Related papers: New Approximation method for the computation of pa…
We consider a class of sums over products of Z-sums whose arguments differ by a symbolic integer. Such sums appear, for instance, in the expansion of Gauss hypergeometric functions around integer indices that depend on a symbolic parameter.…
For a linearly recurrent vector sequence P[n+1] = A(n) * P[n], consider the problem of calculating either the n-th term P[n] or L<=n arbitrary terms P[n_1],...,P[n_L], both for the case of constant coefficients A(n)=A and for a matrix A(n)…
The polylogarithm function is one of the constellation of important mathematical functions. It has a long history, and many connections to other special functions and series, and many applications, for instance in statistical physics.…
Let $P$ be a set of $n$ points in $d$-dimensions. The simplicial depth, $\sigma_P(q)$ of a point $q$ is the number of $d$-simplices with vertices in $P$ that contain $q$ in their convex hulls. The simplicial depth is a notion of data depth…
Logarithmic Number Systems (LNS) hold considerable promise in helping reduce the number of bits needed to represent a high dynamic range of real-numbers with finite precision, and also efficiently support multiplication and division.…
A new algorithms for computing discrete logarithms on elliptic curves defined over finite fields is suggested. It is based on a new method to find zeroes of summation polynomials. In binary elliptic curves one is to solve a cubic system of…
A method of representation of a solution as segments of the series in powers of the step of the independent variable is expanded for solving complex systems of ordinary differential equations (ODE): the Lorenz system and other systems. A…
A new approach is presented for the calculation of p_n and pi_n which uses the Lambert W function. An approximation is first found and using a calculation technique it makes it possible to have an estimate of these two quantities more…
Longest Increasing Subsequence (LIS) is a fundamental statistic of a sequence, and has been studied for decades. While the LIS of a sequence of length $n$ can be computed exactly in time $O(n\log n)$, the complexity of estimating the…
We present an algorithm for computing approximate $\ell_p$ Lewis weights to high precision. Given a full-rank $\mathbf{A} \in \mathbb{R}^{m \times n}$ with $m \geq n$ and a scalar $p>2$, our algorithm computes $\epsilon$-approximate…
In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical…
We present algorithms to evaluate two types of multiple sums, which appear in higher-order loop computations. We consider expansions of a generalized hypergeometric-type sums, $\sum_{n_1,...,n_N} [Gamma(a1.n+c1) Gamma(a2.n}+c2) ...…
Recently, Greg\'orio and Oliveira developed a proximal point scalarization method (applied to multi-objective optimization problems) for an abstract strict scalar representation with a variant of the logarithmic-quadratic function of…
We propose hypergeometric constructions of simultaneous approximations to polylogarithms. These approximations suit for computing the values of polylogarithms and satisfy 4-term Apery-like (polynomial) recursions.
We consider the complexity for computing the approximate sum $a_1+a_2+...+a_n$ of a sorted list of numbers $a_1\le a_2\le ...\le a_n$. We show an algorithm that computes an $(1+\epsilon)$-approximation for the sum of a sorted list of…
A new procedure is presented for computing the matrix cosine and sine simultaneously by means of Taylor polynomial approximations. These are factorized so as to reduce the number of matrix products involved. Two versions are developed to be…
We present an explicit polynomial formula for evaluating the principal logarithm of all matrices lying on the line segment $\{I(1-t)+At:t\in [0,1]\}$ joining the identity matrix $I$ (at $t=0$) to any real matrix $A$ (at $t=1$) having no…
A sum-of-squares is a polynomial that can be expressed as a sum of squares of other polynomials. Determining if a sum-of-squares decomposition exists for a given polynomial is equivalent to a linear matrix inequality feasibility problem.…
Polynomial optimization problems represent a wide class of optimization problems, with a large number of real-world applications. Current approaches for polynomial optimization, such as the sum of squares (SOS) method, rely on large-scale…
This paper considers fast algorithms for operations on linearized polynomials. We propose a new multiplication algorithm for skew polynomials (a generalization of linearized polynomials) which has sub-quadratic complexity in the polynomial…