System of linear equations
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Solving Random Quadratic Systems of Equations Is Nearly as Easy as Solving Linear Systems Open
We consider the fundamental problem of solving quadratic systems of equations in , and is unknown. We propose a novel method, which starts with an initial guess computed by means of a spectral method and proceeds by minimizing a nonconvex …
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Mean-field message-passing equations in the Hopfield model and its generalizations Open
Motivated by recent progress in using restricted Boltzmann machines as preprocessing algorithms for deep neural network, we revisit the mean-field equations [belief-propagation and Thouless-Anderson Palmer (TAP) equations] in the best unde…
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Sparsified Cholesky and multigrid solvers for connection laplacians Open
We introduce the sparsified Cholesky and sparsified multigrid algorithms for solving systems of linear equations. These algorithms accelerate Gaussian elimination by sparsifying the nonzero matrix entries created by the elimination process…
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Solving Random Quadratic Systems of Equations Is Nearly as Easy as\n Solving Linear Systems Open
We consider the fundamental problem of solving quadratic systems of equations\nin $n$ variables, where $y_i = |\\langle \\boldsymbol{a}_i, \\boldsymbol{x}\n\\rangle|^2$, $i = 1, \\ldots, m$ and $\\boldsymbol{x} \\in \\mathbb{R}^n$ is\nunkn…
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Preconditioned GSOR iterative method for a class of complex symmetric system of linear equations Open
Summary In this paper, we present a preconditioned variant of the generalized successive overrelaxation (GSOR) iterative method for solving a broad class of complex symmetric linear systems. We study conditions under which the spectral rad…
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Parallel iterative solution method for large sparse linear equation systems Open
Solving sparse systems of linear equations is at the heart of scientific computing. Large sparse systems often arise in science and engineering problems. One such problem we consider in this paper is the steady-state analysis of Continuous…
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An Efficient Reduced Basis Solver for Stochastic Galerkin Matrix Equations Open
Stochastic Galerkin finite element approximation of PDEs with random inputs leads to linear systems of equations with coefficient matrices that have a characteristic Kronecker product structure. By reformulating the systems as multi-term l…
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Solving a Mixture of Many Random Linear Equations by Tensor Decomposition and Alternating Minimization Open
We consider the problem of solving mixed random linear equations with $k$ components. This is the noiseless setting of mixed linear regression. The goal is to estimate multiple linear models from mixed samples in the case where the labels …
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Truncated low‐rank methods for solving general linear matrix equations Open
Summary This work is concerned with the numerical solution of large‐scale linear matrix equations . The most straightforward approach computes from the solution of an m n × m n linear system, typically limiting the feasible values of m , n…
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A generalized approach for implicit time integration of piecewise linear/nonlinear systems Open
A generalized solution scheme using implicit time integrators for piecewise linear and nonlinear systems is developed. The piecewise linear characteristic has been well‐discussed in previous studies, in which the original problem has been …
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Quantum circulant preconditioner for a linear system of equations Open
We consider the quantum linear solver for $Ax=b$ with the circulant\npreconditioner $C$. The main technique is the singular value estimation (SVE)\nintroduced in [I. Kerenidis and A. Prakash, Quantum recommendation system, in\nITCS 2017]. …
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Linear Searching Method for Solving Approximate Solution to System of Max-Min Fuzzy Relation Equations With Application in the Instructional Information Resources Allocation Open
Max-min fuzzy relation equations are introduced to describe the peer-to-peer (P2P) data transmission mechanism in instructional information resources' sharing system. In some cases, it is not able to satisfy the download requirements of th…
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A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle Open
In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extensio…
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An efficient numerical algorithm for solving system of Lane–Emden type equations arising in engineering Open
The purpose of this paper is to propose an efficient numerical method for solving system of Lane–Emden type equations using Chebyshev operational matrix method. This method transforms the system of Lane-Emden type equation into the system …
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Interactively Cutting and Constraining Vertices in Meshes Using Augmented Matrices Open
We present a finite-element solution method that is well suited for interactive simulations of cutting meshes in the regime of linear elastic models. Our approach features fast updates to the solution of the stiffness system of equations t…
