Relevant bibliographies by topics / Approximation of derivatives of functions

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  1. Journal articles
  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers
  6. Reports

Academic literature on the topic 'Approximation of derivatives of functions'

Author: Grafiati
Published: 5 June 2025
Last updated: 16 July 2025

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Journal articles on the topic "Approximation of derivatives of functions"

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Jetpisbayeva, A. E., and A. A. Jumabayeva. "Upper Estimates of the angle best approximations of generalized Liouville-Weyl derivatives." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 103, no. 3 (2021): 54–67. http://dx.doi.org/10.31489/2021m3/54-67.

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In this article we consider continuous functions f with period 2π and their approximation by trigonometric polynomials. This article is devoted to the study of estimates of the best angular approximations of generalized Liouville-Weyl derivatives by angular approximation of functions in the three-dimensional case. We consider generalized Liouville-Weyl derivatives instead of the classical mixed Weyl derivative. In choosing the issues to be considered, we followed the general approach that emerged after the work of the second author of this article. Our main goal is to prove analogs of the results of in the three-dimensional case. The concept of general monotonic sequences plays a key role in our study. Several well-known inequalities are indicated for the norms, best approximations of the r-th derivative with respect to the best approximations of the function f. The issues considered in this paper are related to the range of issues studied in the works of Bernstein. Later Stechkin and Konyushkov obtained an inequality for the best approximation f^(r). Also, in the works of Potapov, using the angle approximation, some classes of functions are considered. In subsection 1 we give the necessary notation and useful lemmas. Estimates for the norms and best approximations of the generalized Liouville-Weyl derivative in the three-dimensional case are obtained.
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Lamboni, Matieyendou. "Optimal Estimators of Cross-Partial Derivatives and Surrogates of Functions." Stats 7, no. 3 (2024): 697–718. http://dx.doi.org/10.3390/stats7030042.

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Computing cross-partial derivatives using fewer model runs is relevant in modeling, such as stochastic approximation, derivative-based ANOVA, exploring complex models, and active subspaces. This paper introduces surrogates of all the cross-partial derivatives of functions by evaluating such functions at N randomized points and using a set of L constraints. Randomized points rely on independent, central, and symmetric variables. The associated estimators, based on NL model runs, reach the optimal rates of convergence (i.e., O(N−1)), and the biases of our approximations do not suffer from the curse of dimensionality for a wide class of functions. Such results are used for (i) computing the main and upper bounds of sensitivity indices, and (ii) deriving emulators of simulators or surrogates of functions thanks to the derivative-based ANOVA. Simulations are presented to show the accuracy of our emulators and estimators of sensitivity indices. The plug-in estimates of indices using the U-statistics of one sample are numerically much stable.
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Mamman, John Ojima. "Computational Algorithm for Approximating Fractional Derivatives of Functions." Journal of Modeling and Simulation of Materials 5, no. 1 (2022): 31–38. http://dx.doi.org/10.21467/jmsm.5.1.31-38.

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This paper presents an algorithmic approach for numerically solving Caputo fractional differentiation. The trapezoidal rule was modified, the new modification was used to derive an algorithm to approximate fractional derivatives of order α > 0, the fractional derivative used was based on Caputo definition for a given function by a weighted sum of function and its ordinary derivatives values at specified points. The trapezoidal rule was used in conjunction with the finite difference scheme which is the forward, backward and central difference to derive the computational algorithm for the numerical approximation of Caputo fractional derivative for evaluating functions of fractional order. The study was conducted through some illustrative examples and analysis of error.
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Arsent’eva, E. P., and Yu K. Dem’yanovich. "Approximation of functions with growing derivatives." Journal of Mathematical Sciences 178, no. 6 (2011): 565–75. http://dx.doi.org/10.1007/s10958-011-0570-9.

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Vinogradov, Oleg L. "On the constants in the inverse theorems for the first derivative." Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy 8, no. 4 (2021): 559–71. http://dx.doi.org/10.21638/spbu01.2021.401.

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The known proofs of the inverse theorems of the theory of approximation by trigonometric polynomials and by functions of exponential type are based on the idea of S. N. Bernstein to expand a function in a series containing its functions of best approximation. In this paper, a new method to establish the inverse theorems is introduced. We establish simple identities that immediately imply the inverse theorems mentioned and, moreover, with better constants. This method can be applied to derivatives of arbitrary order (not necessarily an integer one) and (with certain modifications) to estimates of some other functionals in terms of best approximations. In this paper, the case of the first derivative of a function itself and of its trigonometrically conjugate is considered.
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Singh, Satwinder Jit, and Anindya Chatterjee. "Beyond fractional derivatives: local approximation of other convolution integrals." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 466, no. 2114 (2009): 563–81. http://dx.doi.org/10.1098/rspa.2009.0378.

