Interval analysis: Theory and applications

Journal of Computational and Applied Mathematics

Volumn 121, Issue 1, 2000, Pages 421-464

  Alefeld, Götz ^a   Mayer, Günter ^b  

^a UNIVERSITY OF KARLSRUHE   (Germany)

^b UNIVERSITY OF ROSTOCK   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY VALUE PROBLEMS; EIGENVALUES AND EIGENFUNCTIONS; INITIAL VALUE PROBLEMS; LINE CONCENTRATORS; NONLINEAR EQUATIONS; ORDINARY DIFFERENTIAL EQUATIONS; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING;

INTERVAL ANALYSIS;

COMPUTATIONAL METHODS;

EID: 0034276022     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00342-3     Document Type: Article

References (120)

1. E. Adams U. Kulisch (Eds.) Scientific Computing with Automatic Result Verification 1993 Academic Press Boston
2. R. Albrecht G. Alefeld H.J. Stetter (Eds.) Validation Numerics, Theory and Applications 1993 Springer Wien
3. G. Alefeld, Intervallrechnung über den komplexen Zahlen und einige Anwendungen, Ph.D. Thesis, Universität Karlsruhe, Karlsruhe, 1968.
4. G. Alefeld Über die Durchführbarkeit des Gaußschen Algorithmus bei Gleichungen mit Intervallen als Koeffizienten Comput. Suppl. 1 1977 15 19
5. G. Alefeld Bounding the slope of polynomials and some applications Computing 26 1981 227 237
6. G. Alefeld Berechenbare Fehlerschranken für ein Eigenpaar unter Einschluß von Rundungsfehlern bei Verwendung des genauen Skalarprodukts Z. Angew. Math. Mech. 67 1987 145 152
7. G. Alefeld Rigorous error bounds for singular values of a matrix using the precise scalar product E. Kaucher U. Kulisch C. Ullrich Computerarithmetic 1987 B.G. Teubner Stuttgart 9 30
8. G. Alefeld Über das Divergenzverhalten des Intervall-Newton-Verfahrens Computing 46 1991 289 294
9. G. Alefeld Inclusion methods for systems of nonlinear equations – the interval Newton method and modifications J. Herzberger Topics in Validated Computations 1994 Elsevier Amsterdam 7 26
10. G. Alefeld A. Gienger G. Mayer Numerical validation for an inverse matrix eigenvalue problem Computing 53 1994 311 322
11. G. Alefeld A. Gienger F. Potra Efficient numerical validation of solutions of nonlinear systems SIAM J. Numer. Anal. 31 1994 252 260
12. G. Alefeld J. Herzberger Einführung in die Intervallrechnung 1974 Bibliographisches Institut Mannheim
13. G. Alefeld J. Herzberger Introduction to Interval Computations 1983 Academic Press New York
14. G. Alefeld R. Hoffmann G. Mayer Verification algorithms for generalized singular values Math. Nachr. 208 1999 5 29
15. G. Alefeld V. Kreinovich G. Mayer On the shape of the symmetric, persymmetric, and skew-symmetric solution set SIAM J. Matrix Anal. Appl. 18 1997 693 705
16. G. Alefeld V. Kreinovich G. Mayer The shape of the solution set of linear interval equations with dependent coefficients Math. Nachr. 192 1998 23 36
17. G. Alefeld, V. Kreinovich, G. Mayer, On the solution sets of particular classes of linear systems, submitted for publication.
18. G. Alefeld R. Lohner On higher order centered forms Computing 35 1985 177 184
19. G. Alefeld G. Mayer The Cholesky method for interval data Linear Algebra Appl. 194 1993 161 182
20. G. Alefeld G. Mayer A computer-aided existence and uniqueness proof for an inverse matrix eigenvalue problem Int. J. Interval Comput. 1994 1 1994 4 27
21. G. Alefeld G. Mayer On the symmetric and unsymmetric solution set of interval systems SIAM J. Matrix Anal. Appl. 16 1995 1223 1240
