Approximations and error bounds for computing the inverse mapping

Lucas Jódar; Enrique Ponsoda; G. Rodríguez Sánchez

Applications of Mathematics (1997)

  • Volume: 42, Issue: 2, page 99-110
  • ISSN: 0862-7940

Abstract

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In this paper we propose a procedure to construct approximations of the inverse of a class of 𝒞 m differentiable mappings. First of all we determine in terms of the data a neighbourhood where the inverse mapping is well defined. Then it is proved that the theoretical inverse can be expressed in terms of the solution of a differential equation depending on parameters. Finally, using one-step matrix methods we construct approximate inverse mappings of a prescribed accuracy.

How to cite

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Jódar, Lucas, Ponsoda, Enrique, and Rodríguez Sánchez, G.. "Approximations and error bounds for computing the inverse mapping." Applications of Mathematics 42.2 (1997): 99-110. <http://eudml.org/doc/32971>.

@article{Jódar1997,
abstract = {In this paper we propose a procedure to construct approximations of the inverse of a class of $\{\mathcal \{C\}\}^\{m\}$ differentiable mappings. First of all we determine in terms of the data a neighbourhood where the inverse mapping is well defined. Then it is proved that the theoretical inverse can be expressed in terms of the solution of a differential equation depending on parameters. Finally, using one-step matrix methods we construct approximate inverse mappings of a prescribed accuracy.},
author = {Jódar, Lucas, Ponsoda, Enrique, Rodríguez Sánchez, G.},
journal = {Applications of Mathematics},
keywords = {approximations; inverse mapping; error bounds; Banach space; matrix differential equations; one-step-method; numerical example; inverse mapping; error bounds; Banach space; matrix differential equations; one-step-method; numerical example},
language = {eng},
number = {2},
pages = {99-110},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Approximations and error bounds for computing the inverse mapping},
url = {http://eudml.org/doc/32971},
volume = {42},
year = {1997},
}

TY - JOUR
AU - Jódar, Lucas
AU - Ponsoda, Enrique
AU - Rodríguez Sánchez, G.
TI - Approximations and error bounds for computing the inverse mapping
JO - Applications of Mathematics
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 42
IS - 2
SP - 99
EP - 110
AB - In this paper we propose a procedure to construct approximations of the inverse of a class of ${\mathcal {C}}^{m}$ differentiable mappings. First of all we determine in terms of the data a neighbourhood where the inverse mapping is well defined. Then it is proved that the theoretical inverse can be expressed in terms of the solution of a differential equation depending on parameters. Finally, using one-step matrix methods we construct approximate inverse mappings of a prescribed accuracy.
LA - eng
KW - approximations; inverse mapping; error bounds; Banach space; matrix differential equations; one-step-method; numerical example; inverse mapping; error bounds; Banach space; matrix differential equations; one-step-method; numerical example
UR - http://eudml.org/doc/32971
ER -

References

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  1. Introduction to Higher Algebra, New York, McMillan, 1947. (1947) 
  2. Foundations of Modern Analysis, New York, Academic Press, 1960. (1960) MR0120319
  3. Linear Operators, Part I, New York, Interscience, 1957. (1957) 
  4. Differential Analysis, Cambridge, Cambrigde Univ. Press, 1980. (1980) Zbl0442.34002MR0561908
  5. Matrix Computations, Baltimore, John Hopkins Univ. Press, 1983. MR0733103
  6. Kronecker Products and Matrix Calculus with Applications, New York, John Wiley, 1981. (1981) Zbl0497.26005MR0640865
  7. Discrete Variable Methods in Ordinary Differential Equations, New York, John Wiley, 1962. (1962) Zbl0112.34901MR0135729
  8. 10.1137/0116083, SIAM J. Appl. Math. 16 (1968)), 1020–1023. (1968)) Zbl0169.35202MR0234974DOI10.1137/0116083
  9. 10.1093/imanum/15.1.61, IMA J. Numer. Anal. 15 (1995), 61–74. (1995) MR1311337DOI10.1093/imanum/15.1.61
  10. Computational Methods in Ordinary Differential Equations, New York, John Wiley, 1962. (1962) MR0423815
  11. The Theory of Matrices. 2nd. ed, New York, Academic Press, 1985. (1985) MR0792300
  12. 10.1137/0118061, SIAM J. Appl. Math. 18 (1970), no. 3, 682–687. (1970) MR0260158DOI10.1137/0118061
  13. Numerical Analysis, a Second Course, New York, Academic Press, 1972. (1972) Zbl0248.65001MR0403154
  14. Derivative operations on matrices, IEEE Trans. Aut. Control. AC-15 (1970), 241–244. (1970) MR0272801
  15. Mathematica, a System for Doing Mathematics by Computer, Redwood City, Addison Wesley Publishing Co., 1989. (1989) 

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