The numerical inversion of functions from the plane to the plane
HTML articles powered by AMS MathViewer
- by Iaci Malta, Nicolau C. Saldanha and Carlos Tomei PDF
- Math. Comp. 65 (1996), 1531-1552 Request permission
Abstract:
This paper contains a description of a program designed to find all the solutions of systems of two real equations in two real unknowns which uses detailed information about the critical set of the associated function from the plane to the plane. It turns out that the critical set and its image are highly structured, and this is employed in their numerical computation. The conceptual background and details of implementation are presented. The most important features of the program are the ability to provide global information about the function and the robustness derived from such topological information.References
- Eugene L. Allgower, Kurt Georg, and Rick Miranda (eds.), Exploiting symmetry in applied and numerical analysis, Lectures in Applied Mathematics, vol. 29, American Mathematical Society, Providence, RI, 1993. MR 1247709
- R. Fletcher, Practical methods of optimization. Vol. 1, A Wiley-Interscience Publication, John Wiley & Sons, Ltd., Chichester, 1980. Unconstrained optimization. MR 585160
- George K. Francis and Stephanie F. Troyer, Excellent maps with given folds and cusps, Houston J. Math. 3 (1977), no. 2, 165–194. MR 516183
- Philip E. Gill, Walter Murray, and Margaret H. Wright, Practical optimization, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1981. MR 634376
- Herbert B. Keller, Global homotopies and Newton methods, Recent advances in numerical analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1978) Publ. Math. Res. Center Univ. Wisconsin, vol. 41, Academic Press, New York-London, 1978, pp. 73–94. MR 519057
- M. Kubíček and M. Marek, Computational methods in bifurcation theory and dissipative structures, Springer Series in Computational Physics, Springer-Verlag, New York, 1983. MR 719370, DOI 10.1007/978-3-642-85957-1
- I. Malta, N. C. Saldanha and C. Tomei, Critical sets of proper Whitney functions in the plane (to appear)
- Iaci Malta and Carlos Tomei, Singularities of vector fields arising from one-dimensional Riemann problems, J. Differential Equations 94 (1991), no. 1, 165–190. MR 1133545, DOI 10.1016/0022-0396(91)90107-K
- John W. Milnor, Topology from the differentiable viewpoint, University Press of Virginia, Charlottesville, Va., 1965. Based on notes by David W. Weaver. MR 0226651
- Alexander Morgan, Solving polynomial systems using continuation for engineering and scientific problems, Prentice Hall, Inc., Englewood Cliffs, NJ, 1987. MR 1049872
- V. Poénaru, Extending immersions of the circle (d’après Samuel Blank), Exposé 342, Séminaire Bourbaki 1967–68, Benjamin, NY, 1969.
- J. R. Quine, A global theorem for singularities of maps between oriented $2$-manifolds, Trans. Amer. Math. Soc. 236 (1978), 307–314. MR 474378, DOI 10.1090/S0002-9947-1978-0474378-X
- S. F. Troyer, Extending a boundary immersion to the disk with $n$ holes, PhD Dissertation, Northeastern Univ., Boston, Mass., 1973
- Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Additional Information
- Iaci Malta
- Affiliation: Departamento de Matemática, PUC-Rio, Rua Marquês de São Vicente 225, Rio de Janeiro 22453-900, Brasil
- Email: malta@mat.puc-rio.br
- Nicolau C. Saldanha
- Affiliation: IMPA, Estr. Dona Castorina 110, Rio de Janeiro 22460-320, Brasil
- Email: nicolau@impa.br
- Carlos Tomei
- Affiliation: Departamento de Matemática, PUC-Rio, Rua Marquês de São Vicente 225, Rio de Janeiro 22453-900, Brasil
- Email: tomei@mat.puc-rio.br
- Received by editor(s): May 31, 1994
- Received by editor(s) in revised form: July 10, 1995
- Additional Notes: The authors received support from MCT and CNPq, Brazil.
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp. 65 (1996), 1531-1552
- MSC (1991): Primary 57R45, 65H10; Secondary 57R42, 65H20
- DOI: https://doi.org/10.1090/S0025-5718-96-00770-3
- MathSciNet review: 1361809