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Epsilon-inflation in verification algorithms. (English) Zbl 0839.65059
The paper deals with the problem of finite termination of verification algorithms using an \(\varepsilon\)-inflation (introduced by S. M. Rump). Sufficient conditions for finite termination are given, based essentially on the assumption of the \(P\)-contraction property of the response mapping. The results are illustrated on several examples including computations of verified enclosures for eigenvalues and singular values.
Reviewer: J.Rohn (Praha)

65G30 Interval and finite arithmetic
65H10 Numerical computation of solutions to systems of equations
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
Full Text: DOI
[1] Alefeld, G., Berechenbare fehlerschranken für ein eigenpaar unter einschluß von rundungsfehlern bei verwendung des genauen skalarprodukts, Z. angew. math. mech., 67, 145-152, (1987) · Zbl 0622.65028
[2] Alefeld, G., Rigorous error bounds for singular values of a matrix using the precise scalar product, (), 9-30
[3] Alefeld, G., Berechenbare fehlerschranken für ein eigenpaar beim verallgemeinerten eigenwertproblem, Z. angew. math. mech., 68, 181-184, (1988) · Zbl 0647.65024
[4] Alefeld, G.; Herzberger, J., Introduction to interval computations, (1983), Academic Press New York
[5] Bauch, H.; Jahn, K.-U.; Oelschlägel, D.; Süsse, H.; Wiebigke, V., Intervallmathematik, (1987), BSB Teubner Leipzig · Zbl 0655.65070
[6] Caprani, O.; Madsen, K., Iterative methods for interval inclusion of fixed points, Bit, 18, 42-51, (1978) · Zbl 0401.65035
[7] Grüner, K., Solving the complex algebraic eigenvalue problem with verified high accuracy, (), 59-78 · Zbl 0675.65024
[8] Heindl, G., Computable hypernorm bounds for the error of an approximate solution to a system of nonlinear equations, (), 451-461 · Zbl 0839.65069
[9] Klatte, R.; Kulisch, U.; Neaga, M.; Ratz, D.; Ullrich, Ch., PASCAL-XSC, sprachbeschreibung mit beispielen, (1991), Springer Berlin · Zbl 0757.68023
[10] M. Koeber, Private communication, Karlsruhe, 1993
[11] König, S., On the inflation parameter used in self-validating methods, (), 127-132 · Zbl 0784.65024
[12] Mayer, G., Grundbegriffe der intervallrechnung, (), 101-117
[13] Mayer, G., Enclosures for eigenvalues and eigenvectors, (), 49-67 · Zbl 0838.65036
[14] Mayer, G., Taylor-verfahren für das algebraische eigenwertproblem, Z. angew. math. mech., 73, T921-T924, (1993)
[15] Mayer, G., A unified approach to enclosure methods for eigenpairs, Z. angew. math. mech., 74, 115-128, (1994) · Zbl 0817.65028
[16] Moore, R.E., Interval analysis, (1966), Prentice-Hall Englewood Cliffs, NJ · Zbl 0176.13301
[17] Moore, R.E., Methods and applications of interval analysis, (1979), SIAM Philadelphia, PA · Zbl 0417.65022
[18] Neumaier, A., Interval methods for systems of equations, (1990), Cambridge Univ. Press Cambridge · Zbl 0706.15009
[19] Ortega, J.M., Numerical analysis: A second course, () · Zbl 0701.65002
[20] Ortega, J.M.; Rheinboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[21] Rump, S.M., Kleine fehlerschranken bei matrixproblemen, () · Zbl 0437.65036
[22] Rump, S.M., Solving algebraic problems with high accuracy, (), 53-120
[23] Rump, S.M., On the solution of interval linear systems, Computing, 47, 337-353, (1992) · Zbl 0753.65030
[24] Varga, R.S., Matrix iterative analysis, (1962), Prentice-Hall Englewood Cliffs, NJ · Zbl 0133.08602
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