Iterative Matrix Inversion and the Iterative Solution of Linear Equations

1984 ◽  
pp. 162-179
Author(s):  
R. T. Gregory ◽  
E. V. Krishnamurthy
Keyword(s):  
Linear Equations ◽  
Iterative Solution ◽  
Matrix Inversion

scholarly journals Krylov Subspace Solvers and Preconditioners

2018 ◽  
Vol 63 ◽  
pp. 1-43
Author(s):  
C. Vuik
Keyword(s):  
Iterative Methods ◽  
Krylov Subspace ◽  
State Of The Art ◽  
Linear Equations ◽  
Iterative Solution ◽  
Krylov Subspace Methods ◽  
Numerical Linear Algebra ◽  
Subspace Methods ◽  
Starting Point ◽  
Lecture Notes

In these lecture notes an introduction to Krylov subspace solvers and preconditioners is presented. After a discretization of partial differential equations large, sparse systems of linear equations have to be solved. Fast solution of these systems is very urgent nowadays. The size of the problems can be 1013 unknowns and 1013 equations. Iterative solution methods are the methods of choice for these large linear systems. We start with a short introduction of Basic Iterative Methods. Thereafter preconditioned Krylov subspace methods, which are state of the art, are describeed. A distinction is made between various classes of matrices. At the end of the lecture notes many references are given to state of the art Scientific Computing methods. Here, we will discuss a number of books which are nice to use for an overview of background material. First of all the books of Golub and Van Loan [19] and Horn and Johnson [26] are classical works on all aspects of numerical linear algebra. These books also contain most of the material, which is used for direct solvers. Varga [50] is a good starting point to study the theory of basic iterative methods. Krylov subspace methods and multigrid are discussed in Saad [38] and Trottenberg, Oosterlee and Schüller [42]. Other books on Krylov subspace methods are [1, 6, 21, 34, 39].

Introducing new global electromagnetic modeling solver

2020 ◽  
Author(s):  
Mikhail Kruglyakov ◽  
Alexey Kuvshinov
Keyword(s):  
Integral Equation ◽  
Galerkin Method ◽  
Numerical Solution ◽  
Linear Equations ◽  
Numerical Algorithms ◽  
Iterative Solution ◽  
Parallel Computations ◽  
Electromagnetic Modeling ◽  
System Of Linear Equations

<p> In this contribution, we present novel global 3-D electromagnetic forward solver based on a numerical solution of integral equation (IE) with contracting kernel. Compared to widely used x3dg code which is also based on IE approach, new solver exploits alternative (more efficient and accurate) numerical algorithms to calculate Green’s tensors, as well as an alternative (Galerkin) method to construct the system of linear equations (SLE). The latter provides guaranteed convergence of the iterative solution of SLE. The solver outperforms x3dg in terms of accuracy, and, in contrast to (sequential) x3dg, it allows for efficient parallel computations, meaning that the code has practically linear scalability up to the hundreds of processors.</p>

Electromagnetic modeling of 3-D structures by the method of system iteration using integral equations

Geophysics ◽  
1992 ◽  
Vol 57 (12) ◽  
pp. 1556-1561 ◽  
Author(s):  
Zonghou Xiong
Keyword(s):  
Integral Equations ◽  
Three Dimensional ◽  
Computation Time ◽  
Iterative Solution ◽  
Matrix Inversion ◽  
Electromagnetic Modeling ◽  
Direct Inversion ◽  
Conductivity Structure ◽  
The Matrix ◽  
Earth Conductivity

A new approach for electromagnetic modeling of three‐dimensional (3-D) earth conductivity structures using integral equations is introduced. A conductivity structure is divided into many substructures and the integral equation governing the scattering currents within a substructure is solved by a direct matrix inversion. The influence of all other substructures are treated as external excitations and the solution for the whole structure is then found iteratively. This is mathematically equivalent to partitioning the scattering matrix into many block submatrices and solving the whole system by a block iterative method. This method reduces computer memory requirements since only one submatrix at a time needs to be stored. The diagonal submatrices that require direct inversion are defined by local scatterers only and thus are generally better conditioned than the matrix for the whole structure. The block iterative solution requires much less computation time than direct matrix inversion or conventional point iterative methods as the convergence depends on the number of the submatrices, not on the total number of unknowns in the solution. As the submatrices are independent of each other, this method is suitable for parallel processing.

scholarly journals Improved Computing-Efficiency Least-Squares Algorithm with Application to All-Pass Filter Design

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Lo-Chyuan Su ◽  
Yue-Dar Jou ◽  
Fu-Kun Chen
Keyword(s):  
Least Squares ◽  
Filter Design ◽  
Linear Equations ◽  
Hankel Matrix ◽  
Cholesky Decomposition ◽  
Matrix Inversion ◽  
System Of Linear Equations ◽  
Computationally Efficient ◽  
Pass Filter ◽  
Simulation Results

All-pass filter design can be generally achieved by solving a system of linear equations. The associated matrices involved in the set of linear equations can be further formulated as a Toeplitz-plus-Hankel form such that a matrix inversion is avoided. Consequently, the optimal filter coefficients can be solved by using computationally efficient Levinson algorithms or Cholesky decomposition technique. In this paper, based on trigonometric identities and sampling the frequency band of interest uniformly, the authors proposed closed-form expressions to compute the elements of the Toeplitz-plus-Hankel matrix required in the least-squares design of IIR all-pass filters. Simulation results confirm that the proposed method achieves good performance as well as effectiveness.

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