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Dynamic systems involving convolution integrals with decaying kernels, of which fractionally damped systems form a special case, are non-local in time and hence infinite dimensional. Straightforward numerical solution of such systems up to time t needs computations owing to the repeated evaluation of integrals over intervals that grow like t . Finite-dimensional and local approximations are thus desirable. We present here an approximation method which first rewrites the evolution equation as a coupled infinite-dimensional system with no convolution, and then uses Galerkin approximation with finite elements to obtain linear, finite-dimensional, constant coefficient approximations for the convolution. This paper is a broad generalization, based on a new insight, of our prior work with fractional order derivatives ( Singh & Chatterjee 2006 Nonlinear Dyn. 45 , 183–206). In particular, the decaying kernels we can address are now generalized to the Laplace transforms of known functions; of these, the power law kernel of fractional order differentiation is a special case. The approximation can be refined easily. The local nature of the approximation allows numerical solution up to time t with computations. Examples with several different kernels show excellent performance. A key feature of our approach is that the dynamic system in which the convolution integral appears is itself approximated using another system, as distinct from numerically approximating just the solution for the given initial values; this allows non-standard uses of the approximation, e.g. in stability analyses.
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Ditzian, Z., and D. Jiang. "Approximation of Functions by Polynomials in C[-L, 1]." Canadian Journal of Mathematics 44, no. 5 (1992): 924–40. http://dx.doi.org/10.4153/cjm-1992-057-2.

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AbstractA pointwise estimate for the rate of approximation by polynomials , For 0 ≤ ƛ ≤ 1, integer r, and δn(x) = n-1 + φ(x), is achieved here. This formula bridges the gap between the classical estimate mentioned in most texts on approximation and obtained by Timan and others (ƛ = 0) and the recently developed estimate by Totik and first author (ƛ = 1 ). Furthermore, a matching converse result and estimates on derivatives of the approximating polynomials and their rate of approximation are derived. These results also cover the range between the classical pointwise results and the modern norm estimates for C[— 1,1].
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Hornik, Kurt, Maxwell Stinchcombe, Halbert White, and Peter Auer. "Degree of Approximation Results for Feedforward Networks Approximating Unknown Mappings and Their Derivatives." Neural Computation 6, no. 6 (1994): 1262–75. http://dx.doi.org/10.1162/neco.1994.6.6.1262.

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Recently Barron (1993) has given rates for hidden layer feedforward networks with sigmoid activation functions approximating a class of functions satisfying a certain smoothness condition. These rates do not depend on the dimension of the input space. We extend Barron's results to feedforward networks with possibly nonsigmoid activation functions approximating mappings and their derivatives simultaneously. Our conditions are similar but not identical to Barron's, but we obtain the same rates of approximation, showing that the approximation error decreases at rates as fast as n−1/2, where n is the number of hidden units. The dimension of the input space appears only in the constants of our bounds.
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Jacobs, B. A. "A New Grünwald-Letnikov Derivative Derived from a Second-Order Scheme." Abstract and Applied Analysis 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/952057.

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A novel derivation of a second-order accurate Grünwald-Letnikov-type approximation to the fractional derivative of a function is presented. This scheme is shown to be second-order accurate under certain modifications to account for poor accuracy in approximating the asymptotic behavior near the lower limit of differentiation. Some example functions are chosen and numerical results are presented to illustrate the efficacy of this new method over some other popular choices for discretizing fractional derivatives.
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Chang, Xirong. "Approximation of locally integrable functions on the real line." International Journal of Wavelets, Multiresolution and Information Processing 12, no. 05 (2014): 1461002. http://dx.doi.org/10.1142/s0219691314610025.

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The aim of this paper is to extend (ψ, β)-derivatives to [Formula: see text]-derivatives for locally integrable functions on the real line and then investigate problems of approximation of the classes of functions determined by these derivatives with the use of entire functions of exponential type.
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Dissertations / Theses on the topic "Approximation of derivatives of functions"

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Deterding, Stephen. "Bounded Point Derivations on Certain Function Spaces." UKnowledge, 2018. https://uknowledge.uky.edu/math_etds/51.

