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Computation and Application of Taylor Polynomials with Interval Remainder Bounds
 Reliable Computing
, 1998
"... . The expansion of complicated functions of many variables in Taylor polynomials is an important problem for many applications, and in practice can be performed rather conveniently (even to high orders) using polynomial algebras. An important application of these methods is the field of beam physics ..."
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Cited by 34 (2 self)
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. The expansion of complicated functions of many variables in Taylor polynomials is an important problem for many applications, and in practice can be performed rather conveniently (even to high orders) using polynomial algebras. An important application of these methods is the field of beam physics, where often expansions in about six variables to orders between five and ten are used. However, often it is necessary to also know bounds for the remainder term of the Taylor formula if the arguments lie within certain intervals. In principle such bounds can be obtained by interval bounding of the (n+1)st derivative, which in turn can be obtained with polynomial algebra; but in practice the method is rather inefficient and susceptible to blowup because of the need of repeated interval evaluations of the derivative. Here we present a new method that allows the computation of sharp remainder intervals in parallel with the accumulation derivatives up to order n. The method is useful for a...
Calculus and Numerics on LeviCivita Fields
, 1996
"... The formal process of the evaluation of derivatives using some of the various modern methods of computational differentiation can be recognized as an example for the application of conventional "approximate" numerical techniques on a nonarchimedean extension of the real numbers. In many cases, the ..."
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Cited by 16 (6 self)
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The formal process of the evaluation of derivatives using some of the various modern methods of computational differentiation can be recognized as an example for the application of conventional "approximate" numerical techniques on a nonarchimedean extension of the real numbers. In many cases, the application of "infinitely small" numbers instead of "small but finite" numbers allows the use of the old numerical algorithm, but now with an error that in a rigorous way can be shown to become infinitely small (and hence irrelevant). While intuitive ideas in this direction have accompanied analysis from the early days of Newton and Leibniz, the first rigorous work goes back to LeviCivita, who introduced a number field that in the past few years turned out to be particularly suitable for numerical problems. While LeviCivita's field appears to be of fundamental importance and simplicity, efforts to introduce advanced concepts of calculus on it are only very new. In this paper, we address s...
Convergence on the LeviCivita Field and Study of Power Series
 In Lecture Notes in Pure and Applied Mathematics
"... Abstract. Convergence under various topologies and analytical properties of power series on LeviCivita fields are studied. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asser ..."
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Cited by 12 (7 self)
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Abstract. Convergence under various topologies and analytical properties of power series on LeviCivita fields are studied. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points the distance of which from the center is infinitely smaller than the radius of convergence. In addition to allowing the introduction of common transcendental functions, power series are shown to behave similar to real power series. Besides being infinitely often differentiable and reexpandable around other points, it is shown that power series satisfy a general intermediate value theorem as well as a maximum theorem and a mean value theorem. 1.
Analytical properties of power series on LeviCivita fields
 Ann. Math. Blaise Pascal
, 2005
"... A detailed study of power series on the LeviCivita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are i ..."
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Cited by 8 (4 self)
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A detailed study of power series on the LeviCivita fields is presented. After reviewing two types of convergence on those fields, including convergence criteria for power series, we study some analytical properties of power series. We show that within their domain of convergence, power series are infinitely often differentiable and reexpandable around any point within the radius of convergence from the origin. Then we study a large class of functions that are given locally by power series and contain all the continuations of real power series. We show that these functions have similar properties as real analytic functions. In particular, they are closed under arithmetic operations and composition and they are infinitely often differentiable. 1
OneDimensional Optimization on NonArchimedean Fields. Journal of Nonlinear and Convex Analysis, 2:351–361
, 2001
"... Abstract. One dimensional optimization on nonArchimedean fields is presented. We derive first and second order necessary and sufficient optimality conditions. For first order optimization, these conditions are similar to the corresponding real ones; but this is not the case for higher order optimiz ..."
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Cited by 7 (5 self)
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Abstract. One dimensional optimization on nonArchimedean fields is presented. We derive first and second order necessary and sufficient optimality conditions. For first order optimization, these conditions are similar to the corresponding real ones; but this is not the case for higher order optimization. This is due to the total disconnectedness of the given nonArchimedean field in the order topology, which renders the usual concept of differentiability weak. We circumvent this difficulty by using a stronger concept of differentiability based on the derivate approach, which entails a Taylor formula with remainder and hence a similar local behavior as in the real case. 1.
On the topological structure of the LeviCivita field
 J. Math. Anal. Appl
"... This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or sel ..."
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Cited by 5 (2 self)
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This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal noncommercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are
The implicit function theorem in a nonArchimedean setting ✩
"... In this paper, the inverse function theorem and the implicit function theorem in a nonArchimedean setting will be discussed. We denote by N any nonArchimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properti ..."
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Cited by 1 (1 self)
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In this paper, the inverse function theorem and the implicit function theorem in a nonArchimedean setting will be discussed. We denote by N any nonArchimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properties of locally uniformly differentiable functions from N n to N m. Then we use that concept of local uniform differentiability to formulate and prove the inverse function theorem for functions from N n to N n and the implicit function theorem for functions from N n to N m with m<n. 1.
Computational Divided Differencing and DividedDifference Arithmetics
, 2000
"... Tools for computational differentiation transform a program that computes a numerical function F (x) into a related program that computes F 0 (x) (the derivative of F ). This paper describes how techniques similar to those used in computationaldifferentiation tools can be used to implement other pr ..."
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Cited by 1 (0 self)
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Tools for computational differentiation transform a program that computes a numerical function F (x) into a related program that computes F 0 (x) (the derivative of F ). This paper describes how techniques similar to those used in computationaldifferentiation tools can be used to implement other program transformations  in particular, a variety of transformations for computational divided differencing . The specific technical contributions of the paper are as follows: It presents a program transformation that, given a numerical function F (x) de ned by a program, creates a program that computes F [x0 ; x1 ], the first divided difference of F(x), where F [x0 ; x1 ] def = F (x 0 ) F (x 1 ) x 0 x 1 if x0 6= x1 d dz F (z); evaluated at z = x0 if x0 = x1 It shows how computational first divided differencing generalizes computational differentiation. It presents a second program transformation that permits the creation of higherorder divided differences of a numerical function de ...
Generalized power series on a nonArchimedean field
, 2005
"... Power series with rational exponents on the real numbers field and the LeviCivita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we ..."
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Power series with rational exponents on the real numbers field and the LeviCivita field are studied. We derive a radius of convergence for power series with rational exponents over the field of real numbers that depends on the coefficients and on the density of the exponents in the series. Then we generalize that result and study power series with rational exponents on the LeviCivita field. A radius of convergence is established that asserts convergence under a weak topology and reduces to the conventional radius of convergence for real power series. It also asserts strong (order) convergence for points whose distance from the center is infinitely smaller than the radius of convergence. Then we study a class of functions that are given locally by power series with rational exponents, which are shown to form a commutative algebra over the LeviCivita field; and we study the differentiability properties of such functions within their domain of convergence. 1.