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Computing the matrix square root or 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…
In this paper, we introduce novel fast matrix inversion algorithms that leverage triangular decomposition and recurrent formalism, incorporating Strassen's fast matrix multiplication. Our research places particular emphasis on triangular…
A novel fast recursive coding technique is proposed. It operates with only integer values not longer 8 bits and is multiplication free. Recursion the algorithm is based on indirectly provides rather effective coding of symbols for very…
Greatest Common Divisor (GCD) computation is one of the most important operation of algorithmic number theory. In this paper we present the algorithms for GCD computation of $n$ integers. We extend the Euclid's algorithm and binary GCD…
We present a new computation method for simulating reflection high-energy electron diffraction and the total-reflection high-energy positron diffraction experiments. The two experiments are used commonly for the structural analysis of…
The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…
Permutation polynomials and their compositional inverses have wide applications in cryptography, coding theory, and combinatorial designs. Motivated by several previous results on finding compositional inverses of permutation polynomials of…
A concise analytical formula is developed for the inverse of an invertible 3 x 3 matrix using a telescoping method, and is generalized to larger square matrices. The formula is confirmed using randomly generated matrices in Matlab
Fast Fourier Transform (FFT) is an efficient algorithm to compute the Discrete Fourier Transform (DFT) and its inverse. In this paper, we pay special attention to the description of complex-data FFT. We analyze two common descriptions of…
Integer factorization is a very hard computational problem. Currently no efficient algorithm for integer factorization is publicly known. However, this is an important problem on which it relies the security of many real world cryptographic…
Obtaining the inverse of a large symmetric positive definite matrix $\mathcal{A}\in\mathbb{R}^{p\times p}$ is a continual challenge across many mathematical disciplines. The computational complexity associated with direct methods can be…
Recently, Ko\c{c} proposed a neat and efficient algorithm for computing \[ x = a^{-1} \pmod {p^k} \] for a prime $p$ based on the exact solution of linear equations using $p$-adic expansions. The algorithm requires only addition and right…
Incremental computation aims to compute more efficiently on changed input by reusing previously computed results. We give a high-level overview of works on incremental computation, and highlight the essence underlying all of them, which we…
A cumbersome operation in numerical analysis and linear algebra, optimization, machine learning and engineering algorithms; is inverting large full-rank matrices which appears in various processes and applications. This has both numerical…
Square matrices of the form $\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^*$ are considered. An explicit expression for the inverse is given, provided $\widetilde{\mathbf{A}}$ and $D$ are invertible with…
We propose to store several integers modulo a small prime into a single machine word. Modular addition is performed by addition and possibly subtraction of a word containing several times the modulo. Modular Multiplication is not directly…
This paper introduced a way of fractal to solve the problem of taking count of the integer partitions, furthermore, using the method in this paper some recurrence equations concerning the integer partitions can be deduced, including the…
In some of the problems, complicated functions of the Z-transform variable, $z$, appear which either cannot be inverted analytically or the required calculations are quite tedious. In such cases numerical methods should be used to find the…
There are thousands of papers on rootfinding for nonlinear scalar equations. Here is one more, to talk about an apparently new method, which I call ``Inverse Cubic Iteration'' (ICI) in analogy to the Inverse Quadratic Iteration in Richard…
Counting distinct permutations with replacement, especially when involving multiple subwords, is a longstanding challenge in combinatorial analysis, with critical applications in cryptography, bioinformatics, and statistical modeling. This…