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Let 𝑋 be a compact subset of the complex plane and denote by 𝑅𝑝(𝑋) the closure of rational functions with poles off 𝑋 in the 𝐿𝑝(𝑋) norm. We show that if a point 𝑥0 admits a bounded point derivation on 𝑅𝑝(𝑋) for 𝑝 > 2, then there is an approximate derivative at 𝑥0. We also prove a similar result for higher order bounded point derivations. This extends a result of Wang, which was proven for 𝑅(𝑋), the uniform closure of rational functions with poles off 𝑋. In addition, we show that if a point 𝑥0 admits a bounded point derivation on 𝑅(𝑋) and if 𝑋 contains an interior cone, then the bounded point derivation can be represented by the difference quotient if the limit is taken over a non-tangential ray to 𝑥0. We also extend this result to the case of higher order bounded point derivations. These results were first shown by O'Farrell; however, we prove them constructively by explicitly using the Cauchy integral formula.
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Al-Mohy, Awad. "Algorithms for the matrix exponential and its Fréchet derivative." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/algorithms-for-the-matrix-exponential-and-its-frechet-derivative(4de9bdbd-6d79-4e43-814a-197668694b8e).html.

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New algorithms for the matrix exponential and its Fréchet derivative are presented. First, we derive a new scaling and squaring algorithm (denoted expm[new]) for computing eA, where A is any square matrix, that mitigates the overscaling problem. The algorithm is built on the algorithm of Higham [SIAM J.Matrix Anal. Appl., 26(4): 1179-1193, 2005] but improves on it by two key features. The first, specific to triangular matrices, is to compute the diagonal elements in the squaring phase as exponentials instead of powering them. The second is to base the backward error analysis that underlies the algorithm on members of the sequence {||Ak||1/k} instead of ||A||. The terms ||Ak||1/k are estimated without computing powers of A by using a matrix 1-norm estimator. Second, a new algorithm is developed for computing the action of the matrix exponential on a matrix, etAB, where A is an n x n matrix and B is n x n₀ with n₀ << n. The algorithm works for any A, its computational cost is dominated by the formation of products of A with n x n₀ matrices, and the only input parameter is a backward error tolerance. The algorithm can return a single matrix etAB or a sequence etkAB on an equally spaced grid of points tk. It uses the scaling part of the scaling and squaring method together with a truncated Taylor series approximation to the exponential. It determines the amount of scaling and the Taylor degree using the strategy of expm[new].Preprocessing steps are used to reduce the cost of the algorithm. An important application of the algorithm is to exponential integrators for ordinary differential equations. It is shown that the sums of the form $\sum_{k=0}^p\varphi_k(A)u_k$ that arise in exponential integrators, where the $\varphi_k$ are related to the exponential function, can be expressed in terms of a single exponential of a matrix of dimension $n+p$ built by augmenting $A$ with additional rows and columns. Third, a general framework for simultaneously computing a matrix function, $f(A)$, and its Fréchet derivative in the direction $E$, $L_f(A,E)$, is established for a wide range of matrix functions. In particular, we extend the algorithm of Higham and $\mathrm{expm_{new}}$ to two algorithms that intertwine the evaluation of both $e^A$ and $L(A,E)$ at a cost about three times that for computing $e^A$ alone. These two extended algorithms are then adapted to algorithms that simultaneously calculate $e^A$ together with an estimate of its condition number. Finally, we show that $L_f(A,E)$, where $f$ is a real-valued matrix function and $A$ and $E$ are real matrices, can be approximated by $\Im f(A+ihE)/h$ for some suitably small $h$. This approximation generalizes the complex step approximation known in the scalar case, and is proved to be of second order in $h$ for analytic functions $f$ and also for the matrix sign function. It is shown that it does not suffer the inherent cancellation that limits the accuracy of finite difference approximations in floating point arithmetic. However, cancellation does nevertheless vitiate the approximation when the underlying method for evaluating $f$ employs complex arithmetic. The complex step approximation is attractive when specialized methods for evaluating the Fréchet derivative are not available.
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Yang, Qianqian. "Novel analytical and numerical methods for solving fractional dynamical systems." Thesis, Queensland University of Technology, 2010. https://eprints.qut.edu.au/35750/1/Qianqian_Yang_Thesis.pdf.