22. H. Beeck Über Struktur und Abschätzungen der Lösungsmenge von linearen Gleichungssystemen mit Intervallkoeffizienten Computing 10 1972 231 244
23. H. Behnke F. Goerisch Inclusions for eigenvalues of selfadjoint problems J. Herzberger Topics in Validated Computations 1994 Elsevier Amsterdam 277 322
24. X. Chen A verification method for solutions of nonsmooth equations Computing 58 1997 281 294
25. G.F. Corliss, Computing narrow inclusions for definite integrals, MRC Technical Summary Report # 2913, University of Madison, Madison, Wisconsin, February 1986.
26. G.F. Corliss, Introduction to validated ODE solving, Technical Report No. 416. Marquette University, Milwaukee, Wisconsin, March 1995.
27. H. Cornelius R. Lohner Computing the range of values of real functions with accuracy higher than second order Computing 33 1984 331 347
28. H.-J. Dobner, Einschließungsalgorithmen für hyperbolische Differentialgleichungen, Thesis, Universität Karlsruhe, Karlsruhe, 1986.
29. P.S. Dwyer Linear Computations 1951 Wiley New York
30. P. Eijgenraam, The solution of initial value problems using interval arithmetic. Formulation and analysis of an algorithm, Thesis, Mathematisch Centrum, Amsterdam, 1981.
31. W. Enger, Intervall Ray Tracing – Ein Divide-and-Conquer Verfahren für photorealistische Computergrafik, Thesis, Universität Freiburg, Freiburg, 1990.
32. A. Gienger, Zur Lösungsverifikation bei Fredholmschen Integralgleichungen zweiter Art, Thesis, Universität Karlsruhe, Karlsruhe, 1997.
33. G.H. Golub C.F. van Loan Matrix Computations 3rd Edition 1995 John Hopkins Baltimore
34. A. Griewank G.F. Corliss (Eds.) Automatic Differentiation of Algorithms 1992 SIAM Philadelphia, PA
35. H. Grell, K. Maruhn, W. Rinow (Eds.), Enzyklopädie der Elementarmathematik, Band I Arithmetik, Dritte Auflage, VEB Deutscher Verlag der Wissenschaften, Berlin, 1966.
36. R. Hammer M. Hocks U. Kulisch D. Ratz Numerical Toolbox for Verified Computing I 1993 Springer Berlin
37. E. Hansen An overview of global optimization using interval analysis R.E. Moore Reliability in Computing The Role of Interval Methods in Scientific Computing, Perspectives in Computing Vol. 19 1988 Academic Press Boston 289 307
38. E. Hansen Global Optimization Using Interval Analysis 1992 Dekker New York
39. P. Hertling, A Lower Bound for Range Enclosure in Interval Arithmetic, Centre for Discrete Mathematics and Theoretical Computer Science Research Report Series, Department of Computer Science, University of Auckland, January 1998.
40. IBM High Accuracy Arithmetic-Extended Scientific Computation (ACRITH-XSC), General Information GC 33-646-01, IBM Corp., 1990.
41. C. Jansson, Rigorous sensitivity analysis for real symmetric matrices with interval data, in: E. Kaucher, S.M. Markov, G. Mayer (Eds.), Computer Arithmetic, Scientific Computation and Mathematical Modelling, IMACS Annals on Computing and Applied Mathematics, Vol. 12, J.C. Baltzer AG, Scientific Publishing, Basel, 1994, pp. 293–316.
42. C. Jansson On self-validating methods for optimization problems J. Herzberger Topics in Validated Computations 1994 Elsevier Amsterdam 381 438
43. E.W. Kaucher W.L. Miranker Self-Validating Numerics for Function Space Problems Computations with Guarantees for Differential and Integral Equations 1984 Academic Press Orlando
