New Research Results of Algorithms for Analysis of Changing Electric Circuits

2020 ◽  
Vol 63 (6) ◽  
pp. 29-36
Author(s):  
Ivan Lebedev ◽  
◽  
Nikolai Savelov ◽  
Keyword(s):  
Numerical Experiments ◽  
Linear Equations ◽  
Iterative Refinement ◽  
Floating Point ◽  
Electrical Circuits ◽  
Electric Circuits ◽  
Research Results ◽  
Systems Of Linear Equations ◽  
New Research ◽  
Refinement Algorithm

New research results in the field of accelerated analysis of linear and linearizable electrical circuits with variable elements are presented. The considered algorithms are based on a new modification of Gaussian elim-ination for solution of systems of linear equations. The researches was directed on a systematic thorough study of the phenomenon of quasi-stabilization of the error discovered by the authors during multiple repeated cir-cuit analyzes. The Sherman-Morrison algorithm and the iterative refinement algorithm are considered as alter-natives. To control the results of numerical experiments, the representation of floating-point numbers in the bi-nary128 format of the IEEE 754 standard is used. A method for the machine formation of mathematical models of circuits with an unlimited number of elements is proposed.

Accelerated GPMHSS Method for Solving Complex Systems of Linear Equations

2017 ◽  
Vol 7 (1) ◽  
pp. 143-155 ◽  
Author(s):  
Jing Wang ◽  
Xue-Ping Guo ◽  
Hong-Xiu Zhong
Keyword(s):  
Complex Systems ◽  
Linear Systems ◽  
Iteration Method ◽  
Numerical Experiments ◽  
Linear Equations ◽  
Splitting Method ◽  
Complex Symmetric Linear Systems ◽  
Systems Of Linear Equations ◽  
Complex Symmetric ◽  
Symmetric Systems

AbstractPreconditioned modified Hermitian and skew-Hermitian splitting method (PMHSS) is an unconditionally convergent iteration method for solving large sparse complex symmetric systems of linear equations, and uses one parameter α. Adding another parameter β, the generalized PMHSS method (GPMHSS) is essentially a twoparameter iteration method. In order to accelerate the GPMHSS method, using an unexpected way, we propose an accelerated GPMHSS method (AGPMHSS) for large complex symmetric linear systems. Numerical experiments show the numerical behavior of our new method.

scholarly journals Clock Math — a System for Solving SLEs Exactly

10.14311/1795 ◽  
2013 ◽  
Vol 53 (2) ◽  
Author(s):  
Jakub Hladík ◽  
Róbert Lórencz ◽  
Ivan Šimeček
Keyword(s):  
Hybrid System ◽  
Chinese Remainder Theorem ◽  
Linear Equations ◽  
Floating Point ◽  
Rounding Errors ◽  
Systems Of Linear Equations ◽  
Floating Point Numbers

In this paper, we present a GPU-accelerated hybrid system that solves ill-conditioned systems of linear equations exactly. Exactly means without rounding errors due to using integer arithmetics. First, we scale floating-point numbers up to integers, then we solve dozens of SLEs within different modular arithmetics and then we assemble sub-solutions back using the Chinese remainder theorem. This approach effectively bypasses current CPU floating-point limitations. The system is capable of solving Hilbert’s matrix without losing a single bit of precision, and with a significant speedup compared to existing CPU solvers.

scholarly journals The Mechanization of the Comrade Matrix Approach in Determining the GCD of Orthogonal Polynomials

MATEMATIKA ◽  
2018 ◽  
Vol 34 (3) ◽  
pp. 25-32
Author(s):  
Siti Nor Asiah Isa ◽  
Nor’aini Aris ◽  
Shazirawati Mohd Puzi ◽  
Yeak Su Hoe
Keyword(s):  
Orthogonal Polynomials ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Qr Decomposition ◽  
Iterative Refinement ◽  
Matrix Approach ◽  
Alternative Approach ◽  
The Matrix ◽  
Systems Of Linear Equations

This paper revisits the comrade matrix approach in finding the greatest common divisor (GCD) of two orthogonal polynomials. The present work investigates on the applications of the QR decomposition with iterative refinement (QRIR) to solve certain systems of linear equations which is generated from the comrade matrix. Besides iterative refinement, an alternative approach of improving the conditioning behavior of the coefficient matrix by normalizing its columns is also considered. As expected the results reveal that QRIR is able to improve the solutions given by QR decomposition while the normalization of the matrix entries do improves the conditioning behavior of the coefficient matrix leading to a good approximate solutions of the GCD.