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During the past three decades, the subject of fractional calculus (that is, calculus of integrals and derivatives of arbitrary order) has gained considerable popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields in science and engineering. For example, fractional calculus has been successfully applied to problems in system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance. In many cases these new fractional-order models are more adequate than the previously used integer-order models, because fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes that are governed by anomalous diffusion. Hence, there is a growing need to find the solution behaviour of these fractional differential equations. However, the analytic solutions of most fractional differential equations generally cannot be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behaviour of such fractional equations and exploring their applications. The main objective of this thesis is to develop new effective numerical methods and supporting analysis, based on the finite difference and finite element methods, for solving time, space and time-space fractional dynamical systems involving fractional derivatives in one and two spatial dimensions. A series of five published papers and one manuscript in preparation will be presented on the solution of the space fractional diffusion equation, space fractional advectiondispersion equation, time and space fractional diffusion equation, time and space fractional Fokker-Planck equation with a linear or non-linear source term, and fractional cable equation involving two time fractional derivatives, respectively. One important contribution of this thesis is the demonstration of how to choose different approximation techniques for different fractional derivatives. Special attention has been paid to the Riesz space fractional derivative, due to its important application in the field of groundwater flow, system biology and finance. We present three numerical methods to approximate the Riesz space fractional derivative, namely the L1/ L2-approximation method, the standard/shifted Gr¨unwald method, and the matrix transform method (MTM). The first two methods are based on the finite difference method, while the MTM allows discretisation in space using either the finite difference or finite element methods. Furthermore, we prove the equivalence of the Riesz fractional derivative and the fractional Laplacian operator under homogeneous Dirichlet boundary conditions – a result that had not previously been established. This result justifies the aforementioned use of the MTM to approximate the Riesz fractional derivative. After spatial discretisation, the time-space fractional partial differential equation is transformed into a system of fractional-in-time differential equations. We then investigate numerical methods to handle time fractional derivatives, be they Caputo type or Riemann-Liouville type. This leads to new methods utilising either finite difference strategies or the Laplace transform method for advancing the solution in time. The stability and convergence of our proposed numerical methods are also investigated. Numerical experiments are carried out in support of our theoretical analysis. We also emphasise that the numerical methods we develop are applicable for many other types of fractional partial differential equations.
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Kárský, Vilém. "Modelování LTI SISO systémů zlomkového řádu s využitím zobecněných Laguerrových funkcí." Master's thesis, Vysoké učení technické v Brně. Fakulta elektrotechniky a komunikačních technologií, 2017. http://www.nusl.cz/ntk/nusl-316278.

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This paper concentrates on the description of fractional order LTI SISO systems using generalized Laguerre functions. There are properties of generalized Laguerre functions described in the paper, and an orthogonal base of these functions is shown. Next the concept of fractional derivatives is explained. The last part of this paper deals with the representation of fractional order LTI SISO systems using generalized Laguerre functions. Several examples were solved to demonstrate the benefits of using these functions for the representation of LTI SISO systems.
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Ugolotti, Alessandro. "Alternative derivative expansion in Functional RG and application." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2016. http://amslaurea.unibo.it/10434/.

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We give a brief review of the Functional Renormalization method in quantum field theory, which is intrinsically non perturbative, in terms of both the Polchinski equation for the Wilsonian action and the Wetterich equation for the generator of the proper verteces. For the latter case we show a simple application for a theory with one real scalar field within the LPA and LPA' approximations. For the first case, instead, we give a covariant "Hamiltonian" version of the Polchinski equation which consists in doing a Legendre transform of the flow for the corresponding effective Lagrangian replacing arbitrary high order derivative of fields with momenta fields. This approach is suitable for studying new truncations in the derivative expansion. We apply this formulation for a theory with one real scalar field and, as a novel result, derive the flow equations for a theory with N real scalar fields with the O(N) internal symmetry. Within this new approach we analyze numerically the scaling solutions for N=1 in d=3 (critical Ising model), at the leading order in the derivative expansion with an infinite number of couplings, encoded in two functions V(phi) and Z(phi), obtaining an estimate for the quantum anomalous dimension with a 10% accuracy (confronting with Monte Carlo results).
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Martin, David Royce. "Quadrature Approximation of Matrix Functions, with Applications." Kent State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=kent1337658656.

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Hed, Lisa. "Approximation and Subextension of Negative Plurisubharmonic Functions." Licentiate thesis, Umeå : Department of Mathematics and Mathematical Statistics, Umeå University, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-1799.

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Zhu, Shengxin. "Numerical linear approximation involving radial basis functions." Thesis, University of Oxford, 2014. http://ora.ox.ac.uk/objects/uuid:b870646b-5155-45f8-b38c-ae6cf4d22f27.