44. R.B. Kearfott A review of techniques in the verified solution of constrained global optimization problems R.B. Kearfott V. Kreinovich Applications of Interval Computations 1996 Kluwer Dordrecht 23 59
45. ++ Class Library for Extended Scientific Computing 1993 Springer Berlin
46. R. Klatte U. Kulisch M. Neaga D. Ratz C. Ullrich PASCAL-XSC, Language Reference with Examples 1992 Springer Berlin
47. M. Koeber, Lösungseinschließung bei Anfangswertproblemen für quasilineare hyperbolische Differentialgleichungen, Thesis, Universität Karlsruhe, Karlsruhe, 1997.
48. R. Krawczyk Newton-Algorithmen zur Bestimmung von Nullstellen mit Fehlerschranken Computing 4 1969 187 201
49. U. Kulisch, Grundzüge der Intervallrechnung, in: Jahrbuch Überblicke Mathematik, Vol. 2, Bibliographisches Institut, Mannheim, 1969.
50. U. Kulisch, Numerical algorithms with automatic result verification, Lectures in Applied Mathematics, Vol. 32, 1996, pp. 471–502.
51. J.-R. Lahmann, Eine Methode zur Einschließung von Eigenpaaren nichtselbstadjungierter Eigenwertprobleme und ihre Anwendung auf die Orr-Sommerfeld-Gleichung, Thesis, Universität Karlsruhe, Karlsruhe, 1999.
52. B. Lang, Lokalisierung und Darstellung von Nullstellenmengen einer Funktion, Diploma Thesis, Universität Karlsruhe, Karlsruhe, 1989.
53. R. Lohner, Einschließung der Lösung gewöhnlicher Anfangs- und Randwertaufgaben und Anwendungen, Thesis, Universität Karlsruhe, Karlsruhe, 1988.
54. R. Lohner Enclosing all eigenvalues of symmetric matrices C. Ullrich J. Wolff von Gudenberg Accurate Numerical Algorithms. A Collection of Research Papers, Research Reports, ESPRIT, Project 1072, Diamond, Vol. 1 1989 Springer Berlin 87 103
55. G. Mayer Old and new aspects for the interval Gaussian algorithm E. Kaucher S.M. Markov G. Mayer Computer Arithmetic, Scientific Computation and Mathematical Modelling IMACS Annals on Computing and Applied Mathematics Vol. 12 1991 J.C. Baltzer AG, Scientific Publishing Basel 329 349
56. G. Mayer Enclosures for eigenvalues and eigenvectors L. Atanassova J. Herzberger Computer Arithmetic and Enclosure Methods 1992 Elsevier Amsterdam 49 67
57. G. Mayer Result verification for eigenvectors and eigenvalues J. Herzberger Topics in Validated Computations 1994 Elsevier Amsterdam 209 276
58. G. Mayer Epsilon-inflation in verification algorithms J. Comput. Appl. Math. 60 1995 147 169
59. G. Mayer Epsilon-inflation with contractive interval functions Appl. Math. 43 1998 241 254
60. G. Mayer J. Rohn On the applicability of the interval Gaussian algorithm Reliable Comput. 4 1998 205 222
61. O. Mayer, Über die in der Intervallrechnung auftretenden Räume und einige Anwendungen, Thesis, Universität Karlsuhe, Karlsruhe, 1968.
62. C. Miranda Un’ osservazione su un teorema di Brouwer Bol. Un. Mat. Ital. Ser. II 3 1941 5 7
63. R.E. Moore, Interval Arithmetic and Automatic Error Analysis in Digital Computing, Thesis, Stanford University, October 1962.
64. R.E. Moore Interval Analysis 1966 Prentice-Hall Englewood Cliffs, NJ
65. M.T. Nakao State of the art for numerical computations with guaranteed accuracy Math. Japanese 48 1998 323 338
66. N.S. Nedialkov, Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation, Thesis, University of Toronto, Toronto, 1999.
67. M. Neher An enclosure method for the solution of linear ODEs with polynomial coefficients Numer. Funct. Anal. Optim. 20 1999 779 803
