Abstract
Algorithms for determining computationally rigorous componentwise bounds on the solutions x ∈ R n of equations F(x, t) = 0 ∈ R m containing parameters t ∈ R l due to small perturbations in t when m ≤ n and when F is at most twice continuously differentiable in x and in t are described. Numerical results which illustrate the behaviour of the algorithms are presented.
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References
Alefeld, G. and Herzberger, J.: Introduction to Interval Computations, Academic Press, London, 1983.
Ben-Israel, A. and Greville, T. N. E.: Generalized Inverses: Theory and Applications, John Wiley & Sons, New York, 1974.
Campbell, S. L. and Meyer, C. D.: Generalized Inverses of Linear Transformations, Dover Publications Inc., New York, 1991.
Gay, D. M.: Computing Perturbation Bounds for Nonlinear Algebraic Equations, SIAM J. Numer. Anal. 20 (1983), pp. 638–651.
Greville, T. N. E.: The Pseudoinverse of a Rectangular or Singular Matrix and Its Application to the Solution of Systems of Linear Equations, SIAM Review 1 (1959), pp. 38–43.
Hammer, R., Hocks, M., Kulisch, U., and Ratz, D.: Numerical Toolbox for Verified Computing I, Springer-Verlag, Berlin Heidelberg, 1993.
Kearfott, R. B.: Rigorous Global Search: Continuous Problems, Kluwer Academic Publishers, Dordrecht, 1996.
Neumaier, A.: Interval Methods for Systems of Equations, Cambridge University Press, Cambridge, 1990.
Neumaier, A.: Rigorous Sensitivity Analysis for Parameter-Dependent Systems of Equations, Journal of Mathematical Analysis and Applications 144 (1989), pp. 16–25.
Ortega, J. M. and Rheinboldt, W. C.: Iterative Solution of Systems of Nonlinear AlgebraicEquations, Academic Press, London, 1970.
Penrose, R.: A Generalized Inverse for Matrices, Proc. Camb. Phil. Soc. 51 (1955), pp. 406–413.
Rheinboldt, W. C.: Numerical Analysis of Parametrized Nonlinear Equations, John Wiley & Sons, New York, 1986.
Walker, H. F.: Newton-Like Methods for Underdetermined Systems, in: Algower, E. L. and Georg, K. (eds), Computational Solutions of Nonlinear Systems of Equations, Lecture Notes in Applied Mathematics 26, American Mathematical Society, Providence, 1990, pp. 679–699.
Wolfe, M. A.: Interval Mathematics, Algebraic Equations and Optimization, Journal of Computational and Applied Mathematics 124 (2000), pp. 263–280.
Wolfe, M. A.: On Bounding Solutions of Underdetermined Systems, Reliable Computing 7 (3) (2001), pp. 195–207.
Yamamura, K., Kawata, H., and Tokue, A.: Interval Solution of Nonlinear Equations Using Linear Programming, BIT 38 (1) (1998), pp. 186–199.
Zhang, D., Li, W., and Shen, Z.: Solving Undetermined Systems with Interval Methods, Reliable Computing 5(1) (1999), pp. 23–33.
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Wolfe, M.A. Bounding Perturbations in Zeros of Nonlinear Systems. Reliable Computing 8, 177–188 (2002). https://doi.org/10.1023/A:1015559628657
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DOI: https://doi.org/10.1023/A:1015559628657