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This thesis aims to acquire, deepen and promote understanding of computing techniques for high dimensional scattered data approximation with radial basis functions. The main contributions of this thesis include sufficient conditions for the sovability of compactly supported radial basis functions with different shapes, near points preconditioning techniques for high dimensional interpolation systems with compactly supported radial basis functions, a heterogeneous hierarchical radial basis function interpolation scheme, which allows compactly supported radial basis functions of different shapes at the same level, an O(N) algorithm for constructing hierarchical scattered data set andan O(N) algorithm for sparse kernel summation on Cartesian grids. Besides the main contributions, we also investigate the eigenvalue distribution of interpolation matrices related to radial basis functions, and propose a concept of smoothness matching. We look at the problem from different perspectives, giving a systematic and concise description of other relevant theoretical results and numerical techniques. These results are interesting in themselves and become more interesting when placed in the context of the bigger picture. Finally, we solve several real-world problems. Presented applications include 3D implicit surface reconstruction, terrain modelling, high dimensional meteorological data approximation on the earth and scattered spatial environmental data approximation.
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Beaty, Michael Graeme. "Reconstructing functions from values and derivatives." Thesis, University of York, 1989. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329630.

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Liu, Shenhui. "Automorphic L-Functions and Their Derivatives." The Ohio State University, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=osu1499965951825371.

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Books on the topic "Approximation of derivatives of functions"

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Temli͡akov, V. N. Approximation of functions with bounded mixed derivative. American Mathematical Society, 1989.

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Alyukov, Sergey. Approximation of piecewise linear and generalized functions. INFRA-M Academic Publishing LLC., 2024. http://dx.doi.org/10.12737/2104876.

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The monograph is devoted to piecewise linear and generalized functions. They are widely used in various fields of research: in the theory of signal transmission and transformation, quantum field theory, control theory, problems of nonlinear dynamics, structural mechanics, semiconductor theory, economic applications, medicine, description of impulse effects and many others. When creating mathematical models, in some cases it is necessary to approximate these functions using analytical expressions, but not in the form of linear combinations, as in known methods, but in the form of attachments, compositions, using recursive sequences.&#x0D; The considered methods are devoid of the disadvantages of Fourier series and have advantages over other approximation methods. The developed approximation methods help to understand the meaning and content of generalized functions and their derivatives, and contribute to the conscious application of these functions in mathematical modeling problems. These methods can be used in a wide range of applied research, from medicine to quantum electronics. The theoretical material is illustrated by a large number of practical examples from a wide variety of applied fields. The fundamentals of the developed macroeconomic theory with impulse, shock, spasmodic characteristics and other types of rapidly changing processes are presented.&#x0D; For mathematicians, students and teachers, and specialists working in applied research fields.
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Nürnberger, Günther. Approximation by Spline Functions. Springer Berlin Heidelberg, 1989. http://dx.doi.org/10.1007/978-3-642-61342-5.

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Temli͡akov, V. N. Approximation of periodic functions. Nova Science Publishers, 1993.

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Nürnberger, G. Approximation by spline functions. Springer-Verlag, 1989.

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Temli︠a︡kov, V. N. Approximation of periodic functions. Nova Science Publishers, 1993.

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Mashreghi, Javad. Derivatives of Inner Functions. Springer New York, 2013. http://dx.doi.org/10.1007/978-1-4614-5611-7.

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Mashreghi, Javad. Derivatives of Inner Functions. Springer New York, 2013.

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Petrushev, P. P. Rational approximation of real functions. Cambridge University Press, 1987.

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Stepanet͡s, A. I. Classification and approximation of periodic functions. Kluwer Academic Publishers, 1995.

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Book chapters on the topic "Approximation of derivatives of functions"

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Marti, K. "Approximations and Derivatives of Probability Functions." In Approximation, Probability, and Related Fields. Springer US, 1994. http://dx.doi.org/10.1007/978-1-4615-2494-6_28.

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Stepanets, Alexander I. "Simultaneous Approximation of Functions and their Derivatives by Fourier Sums." In Classification and Approximation of Periodic Functions. Springer Netherlands, 1995. http://dx.doi.org/10.1007/978-94-011-0115-8_5.

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Anastassiou, George A., and Oktay Duman. "Weighted Approximation in Statistical Sense to Derivatives of Functions." In Towards Intelligent Modeling: Statistical Approximation Theory. Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-19826-7_7.

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Anderson, Douglas R. "Positive Green’s Functions for Boundary Value Problems with Conformable Derivatives." In Mathematical Analysis, Approximation Theory and Their Applications. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-31281-1_3.

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Sochen, Nir A., Yehoshua Y. Zeevi, and Robert M. Haralick. "A Geometric Functional for Derivatives Approximation." In Scale-Space Theories in Computer Vision. Springer Berlin Heidelberg, 1999. http://dx.doi.org/10.1007/3-540-48236-9_51.