68. A. Neumaier The enclosure of solutions of parameter-dependent systems of equations R.E. Moore Reliability in Computing. The Role of Interval Methods in Scientific Computing, Perspectives in Computing Vol. 19 1988 Academic Press Boston 269 286
69. A. Neumaier Interval Methods for Systems of Equations 1990 University Press Cambridge
70. H.T. Nguyen V. Kreinovich V. Nesterov M. Nakumura On hardware support for interval computations and for soft computing: theorems IEEE Trans. Fuzzy Systems 5 1 1997 108 127
71. J.M. Ortega W.C. Rheinboldt Iterative Solution of Nonlinear Equations in Several Variables 1970 Academic Press New York
72. W. Oettli W. Prager Compatibility of approximate solution of linear equations with given error bounds for coefficients and right-hand sides Numer. Math. 6 1964 405 409
73. M. Petković L.D. Petković Complex Interval Arithmetic and Its Applications 1998 Wiley New York
74. M. Plum Inclusion methods for elliptic boundary value problems J. Herzberger Topics in Validated Computations 1994 Elsevier Amsterdam 323 379
75. L.B. Rall Computational Solution of Nonlinear Operator Equations 1969 Wiley New York
76. L.B. Rall Automatic Differentiation: Techniques and Applications Lecture Notes in Computer Science Vol. 120 1981 Springer Berlin
77. H. Ratschek Centered forms SIAM J. Numer. Anal. 17 1980 656 662
78. H. Ratschek J. Rokne Computer Methods for the Range of Functions 1984 Ellis Horwood Chichester
79. H. Ratschek J. Rokne New Computer Methods for Global Optimization 1988 Ellis Horwood Chichester, UK
80. R. Rihm Interval methods for initial value problems in ODE's J. Herzberger Topics in Validated Computations 1994 Elsevier Amsterdam 173 207
81. S.M. Rump Solving algebraic problems with high accuracy U.W. Kulisch W.L. Miranker A New Approach to Scientific Computation 1983 Academic Press New York 53 120
82. S.M. Rump Rigorous sensitivity analysis for systems of linear and nonlinear equations Math. Comp. 54 1990 721 736
83. S.M. Rump On the solution of interval linear systems Computing 47 1992 337 353
84. S.M. Rump Verification methods for dense and sparse systems of equations J. Herzberger Topics in Validated Computations 1994 Elsevier Amsterdam 63 135
85. A. Schrijver Theory of Linear and Integer Programming 1986 Wiley New York
86. H. Schwandt, Schnelle fast global konvergente Verfahren für die Fünf-Punkte-Diskretisierung der Poissongleichung mit Dirichletschen Randbedingungen auf Rechteckgebieten, Thesis, Fachbereich Mathematik der TU Berlin, Berlin, 1981.
87. P.S. Shary Solving the linear interval tolerance problem Math. Comput. Simulation 39 1995 53 85
88. T. Sunaga Theory of an interval algebra and its application to numerical analysis RAAG Memoirs 2 1958 29 46
89. W.V. Walter FORTRAN-XSC: a portable FORTRAN 90 module library for accurate and reliable scientific computing R. Albrecht G. Alefeld H.J. Stetter Validation Numerics, Theory and Applications 1993 Springer Wien 265 285
90. G. Alefeld, A. Frommer, B. Lang (Eds.), Scientific Computing and Validated Numerics, Mathematical Research, Vol. 90, Akademie Verlag, Berlin, 1996.
91. G. Alefeld, R.D. Grigorieff (Eds.), Fundamentals of Numerical Computation, Computer-Oriented Numerical Analysis, Computing Supplementum, Vol. 2, Springer, Wien, 1980.
92. G. Alefeld, J. Herzberger (Eds.), Numerical Methods and Error Bounds, Mathematical Research, Vol. 89, Akademie Verlag, Berlin, 1996.
93. L. Atanassova, J. Herzberger (Eds.), Computer Arithmetic and Enclosure Methods, Elsevier, Amsterdam, 1992.
