Systems of Equations With Positive Definite Symmetric Coefficient Matrix

1990 ◽  
pp. 53-76
Author(s):  
Heinz Rutishauser
Keyword(s):  
Coefficient Matrix ◽  
Positive Definite ◽  
Systems Of Equations

Optimum Accelerated Overrelaxation (AOR) Method for Systems with Positive Definite Coefficient Matrix

1983 ◽  
Vol 20 (4) ◽  
pp. 774-783 ◽  
Author(s):  
N. Gaïtanos ◽  
A. Hadjidimos ◽  
A. Yeyios
Keyword(s):  
Coefficient Matrix ◽  
Positive Definite ◽  
Aor Method

scholarly journals Stability Analysis of Drilling Inclination System with Time-Varying Delay via Free-Matrix-Based Lyapunov–Krasovskii Functional

2021 ◽  
Vol 25 (6) ◽  
pp. 1031-1038
Author(s):  
Zhen Cai ◽  
Guozhen Hu ◽  
◽  
Keyword(s):  
Stability Analysis ◽  
Asymptotic Stability ◽  
Stability Criterion ◽  
Coefficient Matrix ◽  
Positive Definite ◽  
Time Varying ◽  
Time Varying Delay ◽  
The Stability ◽  
Varying Delay ◽  
Insight Into

This study provides an insight into the asymptotic stability of a drilling inclination system with a time-varying delay. An appropriate Lyapunov–Krasovskii functional (LKF) is essential for the stability analysis of the abovementioned system. In general, an LKF is constructed with each coefficient matrix being positive definite, which results in considerable conservatism. Herein, to relax the conditions of the derived criteria, a novel LKF is proposed by avoiding the positive-definite restriction of some coefficient matrices and introducing additional free matrices simultaneously. Subsequently, this relaxed LKF is applied to derive a less conservative stability criterion for the abovementioned system. Finally, the effect of reducing the conservatism of the proposed LKF is verified based on two examples.

scholarly journals On solving classes of positive-definite quantum linear systems with quadratically improved runtime in the condition number

Quantum ◽  
2021 ◽  
Vol 5 ◽  
pp. 573
Author(s):  
Davide Orsucci ◽  
Vedran Dunjko
Keyword(s):  
Linear Systems ◽  
Condition Number ◽  
Quantum Algorithms ◽  
Coefficient Matrix ◽  
Positive Definite ◽  
Fundamental Parameter ◽  
Worst Case ◽  
Complete Problems ◽  
Matrix Block ◽  
Potential Applications

Quantum algorithms for solving the Quantum Linear System (QLS) problem are among the most investigated quantum algorithms of recent times, with potential applications including the solution of computationally intractable differential equations and speed-ups in machine learning. A fundamental parameter governing the efficiency of QLS solvers is κ, the condition number of the coefficient matrix A, as it has been known since the inception of the QLS problem that for worst-case instances the runtime scales at least linearly in κ [Harrow, Hassidim and Lloyd, PRL 103, 150502 (2009)]. However, for the case of positive-definite matrices classical algorithms can solve linear systems with a runtime scaling as κ, a quadratic improvement compared to the the indefinite case. It is then natural to ask whether QLS solvers may hold an analogous improvement. In this work we answer the question in the negative, showing that solving a QLS entails a runtime linear in κ also when A is positive definite. We then identify broad classes of positive-definite QLS where this lower bound can be circumvented and present two new quantum algorithms featuring a quadratic speed-up in κ: the first is based on efficiently implementing a matrix-block-encoding of A−1, the second constructs a decomposition of the form A=LL† to precondition the system. These methods are widely applicable and both allow to efficiently solve BQP-complete problems.

scholarly journals Accelerated Circulant and Skew Circulant Splitting Methods for Hermitian Positive Definite Toeplitz Systems

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
N. Akhondi ◽  
F. Toutounian
Keyword(s):  
Convergence Property ◽  
Linear Equations ◽  
Coefficient Matrix ◽  
Positive Definite ◽  
Optimal Parameters ◽  
Skew Circulant ◽  
Systems Of Linear Equations ◽  
Toeplitz Systems ◽  
Two Parameters ◽  
Two Parameter

We study the CSCS method for large Hermitian positive definite Toeplitz linear systems, which first appears in Ng's paper published in (Ng, 2003), and CSCS stands for circulant and skew circulant splitting of the coefficient matrix . In this paper, we present a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. The method is a two-parameter generation of the CSCS method such that when the two parameters involved are equal, it coincides with the CSCS method. We discuss the convergence property and optimal parameters of this method. Finally, we extend our method to BTTB matrices. Numerical experiments are presented to show the effectiveness of our new method.

scholarly journals A Relaxed Kaczmarz Method for Fuzzy Linear Systems

2021 ◽  
Author(s):  
Ke Wang ◽  
Shijun Zhang ◽  
Shiheng Wang
Keyword(s):  
Linear Systems ◽  
Convergence Theorem ◽  
Iterative Scheme ◽  
Coefficient Matrix ◽  
Systems Of Equations ◽  
Numerical Examples ◽  
Right Hand ◽  
Kaczmarz Method ◽  
Fuzzy Linear Systems ◽  
Linear Systems Of Equations

Abstract A relaxed Kaczmarz method is presented for solving a class of fuzzy linear systems of equations with crisp coefficient matrix and fuzzy right-hand side. The iterative scheme is established and the convergence theorem is provided. Numerical examples show that the method is effective.

scholarly journals A parallel Cholesky algorithm for the solution of symmetric linear systems

2004 ◽  
Vol 2004 (25) ◽  
pp. 1315-1327
Author(s):  
R. R. Khazal ◽  
M. M. Chawla
Keyword(s):  
Parallel Algorithm ◽  
Linear Systems ◽  
Triangular Matrix ◽  
Coefficient Matrix ◽  
Positive Definite ◽  
Identity Matrix ◽  
Lower Triangular Matrix ◽  
Elimination Method ◽  
Lower Triangular ◽  
Positive Definite System

For the solution of symmetric linear systems, the classical Cholesky method has proved to be difficult to parallelize. In the present paper, we first describe an elimination variant of Cholesky method to produce a lower triangular matrix which reduces the coefficient matrix of the system to an identity matrix. Then, this elimination method is combined with the partitioning method to obtain a parallel Cholesky algorithm. The total serial arithmetical operations count for the parallel algorithm is of the same order as that for the serial Cholesky method. The present parallel algorithm could thus perform withefficiencyclose to 1 if implemented on a multiprocessor machine. We also discuss theexistenceof the parallel algorithm; it is shown that for a symmetric and positive definite system, the presented parallel Cholesky algorithm is well defined and will run to completion.

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