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Ihara, Yasutaka, and Kohji Matsumoto. "On the Value-Distribution of Logarithmic Derivatives of Dirichlet L-Functions." In Analytic Number Theory, Approximation Theory, and Special Functions. Springer New York, 2014. http://dx.doi.org/10.1007/978-1-4939-0258-3_3.

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Trigub, Roald M., and Eduard S. Bellinsky. "Classes of Functions with Bounded Mixed Derivative." In Fourier Analysis and Approximation of Functions. Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-2876-2_11.

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Trigub, Roald M., and Eduard S. Bellinsky. "Lebesgue Constants and Approximation of Classes of Functions with Bounded Derivative by Polynomials." In Fourier Analysis and Approximation of Functions. Springer Netherlands, 2004. http://dx.doi.org/10.1007/978-1-4020-2876-2_9.

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Taş, Emre, and Tuğba Yurdakadim. "Approximation to Derivatives of Functions by Linear Operators Acting on Weighted Spaces by Power Series Method." In Springer Proceedings in Mathematics & Statistics. Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-28443-9_26.

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Ding, Jianqiang, Taoran Wu, Zhen Liang, and Bai Xue. "PyBDR: Set-Boundary Based Reachability Analysis Toolkit in Python." In Lecture Notes in Computer Science. Springer Nature Switzerland, 2024. http://dx.doi.org/10.1007/978-3-031-71177-0_10.

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AbstractWe present PyBDR, a Python reachability analysis toolkit based on set-boundary analysis, which centralizes on widely-adopted set propagation techniques for formal verification, controller synthesis, state estimation, etc. It employs boundary analysis of initial sets to mitigate the wrapping effect during computations, thus improving the performance of reachability analysis algorithms without significantly increasing computational costs. Beyond offering various set representations such as polytopes and zonotopes, our toolkit particularly excels in interval arithmetic by extending operations to the tensor level, enabling efficient parallel interval arithmetic computation and unifying vector and matrix intervals into a single framework. Furthermore, it features symbolic computation of derivatives of arbitrary order and evaluates them as real or interval-valued functions, which is essential for approximating behaviours of nonlinear systems at specific time instants. Its modular architecture design offers a series of building blocks that facilitate the prototype development of reachability analysis algorithms. Comparative studies showcase its strengths in handling verification tasks with large initial sets or long time horizons. The toolkit is available at https://github.com/ASAG-ISCAS/PyBDR.
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Conference papers on the topic "Approximation of derivatives of functions"

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Tichavský, Petr, and Ondřej Straka. "Tensor Train Approximation of Multivariate Functions." In 2024 32nd European Signal Processing Conference (EUSIPCO). IEEE, 2024. http://dx.doi.org/10.23919/eusipco63174.2024.10715191.

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Najmon, Joel C., and Andres Tovar. "Comparing Derivatives of Neural Networks for Regression." In ASME 2023 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2023. http://dx.doi.org/10.1115/detc2023-117571.

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Abstract In the past decades, neural networks have rapidly grown in popularity as a way to model complex non-linear relationships. The computational efficiently and flexibility of neural networks has made them popular for machine learning-based optimization methods. As such the derivative of a neural network’s output is required for gradient-based optimization algorithms. Recently, there have been several works towards improving derivatives of neural network targets, however there is yet to be done a comparative study on the different derivation methods for the derivative of a neural network’s targets with respect to its input features. Consequently, this paper’s objective is to implement and compare common methods for obtaining or approximating the derivative of neural network targets with respect to their inputs. The methods studied include analytical derivatives, finite differences, complex step approximation, and automatic differentiation. The methods are tested by training deep multilayer perceptrons for regression with several analytical functions. The derivatives of the neural network-derived methods are evaluated against the exact derivative of the test functions. Results show that all of the derivation methods provide the same derivative approximation to near working precision of the computer. Implementation of the study is done using the TensorFlow library in a provided Python code.
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Venkataraman, P. "Solving Inverse ODE Using Bezier Functions." In ASME 2009 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2009. http://dx.doi.org/10.1115/detc2009-86331.

