The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! 
We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to KawamuraC and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). 
It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity.

Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

The effectiveness of frequency domain processing algorithms in bistatic synthetic aperture radar (SAR) focusing depends critically on the accuracy of the bistatic slant range function approximation. This letter presents a new Chebyshev slant range function approximation that is shown to increase the accuracy of the analytical approximation of the bistatic point target spectrum. The performance of the new method is compared to the conventional Taylor series approximation approach in the generation of the point target spectrum. The new approach is shown to provide a more accurate approximation of the slant range function with negligible increase in processing requirements compared to the traditional Taylor series approximation. The accuracy improvement is shown to yield a more accurate spectrum that can be exploited in bistatic SAR focusing algorithms.

Approximation of the Bistatic Slant Range Using Chebyshev Polynomials

On the numerical solution of the integro-differential equation of the point kinetics of nuclear reactors

Stability of narrow-gap Taylor-Dean flow with radial heating: Stationary critical modes

This paper puts forward a new learning model based on the collaborative effort of active learning and particle swarm optimization (PSO) in functional link artificial neural networks (FLANNs) to estimate software effort. The active learning uses quick algorithm to detect the essential content of the datasets by which the dataset is reduced and are processed through PSO optimized FLANN. The PSO uses the inertia weight, which is an important parameter in PSO that significantly affects the convergence and exploration-exploitation in the search space while training FLANN. The Chebyshev polynomial has been used for mapping the original feature space from lower to higher dimensional functional space. The method has been evaluated exhaustively on different test suits of PROMISE repository to study the performance. The computational results show that the active learning along with PSO optimized FLANN greatly improves the performance of the model and its variants for software development effort estimation.

Software Effort Estimation Using Functional Link Neural Networks Tuned with Active Learning and Optimized with Particle Swarm Optimization

We show that Chebyshev Polynomials are a practical representation of computable functions on the computable reals. The paper presents error estimates for common operations and demonstrates that Chebyshev Polynomial methods would be more efficient than Taylor Series methods for evaluation of transcendental functions.

/pdf/computable-function-representations-using-effective-41170uy24l.pdf

Computable Function Representations Using Effective Chebyshev Polynomial

When implementing a real function such as sin, cos, etc. on any computing system, polynomial approximations are almost always used to replace the actual function. The reason behind this is that addition, subtraction, and multiplication are efficiently implemented in general-purpose processors [2]. In this paper we extend the idea of Brisebarre, Muller, Tisserand, and Chevillard on machine-efficient Chebyshev approximation [2,3]. Our extensions include standard statistical distribution functions, by which we mean: the normal distribution, the beta distribution, the F-distribution, and the Student’s t-distribution. We choose Chebyshev polynomials as they provide a good polynomial approximation [1,5]. These practical calculations have been performed using Müller’s iRRAM [6]. The standard statistical distribution functions are important functions that are used in probability theory and statistics [4]. Finding an efficient way of approximating these functions would be beneficial. The (cumulative) normal distribution function N(0,1) with mean 0 is defined by the following Equation. N0,1(x) = 1 √ 2π Z x −∞ e− 1 2 u 2 du (1) On the other hand, the beta distribution function is defined in Equation 2. It is defined in the interval [0,1] with two positive shape values α and β. F (x; α, β) = Bx(α, β) B(α, β) = Ix(α, β) (2) where Bx(α, β) is the incomplete beta function, B(α, β) is the beta function, and Ix(α, β) is the normalized incomplete beta function [4].

Machine-Efficient Chebyshev Approximation for Standard Statistical Distribution Functions

We extend the work of Brisebarre, Muller, Tisserand, and Chevillard [2, 3] on machineefficient Chebyshev approximation, so that it is applicable to non-linear PDEs, and in particular we explore the application of this method to the Korteweg-de Vries equation [4, 6]. Current numerical algorithms to solve this differential equation for high accuracy are computationally expensive; and methods that use polynomial approximation such as spectral methods are more efficient in this case. This is because high accuracy addition, subtraction, and multiplication are efficiently implemented in general-purpose processors [2, 5, 7, 9, 11]. The proposed approach adopts the usual Chebyshev method except that coefficients are chosen so that they are efficiently handled by current computer hardware. We implement the machine-efficient Chebyshev approximation as shown in Equation 1.

/pdf/exact-numeric-solutions-of-non-linear-pdes-an-application-of-5a68qtp1np.pdf

Exact numeric solutions of non-linear PDEs: an application of machine efficient Chebyshev methods

Machine-Ecient Chebyshev Approximation is a technique that permits practical evaluation of transcendental functions within a computable arithmetic, such as the computable reals. The approach adopts the usual Chebyshev method so that coecients are eciently handled by current computer hardware. The proposed technique has an application to spectral methods for sobolev spaces. A practical demonstration of this work

Approximation: an Application to Spectral Methods for Sobolev Spaces

In this paper we use the idea of Brisebarre, Muller, Tisserand and Chevillard on machine-efficient Chebyshev approximation. Our aim is to provide high accuracy results for log, Gamma (and related functions), and the solution of Ordinary Differential Equations by Picard iteration using machine-efficient Chebyshev approximations. We demonstrate that these machine-efficient approximations do indeed improve the efficiency with which these operations can be performed. These practical calculations were performed using Muller's iRRAM.

Machine-efficient Chebyshev approximation for exact arithmetic: their use with first-order ordinary differential equations

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