. Shary, A new class of algorithms for optimal solution of interval linear systems, Interval Computations, 2 (1992), pp. 18–29.
S. P. Shary, On controlled solution set of interval algebraic systems, Interval Computations, 6 (1992), pp. 66–75.
S. P. Shary, Solving the tolerance problem for interval linear systems, Interval Computations, 2 (1994), pp. 6–26.
S. P. Shary, Solving the linear interval tolerance problem, Mathematics and Computers in Simulation, 39 (1995), pp. 53–85.
S. P. Shary, Algebraic approach to the interval linear static identification, tolerance and control problems, or One more application of Kaucher arithmetic, Reliable Computing, 2 (1996), pp. 3–33.
S. P. Shary, Algebraic solutions to interval linear equations and their applications, in Numerical Methods and Error Bounds, G. Alefeld and J. Herzberger, eds., Mathematical Research, Vol. 89, Akademie Verlag, Berlin, 1996, pp. 224–233.
S. P. Shary, Controllable solutions sets to interval static systems, Applied Mathematics and Computation, 86 (1997), pp. 185–196.
S. P. Shary, A new technique in systems analysis under interval uncertainty and ambiguity, Reliable Computing, 8 (2002), pp. 321–418.
V. V. Shaydurov and S. P. Shary, Resheniye interval’noi algebraicheskoi zadachi o dopuskakh, Preprint 5, Siberian Branch of the Soviet Academy of Sciences, Krasnoyarsk, 1988.
Y. I. Shokin, Interval’nyj analiz, Nauka, Novosibirsk, 1981.
Y. I. Shokin, On interval problems, interval algorithms and their computational complexity, in Scientific Computing and Validated Numerics, G. Alefeld, A. Frommer, and B. Lang, eds., Akademie Verlag, Berlin, 1996, pp. 314–328.
E. W. Weisstein, Gray code. http://mathworld.wolfram.com/GrayCode.html.
Rights and permissions
Copyright information
© 2006 Springer Science+Business Media, Inc.
About this chapter
Cite this chapter
Rohn, J. (2006). Solvability of systems of interval linear equations and inequalities. In: Linear Optimization Problems with Inexact Data. Springer, Boston, MA. https://doi.org/10.1007/0-387-32698-7_2
Download citation
DOI: https://doi.org/10.1007/0-387-32698-7_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-387-32697-9
Online ISBN: 978-0-387-32698-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)