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The simplest inverse boundary value problem is to identify the differential equation and the boundary conditions from a given set of discrete data points. For an ordinary differential equation, it would involve finding a function, which when expressed through some function of itself and its derivatives, and integrated using particular boundary conditions would generate the given data. Parametric Bezier functions are excellent candidates for these functions. They allow efficient approximation of data and its derivative content. The Bezier function is smooth and continuous to a high degree. In this paper the best Bezier function to fit the data represents this function which is being sought. This Bezier approximation also determines the boundary conditions. Next, a generic form of the differential equation is assumed. The Bezier function and its derivatives are then used in this generic form to establish the exponents and coefficients of the various terms in the actual differential equation. The paper looks at homogeneous ordinary differential equations and shows it can recover the exact form of both linear and nonlinear differential equations. Two examples are presented. The first example uses data from the Bessel equation, representing a linear equation. The second example uses the data from the Blassius equation which is nonlinear. In both cases the exact form of the equation is identified from the given discrete data.
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Marti, K. "Approximation and Derivatives of Probability Functions in Probabilistic Structural Analysis and Design." In ASME 1995 Design Engineering Technical Conferences collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium. American Society of Mechanical Engineers, 1995. http://dx.doi.org/10.1115/detc1995-0048.

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Abstract Yield stresses, allowable stresses, moment capacities (plastic moments), external loadings, manufacturing errors are not given fixed quantities in practice, but have to be modelled as random variables with a certain joint probability distribution. Hence, problems from limit (collapse) load analysis or plastic analysis and from plastic and elastic design of structures are treated in the framework of stochastic optimization. Using especially reliability-oriented optimization methods, the behavioral constraints are quantified by means of the corresponding probability ps of survival. Lower bounds for ps are obtained by selecting certain redundants in the vector of internal forces; moreover, upper bounds for ps are constructed by considering a pair of dual linear pro-prams for the optimizational representation of the yield or safety conditions. Whereas ps can be computed e.g. by sampling methods or by asymptotic expansion techniques based on Laplace integral representations of certain multiple integrals, efficient techniques for the computation of the sensitivities (of various orders) of ps with respect to input or design variables have yet to be developed. Hence several new techniques are suggested for the numerical computation of derivatives of ps.
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Han, Dongkun, and Dimitra Panagou. "Chebyshev approximation and higher order derivatives of Lyapunov functions for estimating the domain of attraction." In 2017 IEEE 56th Annual Conference on Decision and Control (CDC). IEEE, 2017. http://dx.doi.org/10.1109/cdc.2017.8263816.

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Tukhliev, K. "On the joint polynomial approximation of functions and its derivatives partial sums of Fourier-Chebyshev." In Научные тенденции: Вопросы точных и технических наук. ЦНК МОАН, 2018. http://dx.doi.org/10.18411/spc-12-12-2018-03.

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Bortoff, Scott A., Christopher R. Laughman, Vedang Deshpande, and Hongtao Qiao. "Fluid Property Functions in Polar and Parabolic Coordinates." In American Modelica Conference 2024. Linköping University Electronic Press, 2025. https://doi.org/10.3384/ecp20721.

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This paper presents two methods for reallizing fluid property functions in Modelica simulation models. Each makes use of a coordinate transformation that aligns one coordinate with the saturation curve. This provides for a precise representation of the fluid property function at the saturation curve, and for connected domains of interest including the liquid, vapor, supercritical and two-phase regions. Both approaches make use of spline function approximation in the aligned coordinates, and are numerically efficient, well conditioned, and allow for efficient calculation of derivatives up to any desired order that are precise up to processor numerical tolerance.
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Shi, Yuyun, Hui Li, Zhifu Li, and Huilong Ren. "Approximation of Higher-Order Derivatives of the Frequency Domain Free Surface Green Function." In ASME 2015 34th International Conference on Ocean, Offshore and Arctic Engineering. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/omae2015-41558.

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The higher-order derivatives of the free-surface Green Function are critically important in three-dimensional frequency-domain boundary element methods using mixed dipole-source distribution. To improve the accuracy and efficiency of numerical schemes, the computing domain is divided into five areas. Derivatives in four areas are calculated analytically since the Green function is defined analytically. The 5th area is divided into a number of sub-areas in which truncated Double Chebyshev series are used to approximate the Green function. Unlike the usual way in which the derivatives of Green function are obtained by differentiating the series, we re-approximate the derivatives by new Chebyshev series with new coefficients. Numerical results show that the new series are more accurate, in particular, second order derivatives.
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Young, Richard A. "Gaussian derivative model of spatial vision." In OSA Annual Meeting. Optica Publishing Group, 1985. http://dx.doi.org/10.1364/oam.1985.fv4.