scholarly journals An HHL-based algorithm for computing hitting probabilities of quantum walks

2021 ◽  
Vol 21 (5&6) ◽  
pp. 395-404
Author(s):  
Ji Guan ◽  
Qisheng Wang ◽  
Mingsheng Ying
Keyword(s):  
Numerical Experiments ◽  
Linear Equations ◽  
Quantum Algorithm ◽  
Quantum Walks ◽  
One Dimensional ◽  
Simple Observation ◽  
Hitting Probabilities ◽  
Absorbing Boundaries ◽  
Systems Of Linear Equations ◽  
Absorption Probabilities

We present a novel application of the HHL (Harrow-Hassidim-Lloyd) algorithm --- a quantum algorithm solving systems of linear equations --- in solving an open problem about quantum walks, namely computing hitting (or absorption) probabilities of a general (not only Hadamard) one-dimensional quantum walks with two absorbing boundaries. This is achieved by a simple observation that the problem of computing hitting probabilities of quantum walks can be reduced to inverting a matrix. Then a quantum algorithm with the HHL algorithm as a subroutine is developed for solving the problem, which is faster than the known classical algorithms by numerical experiments.

scholarly journals Linear Equation System Study on Electrical Circuits Using Matlab

2018 ◽  
Vol 3 (1) ◽  
pp. 1
Author(s):  
Ulul Ilmi
Keyword(s):  
Everyday Life ◽  
Linear Equations ◽  
Equation System ◽  
Electrical Circuit ◽  
Matrix Form ◽  
System Study ◽  
System Of Linear Equations ◽  
Electrical Circuits ◽  
Linear Equation System ◽  
Systems Of Linear Equations

In everyday life, especially in the electrical circuit, there are many usage of matrix. One use of a matrix is found in the system of linear equations. In the field of electrical circuits there are also problems involving systems of linear equations in matrix form. To solve the system of linear equations in matrix form, in addition to using elementary row operations, also used matlab.

On Some Properties of Semi-rings

2018 ◽  
pp. 35-50
Author(s):  
A. I. Belousov
Keyword(s):  
Linear Equations ◽  
Oriented Graph ◽  
Theoretical Computer Science ◽  
Alternative Methods ◽  
Main Role ◽  
Cost Matrix ◽  
Sequential Elimination ◽  
The Matrix ◽  
Systems Of Linear Equations ◽  
The Cost

The main objective of this paper is to prove a theorem according to which a method of successive elimination of unknowns in the solution of systems of linear equations in the semi-rings with iteration gives the really smallest solution of the system. The proof is based on the graph interpretation of the system and establishes a relationship between the method of sequential elimination of unknowns and the method for calculating a cost matrix of a labeled oriented graph using the method of sequential calculation of cost matrices following the paths of increasing ranks. Along with that, and in terms of preparing for the proof of the main theorem, we consider the following important properties of the closed semi-rings and semi-rings with iteration.We prove the properties of an infinite sum (a supremum of the sequence in natural ordering of an idempotent semi-ring). In particular, the proof of the continuity of the addition operation is much simpler than in the known issues, which is the basis for the well-known algorithm for solving a linear equation in a semi-ring with iteration.Next, we prove a theorem on the closeness of semi-rings with iteration with respect to solutions of the systems of linear equations. We also give a detailed proof of the theorem of the cost matrix of an oriented graph labeled above a semi-ring as an iteration of the matrix of arc labels.The concept of an automaton over a semi-ring is introduced, which, unlike the usual labeled oriented graph, has a distinguished "final" vertex with a zero out-degree.All of the foregoing provides a basis for the proof of the main theorem, in which the concept of an automaton over a semi-ring plays the main role.The article's results are scientifically and methodologically valuable. The proposed proof of the main theorem allows us to relate two alternative methods for calculating the cost matrix of a labeled oriented graph, and the proposed proofs of already known statements can be useful in presenting the elements of the theory of semi-rings that plays an important role in mathematical studies of students majoring in software technologies and theoretical computer science.

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