scholarly journals MINRES Seed Projection Methods for Solving Symmetric Linear Systems with Multiple Right-Hand Sides

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Xin Li ◽  
Hao Liu ◽  
Jingfu Zhu
Keyword(s):  
Linear Systems ◽  
Projection Method ◽  
Symmetric Matrix ◽  
Coefficient Matrix ◽  
Projection Methods ◽  
Numerical Examples ◽  
Right Hand ◽  
Symmetric Systems

We consider the MINRES seed projection method for solving multiple right-hand side linear systemsAX=B, whereA∈Rn×nis a nonsingular symmetric matrix,B∈Rn×p. In general, GMRES seed projection method is one of the effective methods for solving multiple right-hand side linear systems. However, when the coefficient matrix is symmetric, the efficiency of this method would be weak. MINRES seed projection method for solving symmetric systems with multiple right-hand sides is proposed in this paper, and the residual estimation is analyzed. The numerical examples show the efficiency of this 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 Breakdown-Free Block COCG Method for Complex Symmetric Linear Systems with Multiple Right-Hand Sides

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1302
Author(s):  
Hong-Xiu Zhong ◽  
Xian-Ming Gu ◽  
Shao-Liang Zhang
Keyword(s):  
Linear Systems ◽  
Spectral Properties ◽  
Coefficient Matrix ◽  
Complex Symmetric Linear Systems ◽  
Right Hand ◽  
Orthogonality Conditions ◽  
Free Block ◽  
Complex Symmetric ◽  
The Right ◽  
Conjugate Residual

The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm with new parameter matrices to handle rank deficiency. To improve the spectral properties of coefficient matrix A, a precondition version of the breakdown-free BCOCG is proposed in detail. We also give the relative algorithms for the block conjugate A-orthogonal conjugate residual method. Numerical results illustrate that when breakdown occurs, the breakdown-free algorithms yield faster convergence than the non-breakdown-free algorithms.

A two-step iterative method based on diagonal and off-diagonal splitting for solving linear systems

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1441-1452
Author(s):  
Mehdi Dehghana ◽  
Marzieh Dehghani-Madisehb ◽  
Masoud Hajarianc
Keyword(s):  
Numerical Analysis ◽  
Linear Systems ◽  
Iterative Methods ◽  
Classical Problem ◽  
Coefficient Matrix ◽  
New Method ◽  
Step Method ◽  
Numerical Examples ◽  
Special Cases ◽  
Two Parameter

Solving linear systems is a classical problem of engineering and numerical analysis which has various applications in many sciences and engineering. In this paper, we study efficient iterative methods, based on the diagonal and off-diagonal splitting of the coefficient matrix A for solving linear system Ax = b, where A ? Cnxn is nonsingular and x,b ? Cnxm. The new method is a two-parameter two-step method that has some iterative methods as its special cases. Numerical examples are presented to illustrate the effectiveness of the new method.

scholarly journals ON THE PARALLEL SOLUTION OF LARGE INDUSTRIAL WAVE PROPAGATION PROBLEMS

2006 ◽  
Vol 14 (01) ◽  
pp. 83-111 ◽  
Author(s):  
L. GIRAUD ◽  
J. LANGOU ◽  
G. SYLVAND
Keyword(s):  
Linear Systems ◽  
Message Passing ◽  
Integral Equation Method ◽  
Coefficient Matrix ◽  
Industrial Test ◽  
Iterative Solvers ◽  
Test Problems ◽  
Simulation Code ◽  
Approximate Inverse ◽  
Right Hand

The use of Fast Multipole Methods (FMM) combined with embedded Krylov solvers preconditioned by a sparse approximate inverse is investigated for the solution of large linear systems arising in industrial acoustic and electromagnetic simulations. We use a boundary element integral equation method to solve the Helmholtz and the Maxwell equations in the frequency domain. The resulting linear systems are solved by iterative solvers using FMM to accelerate the matrix-vector products. The simulation code is developed in a distributed memory environment using message passing and it has out-of-core capabilities to handle very large calculations. When the calculation involves one incident wave, one linear system has to be solved. In this situation, embedded solvers can be combined with an approximate inverse preconditioner to design extremely robust algorithms. For radar cross section calculations, several linear systems have to be solved. They involve the same coefficient matrix but different right-hand sides. In this case, we propose a block variant of the single right-hand side scheme. The efficiency, robustness and parallel scalability of our approach are illustrated on a set of large academic and industrial test problems.

