Can Computer Algebra be Liberated from its Algebraic Yoke ?

So far, the scope of computer algebra has been needlessly restricted to exact algebraic methods. Its possible extension to approximate analytical methods is discussed. The entangled roles of functional analysis and symbolic programming, especially the functional and transformational paradigms, are put forward. In the future, algebraic algorithms could constitute the core of extended symbolic manipulation systems including primitives for symbolic approximations.Comment: 8 pages, 2-column presentation, 2 figure

Rational approximation for solving an implicitly given Colebrook flow friction equation

The empirical logarithmic Colebrook equation for hydraulic resistance in pipes implicitly considers the unknown flow friction factor. Its explicit approximations, used to avoid iterative computations, should be accurate but also computationally efficient. We present a rational approximate procedure that completely avoids the use of transcendental functions, such as logarithm or non-integer power, which require execution of the additional number of floating-point operations in computer processor units. Instead of these, we use only rational expressions that are executed directly in the processor unit. The rational approximation was found using a combination of a Pade approximant and artificial intelligence (symbolic regression). Numerical experiments in Matlab using 2 million quasi-Monte Carlo samples indicate that the relative error of this new rational approximation does not exceed 0.866%. Moreover, these numerical experiments show that the novel rational approximation is approximately two times faster than the exact solution given by the Wright omega function.Web of Science81art. no. 2

One-log call iterative solution of the Colebrook equation for flow friction based on Pade polynomials

The 80 year-old empirical Colebrook function zeta, widely used as an informal standard for hydraulic resistance, relates implicitly the unknown flow friction factor lambda, with the known Reynolds number Re and the known relative roughness of a pipe inner surface epsilon* ; lambda = zeta(Re, epsilon* ,lambda). It is based on logarithmic law in the form that captures the unknown flow friction factor l in a way that it cannot be extracted analytically. As an alternative to the explicit approximations or to the iterative procedures that require at least a few evaluations of computationally expensive logarithmic function or non-integer powers, this paper offers an accurate and computationally cheap iterative algorithm based on Pade polynomials with only one log-call in total for the whole procedure (expensive log-calls are substituted with Pade polynomials in each iteration with the exception of the first). The proposed modification is computationally less demanding compared with the standard approaches of engineering practice, but does not influence the accuracy or the number of iterations required to reach the final balanced solution.Web of Science117art. no. 182

Thermal explosion analysis of a strong exothermic chemical reaction with variable pre-exponential factor in a spherical vessel

This study is devoted to investigate the analysis of thermal explosion of a strong exothermic chemical reaction with variable pre-exponential factor in a spherical vessel. The steady state solutions for strong exothermic decomposition of a combustible material uniformly distributed in a heated spherical vessel under Bimolecular, Arrhenius and Sensitised reaction rates. Analytical solutions are constructed for the governing nonlinear boundary-value problem using perturbation technique together with a special type of Hermite-Padé approximants and important properties of the temperature field including bifurcations and thermal criticality are discussed

Colebrook’s flow friction explicit approximations based on fixed-point iterative cycles and symbolic regression

The logarithmic Colebrook flow friction equation is implicitly given in respect to an unknown flow friction factor. Traditionally, an explicit approximation of the Colebrook equation requires evaluation of computationally demanding transcendental functions, such as logarithmic, exponential, non-integer power, Lambert W and WrightΩ functions. Conversely, we herein present several computationally cheap explicit approximations of the Colebrook equation that require only one logarithmic function in the initial stage, whilst for the remaining iterations the cheap Padé approximant of the first order is used instead. Moreover, symbolic regression was used for the development of a novel starting point, which significantly reduces the error of internal iterations compared with the fixed value staring point. Despite the starting point using a simple rational function, it reduces the relative error of the approximation with one internal cycle from 1.81% to 0.156% (i.e., by a factor of 11.6), whereas the relative error of the approximation with two internal cycles is reduced from 0.317% to 0.0259% (i.e., by a factor of 12.24). This error analysis uses a sample with 2 million quasi-Monte Carlo points and the Sobol sequence