94. H. Bauch, K.-U. Jahn, D. Oelschlägel, H. Süße, V. Wiebigke, Intervallmathematik, Theorie und Anwendungen, Mathematisch-Naturwissenschaftliche Bibliothek, Vol. 72, BSB B.G. Teubner, Leipzig, 1987.
95. E. Hansen (Ed.), Topics in Interval Analysis, Oxford University Press, Oxford, 1969.
96. J. Herzberger (Ed.), Topics in Validated Computations, Elsevier, Amsterdam, 1994.
97. J. Herzberger (Ed.), Wissenschaftliches Rechnen, Eine Einführung in das Scientific Computing, Akademie Verlag, Berlin, 1995.
98. S.A. Kalmykov, J.I. Shokin, S.C. Yuldashev, Methods of Interval Analysis, Novosibirsk, 1986 (in Russian).
99. E. Kaucher, U. Kulisch, C. Ullrich (Eds.), Computerarithmetic, B.G. Teubner, Stuttgart, 1987.
100. E. Kaucher, S.M. Markov, G. Mayer (Eds.), Computer Arithmetic, Scientific Computation and Mathematical Modelling, IMACS Annals on Computing and Applied Mathematics, Vol. 12, J.C. Baltzer AG, Scientific Publishing, Basel, 1991.
101. R.B. Kearfott, V. Kreinovich (Eds.), Applications of Interval Computations, Kluwer, Dordrecht, 1996.
102. V. Kreinovich, A. Lakeyev, J. Rohn, P. Kahl, Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer, Dordrecht, 1998.
103. U. Kulisch (Ed.), Wissenschaftliches Rechnen mit Ergebnisverifikation, Vieweg, Braunschweig, 1989.
104. U.W. Kulisch, W.L. Miranker, Computer Arithmetic in Theory and Practice, Academic Press, New York, 1981.
105. U.W. Kulisch, W.L. Miranker (Eds.), A New Approach to Scientific Computation, Academic Press, New York, 1983.
106. U. Kulisch, W. Miranker, The arithmetic of the digital computer: a new approach SIAM Rev. 28 (1986) 1–40.
107. U. Kulisch, H.J. Stetter (Eds.), Scientific Computation with Automatic Result Verification, Computing Supplementum, Vol. 6, Springer, Wien, 1988.
108. S.M. Markov (Ed.), Scientific Computation and Mathematical Modelling, DATECS Publishing, Sofia, 1993.
109. W.L. Miranker, R.A. Toupin (Eds.), Accurate Scientific Computations, Lecture Notes in Computer Science, Vol. 235, Springer, Berlin, 1986.
110. R.E. Moore, Methods and Applications of Interval Analysis, SIAM, Philadelphia, 1979.
111. R.E. Moore, Computational Functional Analysis, Ellis Horwood, Chichester, 1985.
112. R.E. Moore (Ed.), Reliability in Computing. The Role of Interval Methods in Scientific Computing, Perspectives in Computing, Vol. 19, Academic Press, Boston, 1988.
113. K. Nickel (Ed.), Interval Mathematics, Lecture Notes in Computer Science, Vol. 29, Springer, Berlin, 1975.
114. K.L.E. Nickel (Ed.), Interval Mathematics 1980, Academic Press, New York, 1980.
115. K. Nickel (Ed.), Interval Mathematics 1985, Lecture Notes in Computer Science, Vol. 212, Springer, Berlin, 1985.
116. M. Petković, Iterative Methods for Simultaneous Inclusion of Polynomial Zeros, Lecture Notes in Mathematics, Vol. 1387, Springer, Berlin, 1989.
117. C. Ullrich (Ed.), Computer Arithmetic and Self-Validating Numerical Methods, Academic Press, Boston, 1990.
118. C. Ullrich (Ed.), Contributions to Computer Arithmetic and Self-Validating Numerical Methods, IMACS Annals on Computing and Applied Mathematics, Vol. 7, J.C. Baltzer AG, Scientific Publishing, Basel, 1990.
119. C. Ullrich, J. Wolff von Gudenberg (Eds.), Accurate Numerical Algorithms. A Collection of Research Papers, Research Reports, ESPRIT, Project 1072, Diamond, Vol. 1, Springer, Berlin, 1989.
120. T. Csendes (Ed.), Developments in Reliable Computing, Kluwer, Dordrecht, 1999.