scholarly journals Deflated BiCG with an Application to Model Reduction

2015 ◽  
Vol 2015 ◽  
pp. 1-8
Author(s):  
Jing Meng ◽  
Pei-Yong Zhu ◽  
Hou-Biao Li
Keyword(s):  
Model Reduction ◽  
Linear Systems ◽  
Invariant Subspaces ◽  
Numerical Examples ◽  
Right Hand ◽  
Ritz Vectors ◽  
Left And Right

Most calculations in model reduction involve the solutions of a sequence of dual linear systems with multiple right-hand sides. To solve such systems efficiently, a new deflated BiCG method is explored in this paper. The proposed algorithm uses harmonic Ritz vectors to approximate left and right invariant subspaces inexpensively via small descenting direction vectors found by subsequent runs of deflated BiCG and then derives the deflated subspaces for the next pair of dual linear systems. This process leads to faster convergence for the next pair of systems. Numerical examples illustrate the effectiveness of the proposed method.

scholarly journals On Existence of Stabilizing Switching Laws within a Class of Unstable Linear Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Sendren Sheng-Dong Xu ◽  
Chih-Chiang Chen
Keyword(s):  
Control Systems ◽  
Linear Systems ◽  
Switched Systems ◽  
The Other ◽  
Linear Control ◽  
Linear Control Systems ◽  
Problem Statement ◽  
Theoretical Derivation ◽  
Control Laws ◽  
Numerical Examples

The equivalence of two conditions, condition (3) and condition (4) stated in Problem Statement section, regarding the existence of stabilizing switching laws between two unstable linear systems first appeared in (Feron 1996). Although Feron never published this result, it has been referenced in almost every survey on switched systems; see, for example, (Liberzon and Morse 1999). This paper proposes another way to prove the equivalence of two conditions regarding the existence of stabilizing switching laws between two unstable linear systems. One is effective for theoretical derivation, while the other is implementable, and a class of stabilizing switching laws have been explicitly constructed by Wicks et al. (1994). With the help of the equivalent relation, a condition for the existence of controllers and stabilizing switching laws between two unstabilizable linear control systems is then proposed. Then, the study is further extended to the issue concerning the construction of quadratically stabilizing switching laws among unstable linear systems and unstabilizable linear control systems. The obtained results are employed to study the existence of control laws and quadratically stabilizing switching laws within a class of unstabilizable linear control systems. The numerical examples are illustrated and simulated to show the feasibility and effectiveness of the proposed methods.

Nondegenerate Piecewise Linear Systems: A Finite Newton Algorithm and Applications in Machine Learning

2012 ◽  
Vol 24 (4) ◽  
pp. 1047-1084 ◽  
Author(s):  
Xiao-Tong Yuan ◽  
Shuicheng Yan
Keyword(s):  
Linear Systems ◽  
Optimization Problems ◽  
Piecewise Linear ◽  
Optimization Methods ◽  
Coefficient Matrix ◽  
Learning Problems ◽  
Support Vector ◽  
Data Sets ◽  
Piecewise Linear Systems ◽  
Vector Machines

We investigate Newton-type optimization methods for solving piecewise linear systems (PLSs) with nondegenerate coefficient matrix. Such systems arise, for example, from the numerical solution of linear complementarity problem, which is useful to model several learning and optimization problems. In this letter, we propose an effective damped Newton method, PLS-DN, to find the exact (up to machine precision) solution of nondegenerate PLSs. PLS-DN exhibits provable semiiterative property, that is, the algorithm converges globally to the exact solution in a finite number of iterations. The rate of convergence is shown to be at least linear before termination. We emphasize the applications of our method in modeling, from a novel perspective of PLSs, some statistical learning problems such as box-constrained least squares, elitist Lasso (Kowalski & Torreesani, 2008 ), and support vector machines (Cortes & Vapnik, 1995 ). Numerical results on synthetic and benchmark data sets are presented to demonstrate the effectiveness and efficiency of PLS-DN on these problems.

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