Related papers: Fast Fraction-Integer Method for Computing Multipl…
Numerous methods have been considered to create a fast integer factorization algorithm. Despite its apparent simplicity, the difficulty to find such an algorithm plays a crucial role in modern cryptography, notably, in the security of RSA…
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating…
On common processors, integer multiplication is many times faster than integer division. Dividing a numerator n by a divisor d is mathematically equivalent to multiplication by the inverse of the divisor (n / d = n x 1/d). If the divisor is…
In this article we provide a fast computational method in order to calculate the Moore-Penrose inverse of singular square matrices and of rectangular matrices. The proposed method proves to be much faster and has significantly better…
We introduce a new numerical method for the computation of the inverse nonlinear Fourier transform and compare its computational complexity and accuracy to those of other methods available in the literature. For a given accuracy, the…
A method of determining two factors of an odd integer without need of multiplication or division operation in iterative portion of computation is presented. It is feasible for an implementing algorithm to use only integer addition and…
We study algorithms for the fast computation of modular inverses. Newton-Raphson iteration over $p$-adic numbers gives a recurrence relation computing modular inverse modulo $p^m$, that is logarithmic in $m$. We solve the recurrence to…
We present improved algorithms for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithms are much more accurate than the famous fast inverse square root algorithm and have the same or…
We address the general mathematical problem of computing the inverse $p$-th root of a given matrix in an efficient way. A new method to construct iteration functions that allow calculating arbitrary $p$-th roots and their inverses of…
It is well known that the repeated square and multiply algorithm is an efficient way of modular exponentiation. The obvious question to ask is if this algorithm has an inverse which would calculate the discrete logarithm efficiently. The…
We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing…
Our paper introduces a novel method for calculating the inverse $\mathcal{Z}$-transform of rational functions. Unlike some existing approaches that rely on partial fraction expansion and involve dividing by $z$, our method allows for the…
We observe structure in the sequences of quotients and remainders of the Euclidean algorithm with two families of inputs. Analyzing the remainders, we obtain new algorithms for computing modular inverses and representating prime numbers by…
This paper presents an algorithm for the integer multiplicative inverse (mod $2^w$) which completes in the fewest cycles known for modern microprocessors, when using the native bit width $w$ for the modulus $2^w$. The algorithm is a…
This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
We believe that there is no real data protection without our own tools. Therefore, our permanent aim is to have more of our own codes. In order to achieve that, it is necessary that a lot of young researchers become interested in…
We propose a generic algorithm for computing the inverses of a multiplicative function under the assumption that the set of inverses is finite. More generally, our algorithm can compute certain functions of the inverses, such as their power…
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
The usual formulation of efficient division uses Newton iteration to compute an inverse in a related domain where multiplicative inverses exist. On one hand, Newton iteration allows quotients to be calculated using an efficient…