Related papers: A square root algorithm faster than Newton's metho…
This paper presents a modification of Secant method for finding roots of equations that uses three points for iteration instead of just two. The development of the mathematical formula to be used in the iteration process is provided…
In this paper, the Newton-Anderson method, which results from applying an extrapolation technique known as Anderson acceleration to Newton's method, is shown both analytically and numerically to provide superlinear convergence to non-simple…
We present a new algorithm to solve polynomial equations, and publish its code, which is 1.6-3 times faster than the ZROOTS subroutine that is commercially available from Numerical Recipes, depending on application. The largest improvement,…
Multi-Pivot Quicksort refers to variants of classical quicksort where in the partitioning step $k$ pivots are used to split the input into $k + 1$ segments. For many years, multi-pivot quicksort was regarded as impractical, but in 2009 a…
We develop an accurate square-root-free algorithm for constructing real Givens rotations. On processors that support the fused multiply-add operation in hardware, the algorithm is competitive with square-root based algorithms using a…
In this paper we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and…
We discuss quantum algorithms that calculate numerical integrals and descriptive statistics of stochastic processes. With either of two distinct approaches, one obtains an exponential speed increase in comparison to the fastest known…
Fast algorithms for integer and polynomial multiplication play an important role in scientific computing as well as in other disciplines. In 1971, Sch{\"o}nhage and Strassen designed an algorithm that improved the multiplication time for…
Many problems in applied mathematics require root finding algorithms. Unfortunately, root finding methods have limitations. Firstly, regarding the convergence, there is a trade-off between the size of it's domain and it's rate. Secondly the…
Factoring integers is considered as a computationally-hard problem for classical methods, whereas there exists polynomial-time Shor's quantum algorithm for solving this task. However, requirements for running the Shor's algorithm for…
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…
This article generalizes a recently introduced procedure to solve nonlinear systems of equations, radically departing from the conventional Newton-Raphson scheme. The original nonlinear system is first unfolded into three simpler…
For scientific computations on a digital computer the set of real number is usually approximated by a finite set F of "floating-point" numbers. We compare the numerical accuracy possible with difference choices of F having approximately the…
In this work, a rationalized algorithm for calculating the quotient of two quaternions is presented which reduces the number of underlying real multiplications. Hardware for fast multiplication is much more expensive than hardware for fast…
Newton's method for polynomial root finding is one of mathematics' most well-known algorithms. The method also has its shortcomings: it is undefined at critical points, it could exhibit chaotic behavior and is only guaranteed to converge…
A general class of Newton algorithms on Gra{\ss}mann and Lagrange-Gra{\ss}mann manifolds is introduced, that depends on an arbitrary pair of local coordinates. Local quadratic convergence of the algorithm is shown under a suitable condition…
In recent work, Lemire (2021) presented a fast algorithm to convert number strings into binary floating-point numbers. The algorithm has been adopted by several important systems: e.g., it is part of the runtime libraries of GCC 12, Rust…
Computing the matrix square root and its inverse in a differentiable manner is important in a variety of computer vision tasks. Previous methods either adopt the Singular Value Decomposition (SVD) to explicitly factorize the matrix or use…
Processors may find some elementary operations to be faster than the others. Although an operation may be conceptually as simple as some other operation, the processing speeds of the two can vary. A clever programmer will always try to…
The problem of efficient multiplication of large numbers has been a long-standing challenge in classical computation and has been extensively studied for centuries. It appears that the existing classical algorithms are close to their…