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By combining and improving recent techniques and results, we provide explicit estimates for the error terms $|\pi(x)-\text{li}(x)|$, $|\theta(x)-x|$ and $|\psi(x)-x|$ appearing in the prime number theorem. For example, we show for all…
Algorithms for numerical tasks in finite precision simultaneously seek to minimize the number of floating point operations performed, and also the number of bits of precision required by each floating point operation. This paper presents an…
The need for recognition/approximation of functions in terms of elementary functions/operations emerges in many areas of experimental mathematics, numerical analysis, computer algebra systems, model building, machine learning, approximation…
We propose a new algorithm that finds an $\varepsilon$-approximate fixed point of a smooth function from the $n$-dimensional $\ell_2$ unit ball to itself. We use the general framework of finding approximate solutions to a variational…
The prime-counting function $\pi(x)$ which computes the number of primes smaller or equal to a given real number has a long-standing interest in number theory. The present manuscript proposes a method to compute $\pi(x)$ with time…
We present a new method for approximating real-valued functions on ${\mathbb R}^+$ by linear combinations of exponential functions with complex coefficients. The approach is based on a multi-point Pad\'e approximation of the Laplace…
We suggest a new optimization technique for minimizing the sum $\sum_{i=1}^n f_i(x)$ of $n$ non-convex real functions that satisfy a property that we call piecewise log-Lipschitz. This is by forging links between techniques in computational…
This article discusses the classical problem of how to calculate $r_n(m)$, the number of ways to represent an integer $m$ by a sum of $n$ squares from a computational efficiency viewpoint. Although this problem has been studied in great…
We obtain new complexity bounds for computing a triangular integral basis of a number field or a function field. We reach for function fields a softly linear cost with respect to the size of the output when the residual characteristic is…
We study function estimation in the empirical Bayes setting for Poisson and normal means. Specifically, given observations $Y_i\sim f(\cdot; \theta_i)$ with latent parameters $\theta_i\sim \pi$, the goal is to estimate…
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 computational practice, most attention is paid to rational approximations of functions and approximations by the sum of exponents. We consider a wide enough class of nonlinear approximations characterized by a set of two required…
#SMT, or model counting for logical theories, is a well-known hard problem that generalizes such tasks as counting the number of satisfying assignments to a Boolean formula and computing the volume of a polytope. In the realm of…
An efficient procedure for the computation of $Li_{s}(z)$ where $s<0$ is here presented. We started with Polylogarithm $Li_{s}(z)$ where $s<0$. The summation of $n^{s}z^{n}$ is evaluated using a new method. An assumption is made that the…
Numerical data processing is a key task across different fields of computer technology use. However, even simple summation of values is not precise due to the floating point representation use. This paper presents a practical algorithm for…
This paper introduces an efficient algorithm for computing the general oscillatory matrix functions. These computations are crucial for solving second-order semi-linear initial value problems. The method is exploited using the scaling and…
We analyse and compare the complexity of several algorithms for computing modular polynomials. We show that an algorithm relying on floating point evaluation of modular functions and on interpolation, which has received little attention in…
We describe an algorithm to numerically evaluate Riemann theta functions in any dimension in quasi-linear time in terms of the required precision, uniformly on reduced input. This algorithm is implemented in the FLINT number theory library…
Computers calculate transcendental functions by approximating them through the composition of a few limited-precision instructions. For example, an exponential can be calculated with a Taylor series. These approximation methods were…
We construct fast algorithms for evaluating transforms associated with families of functions which satisfy recurrence relations. These include algorithms both for computing the coefficients in linear combinations of the functions, given the…