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Approximate Green’s Function Coupled Cluster Method Employing Effective Dimension Reduction Open
The Green's function coupled cluster (GFCC) method, originally proposed in the early 1990s, is a powerful many-body tool for computing and analyzing the electronic structure of molecular and periodic systems, especially when electrons of t…
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Solution of Nonlinear Systems of Equations via Metaheuristics Open
A framework devoted to the solution of nonlinear systems of equations using grey wolf optimization algorithm (GWO) and a multi-objective particle swarm optimization algorithm (MOPSO) is presented in this work. Due to several numerical issu…
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The Sherman–Morrison–Woodbury formula for generalized linear matrix equations and applications Open
We discuss the use of a matrix‐oriented approach for numerically solving the dense matrix equation AX + XA T + M 1 XN 1 + … + M ℓ XN ℓ = F , with ℓ ≥ 1, and M i , N i , i = 1, … , ℓ of low rank. The approach relies on the Sherman–Morrison–…
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New method for solving a class of fractional partial differential equations with applications Open
In this work we suggest a numerical approach based on the B-spline polynomial to obtain the solution of linear fractional partial differential equations. We find the operational matrix for fractional integration and then we convert the mai…
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Transformation Method for Solving System of Boolean Algebraic Equations Open
In recent years, various methods and directions for solving a system of Boolean algebraic equations have been invented, and now they are being very actively investigated. One of these directions is the method of transforming a system of Bo…
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Computer assisted solution of systems of two variable linear functional equations Open
In the present paper, a general class of linear functional equations is considered and a computer program is described, which determines the exact solutions of systems of equations belonging to this class.
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Solving Systems of Polynomial Equations over GF(2) by a Parity-Counting Self-Reduction Open
We consider the problem of finding solutions to systems of polynomial equations over a finite field. Lokshtanov et al. [SODA'17] recently obtained the first worst-case algorithms that beat exhaustive search for this problem. In particular …
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Improved Block-Pulse Functions for Numerical Solution of Mixed Volterra-Fredholm Integral Equations Open
The present paper employs a numerical method based on the improved block–pulse basis functions (IBPFs). This was mainly performed to resolve the Volterra–Fredholm integral equations of the second kind. Those equations are often simplified …
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A new preconditioning approach for an interior point‐proximal method of multipliers for linear and convex quadratic programming Open
In this article, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers,…
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Parameter robust preconditioning by congruence for multiple-network poroelasticity Open
The mechanical behavior of a poroelastic medium permeated by multiple interacting fluid networks can be described by a system of time-dependent partial differential equations known as the multiple-network poroelasticity (MPET) equations or…
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An operational matrix method for solving linear Fredholm--Volterra integro-differential equations Open
The aim of this paper is to propose an efficient method to compute approximate solutions of linear Fredholm‒Volterra integro-differential equations (FVIDEs) using Taylor polynomials. More precisely, we present a method based on operational…
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Biorthogonal-Wavelet-Based Method for Numerical Solution of Volterra Integral Equations Open
Framelets theory has been well studied in many applications in image processing, data recovery and computational analysis due to the key properties of framelets such as sparse representation and accuracy in coefficients recovery in the are…
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A novel study for solving systems of nonlinear fractional integral equations Open
In this study, we explore the solution of a nonlinear system of fractional integro-differential equations based on the operational matrix method. We have modified the operational matrix method to accommodate such systems and have streamlin…
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A machine learning based solver for pressure Poisson equations Open
When using the projection method (or fractional step method) to solve the incompressible Navier-Stokes equations, the projection step involves solving a large-scale pressure Poisson equation (PPE), which is computationally expensive and ti…
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A specialised cyclic reduction algorithm for linear algebraic equation systems with quasi-tridiagonal matrices Open
Extensions have been developed, of several variants of the stride of two cyclic reduction method. The extensions refer to quasi-tridiagonal linear equation systems involving two additional nonzero elements in the first and last rows of the…