Matrix representation of Taylor’s formula for mappings in finite dimensional spaces

P. S. Senio

doi:10.15330/ms.51.1.92-106

In this paper, the matrix representations of derivatives, differentials and Taylor’s formulas for functionals and mappings are presented. The behavior law of intermediate points in the formula’s remainder terms is found for shrinking the expansion interval of mapping to a point. This, in particular, allows one to analyze the local uncertainties in the remainder terms of Taylor’s formula and to develop and explore the new effective methods for solving systems of nonlinear algebraic equations.

1. Kolmogorov A.N., Fomin S.V., Elements of the theory of functions and functional analysis. M.: Nauka, 1989. (in Russian)

2. Ortega J., Ranboldt V., Iterative methods for solving nonlinear systems of equations with many unknowns. M.: Mir, 1975. (in Russian)

3. Senio P.S., A new approach to the construction of interval methods for solving nonlinear equations, Visn. Lviv Univ., Ser. Mekh.-Mat., 31 (1989), 85–92. (in Ukranian)

4. Senio P.S., Interval methods for solving certain classes of deterministic problems, Visn. Lviv Univ., Ser. Prykladn. Mat. Inform., 7 (2003), 86–96. (in Ukranian)

5. Senio P.S., Two-sided approximation finding of a function based on mathematics of functional intervals. Proceedings of XXXII International Conference PDMU. Czech Republic. Prague. August 27-31, 2018, 146–154.

6. Ilin V.A., Sadovnichiy V.A., Sendov B.K., Mathematical Analysis. Beginner course. M.: MGU, 1985. (in Russian)

7. Schwartz L., Analysis, V.1, M.: Mir, 1972. (in Russian) 8. Zorich V.A., Mathematical Analysis II. Berlin-Heidelberg: Springer-Verlag, 2 ed., 2016.

	Matrix representation of Taylor’s formula for mappings in finite dimensional spaces
Author	P. S. Senio senio.petro@gmail.com Ivan Franko National University of Lviv, Lviv, Ukraine
Abstract	In this paper, the matrix representations of derivatives, differentials and Taylor’s formulas for functionals and mappings are presented. The behavior law of intermediate points in the formula’s remainder terms is found for shrinking the expansion interval of mapping to a point. This, in particular, allows one to analyze the local uncertainties in the remainder terms of Taylor’s formula and to develop and explore the new effective methods for solving systems of nonlinear algebraic equations.
Keywords	functional and mapping derivatives; functional and mapping differentials; Taylor’s formula for functionals and mappings; remainder terms of Taylor’s formula
DOI	doi:10.15330/ms.51.1.92-106
Reference	1. Kolmogorov A.N., Fomin S.V., Elements of the theory of functions and functional analysis. M.: Nauka, 1989. (in Russian) 2. Ortega J., Ranboldt V., Iterative methods for solving nonlinear systems of equations with many unknowns. M.: Mir, 1975. (in Russian) 3. Senio P.S., A new approach to the construction of interval methods for solving nonlinear equations, Visn. Lviv Univ., Ser. Mekh.-Mat., 31 (1989), 85–92. (in Ukranian) 4. Senio P.S., Interval methods for solving certain classes of deterministic problems, Visn. Lviv Univ., Ser. Prykladn. Mat. Inform., 7 (2003), 86–96. (in Ukranian) 5. Senio P.S., Two-sided approximation finding of a function based on mathematics of functional intervals. Proceedings of XXXII International Conference PDMU. Czech Republic. Prague. August 27-31, 2018, 146–154. 6. Ilin V.A., Sadovnichiy V.A., Sendov B.K., Mathematical Analysis. Beginner course. M.: MGU, 1985. (in Russian) 7. Schwartz L., Analysis, V.1, M.: Mir, 1972. (in Russian) 8. Zorich V.A., Mathematical Analysis II. Berlin-Heidelberg: Springer-Verlag, 2 ed., 2016.
Pages	92-106
Volume	51
Issue	1
Year	2019
Journal	Matematychni Studii
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