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A new mathematical description of visual receptive fields in the striate cortex is proposed. The hypothesis is advanced that receptive field line­weighting profiles along the axis of maximum response are describable as Gaussian derivative functions. Such functions are a close approximation to the optimum functions theoretically possible for the detection and precise localization of visual stimuli in the presence of noise. The shapes of the spatial frequency sensitivity tuning curves of fifty-five monkey cortical cells were well characterized by a single Gaussian directional derivative, most often the first through fourth derivative, with corresponding half-amplitude bandwidths in the range of 2.6-1.2 octaves. The receptive field shapes of cat and monkey cells in the spatial domain, as measured, with bar stimuli, were also well described by a single Gaussian derivative term. Gaussian derivatives generally described, better than Gabor functions, (1) the locations of the zero crossings of receptive field profiles in the spatial domain; (2) the shape of cell’s spatial frequency tuning curve, at the low and high frequency extremes; and (3) the symmetry properties of fields in the spatial domain as a function of their independently measured bandwidths. A new neural model for the formation of receptive fields described how simple differences of offset Gaussians (DOOGs) with identical standard deviations, such as are found in the monkey X-cell pathway, can easily give rise to Gaussian derivativelike functions of any order.
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Ashyralyyev, Charyyar, and Beyza Öztürk. "Finite difference approximations of first order derivatives of complex-valued functions." In INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2018). Author(s), 2018. http://dx.doi.org/10.1063/1.5049048.

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Reports on the topic "Approximation of derivatives of functions"

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Markova, Oksana, Serhiy Semerikov та Maiia Popel. СoCalc as a Learning Tool for Neural Network Simulation in the Special Course “Foundations of Mathematic Informatics”. Sun SITE Central Europe, 2018. http://dx.doi.org/10.31812/0564/2250.

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The role of neural network modeling in the learning сontent of special course “Foundations of Mathematic Informatics” was discussed. The course was developed for the students of technical universities – future IT-specialists and directed to breaking the gap between theoretic computer science and it’s applied applications: software, system and computing engineering. CoCalc was justified as a learning tool of mathematical informatics in general and neural network modeling in particular. The elements of technique of using CoCalc at studying topic “Neural network and pattern recognition” of the special course “Foundations of Mathematic Informatics” are shown. The program code was presented in a CofeeScript language, which implements the basic components of artificial neural network: neurons, synaptic connections, functions of activations (tangential, sigmoid, stepped) and their derivatives, methods of calculating the network`s weights, etc. The features of the Kolmogorov–Arnold representation theorem application were discussed for determination the architecture of multilayer neural networks. The implementation of the disjunctive logical element and approximation of an arbitrary function using a three-layer neural network were given as an examples. According to the simulation results, a conclusion was made as for the limits of the use of constructed networks, in which they retain their adequacy. The framework topics of individual research of the artificial neural networks is proposed.
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Tong, C. An Adaptive Derivative-based Method for Function Approximation. Office of Scientific and Technical Information (OSTI), 2008. http://dx.doi.org/10.2172/945874.

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Reichenbach, H., K. Behrens, and A. L. Kuhl. Approximation functions for airblast environments from buried charges. Office of Scientific and Technical Information (OSTI), 1993. http://dx.doi.org/10.2172/10141968.

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Buhmann, Martin D., and Amos Ron. Radial Base Functions: Lp-approximation Orders with Scattered Centres. Defense Technical Information Center, 1994. http://dx.doi.org/10.21236/ada274682.

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Franke, Richard, Hans Hagen, and Gregory M. Nielson. Least Squares Surface Approximation to Scattered Data Using Multiquadric Functions. Defense Technical Information Center, 1992. http://dx.doi.org/10.21236/ada259804.

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Yavuz, Enes. Basic Trigonometric Korovkin Approximation for Fuzzy Valued Functions of Two Variables. "Prof. Marin Drinov" Publishing House of Bulgarian Academy of Sciences, 2021. http://dx.doi.org/10.7546/crabs.2021.09.02.

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Bak, Keld L., and Jack Simons. Geometrical Linear Responses and Directional Energy Derivatives for Energetically Degenerate MCSCF Electronic Functions. Defense Technical Information Center, 1991. http://dx.doi.org/10.21236/ada236074.

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Chen, Qi, and Ivo Babuska. Polynomial Interpolation and Approximation of Real Functions 2: Symmetrical Interpolation for the Triangle. Defense Technical Information Center, 1993. http://dx.doi.org/10.21236/ada277345.

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Emre, Erol. Adaptive Estimation and Approximation of Continuously Varying Spectral Density Functions to Airborne Radar. Defense Technical Information Center, 1993. http://dx.doi.org/10.21236/ada277532.

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Franke, Richard. Using Legendre Functions for Spatial Covariance Approximation and Investigation of Radial Nonisotrophy for NOGAPS Data. Defense Technical Information Center, 2001. http://dx.doi.org/10.21236/ada389396.

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