Related papers: Efficient Multi-Accuracy Computations of Complex F…
A robust, fast and accurate method for solving the Colebrook-like equations is presented. The algorithm is efficient for the whole range of parameters involved in the Colebrook equation. The computations are not more demanding than…
This paper presents a new approach in application of the Fourier transform to the complex error function resulting in an efficient rational approximation. Specifically, the computational test shows that with only $17$ summation terms the…
We improve the Modified Winitzki's Approximation of the error function $erf(x)\cong \sqrt{1-e^{-x^2\frac{\frac{4}{\pi}+0.147x^2}{1+0.147x^2}}}$ which has error $|\varepsilon (x)| < 1.25 \cdot 10^{-4}$ $\forall x \ge 0$ till reaching 4…
Fast Fourier transform algorithms are an arsenal of effective tools for solving various problems of analysis and high-speed processing of signals of various natures. Almost all of these algorithms are designed to process sequences of…
We prove simple general formulas for expectations of functions of a L\'evy process and its running extremum. Under additional conditions, we derive analytical formulas using the Fourier/Laplace inversion and Wiener-Hopf factorization, and…
Many practical problems involve estimating low dimensional statistical quantities with high-dimensional models and datasets. Several approaches address these estimation tasks based on the theory of influence functions, such as…
The function $y = g(x) = \mathrm{log}\big(W(e^x)\big)$, where $W()$ denotes the Lambert W function, is the solution to the equation $y + e^y = x$. It appears in various problem situations, for instance the calculation of current-voltage…
In various applications, computers are required to compute approximations to univariate elementary and special functions such as $\exp$ and $\arctan$ to modest accuracy. This paper proposes a new heuristic for automating the design of such…
Various approaches to the numerical representation of the Incomplete Gamma Function F_m(z) for complex arguments z and small integer indexes m are compared with respect to numerical fitness (accuracy and speed). We consider power series,…
In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta. These derivations are then…
Basis Function (BF) expansions are a cornerstone of any engineer's toolbox for computational function approximation which shares connections with both neural networks and Gaussian processes. Even though BF expansions are an intuitive and…
We establish effective mean-value estimates for a wide class of multiplicative arithmetic functions, thereby providing (essentially optimal) quantitative versions of Wirsing's classical estimates and extending those of Hal\'asz. Several…
The computational burden of probabilistic inference remains a hurdle for applying probabilistic programming languages to practical problems of interest. In this work, we provide a semantic and algorithmic foundation for efficient exact…
This article is about twofold arithmetic. Here I introduce algorithms and experimental code for twofold variant of C/C++ standard functions exp() and log(), and expm1() and log1p(). Twofold function $y_0+y_1 \approx f(x_0+x_1)$ is nearly…
We show how rational function approximations to the logarithm, such as $\log z \approx (z^2 - 1)/(z^2 + 6z + 1)$, can be turned into fast algorithms for approximating the determinant of a very large matrix. We empirically demonstrate that…
Our goal is to find accurate and efficient algorithms, when they exist, for evaluating rational expressions containing floating point numbers, and for computing matrix factorizations (like LU and the SVD) of matrices with rational…
In the article we outline the set of Matlab functions that enable the computation of elliptic Integrals and Jacobian elliptic functions for real arguments. Correctness, robustness, efficiency and accuracy of the functions are discussed in…
Bayesian probabilistic numerical methods are a set of tools providing posterior distributions on the output of numerical methods. The use of these methods is usually motivated by the fact that they can represent our uncertainty due to…
In studying the complexity of iterative processes it is usually assumed that the arithmetic operations of addition, multiplication, and division can be performed in certain constant times. This assumption is invalid if the precision…
The plasma dispersion function $Z(s)$ is a fundamental complex special integral function widely used in the field of plasma physics. The simplest and most rapid, yet accurate, approach to calculating it is through rational or equivalent…