相关论文: Speeding up the Karatsuba algorithm
We improve the space complexity of Karatsuba multiplication on a quantum computer from $O(n^{1.427})$ to $O(n)$ while maintaining $O(n^{\lg 3})$ gate complexity. We achieve this by ensuring recursive calls can add their outputs directly…
The traditional Karatsuba algorithm for the multiplication of polynomials and multi-precision integers has a time complexity of $O(n^{1.59})$ and a space complexity of $O(n)$. Roche proposed an improved algorithm with the same $O(n^{1.59})$…
We present COPSIM a parallel implementation of standard integer multiplication for the distributed memory setting, and COPK a parallel implementation of Karatsuba's fast integer multiplication algorithm for a distributed memory setting.…
This article presents a vertical multiplication formula for calculating the multiplication of any two multi-digit integers, which may be not only used to design the multiplier but also to the mental multiplication. Our algorithm is a…
While the Karatsuba algorithm reduces the complexity of large integer multiplication, the extra additions required minimize its benefits for smaller integers of more commonly-used bitwidths. In this work, we propose the extension of the…
This paper describes a new median algorithm and a median approximation algorithm. The former has O(n) average running time and the latter has O(n) worst-case running time. These algorithms are highly competitive with the standard algorithm…
An algorithm for the evaluation of the complex exponential function is proposed which is quasi-linear in time and linear in space. This algorithm is based on a modified binary splitting method for the hypergeometric series and a modified…
In this paper, we explore how numerical calculations can be accelerated by implementing several numerical methods of fractional-order systems using parallel computing techniques. We investigate the feasibility of parallel computing…
A method is presented which allows for a tremendous speed-up of computer simulations of statistical systems by orders of magnitude. This speed-up is achieved by means of a new observable, while the algorithm of the simulation remains…
We demonstrate a multiplication method based on numbers represented as set of polynomial radix 2 indices stored as an integer list. The 'polynomial integer index multiplication' method is a set of algorithms implemented in python code. We…
A New Number Theoretic Transform(NTT), which is a form of FFT, is introduced, that is faster than FFTs. Also, a multiplication algorithm is introduced that uses this to perform integer multiplication faster than O(n log n). It uses…
In this paper, we propose a fast proximal gradient algorithm for multiobjective optimization, it is proved that the convergence rate of the accelerated algorithm for multiobjective optimization developed by Tanabe et al. can be improved…
Consider the problem of estimating the median of N items to a precision epsilon, i.e., the estimate should be such that, with a high probability, the number of items, with values both smaller than and larger than this estimate, is less than…
We introduce primed-PCA (pPCA), a two-step algorithm for speeding up the approximation of principal components. This algorithm first runs any approximate-PCA method to get an initial estimate of the principal components (priming), and then…
An algorithm is given to factor an integer with $N$ digits in $\ln^m N$ steps, with $m$ approximately 4 or 5. Textbook quadratic sieve methods are exponentially slower. An improvement with the aid of an a particular function would provide a…
Integer arithmetic is the underpinning of many quantum algorithms, with applications ranging from Shor's algorithm over HHL for matrix inversion to Hamiltonian simulation algorithms. A basic objective is to keep the required resources to…
We study the early work scheduling problem on identical parallel machines in order to maximize the total early work, i.e., the parts of non-preemptive jobs executed before a common due date. By preprocessing and constructing an auxiliary…
Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer $N$ with a…
K{\"o}hler's method is a useful multi-thresholding technique based on boundary contrast. However, the direct algorithm has a too high complexity-O(N 2) i.e. quadratic with the pixel numbers N-to process images at a sufficient speed for…
The quality of numerical computations can be measured through their forward error, for which finding good error bounds is challenging in general. For several algorithms and using stochastic rounding (SR), probabilistic analysis has been…