A Solution of Simultaneous Linear Equations and Matrix Inversion With High Speed Computing Devices

1953 ◽  
Vol 7 (42) ◽  
pp. 77
Author(s):  
Gabriel G. Bejarano ◽  
Bruce R. Rosenblatt
Keyword(s):  
High Speed ◽  
Linear Equations ◽  
Matrix Inversion ◽  
Simultaneous Linear Equations

TREATMENT OF SINGULARITIES IN SCATTERING FROM PERFECTLY CONDUCTING POLYGONAL CYLINDERS—A NUMERICAL TECHNIQUE

1967 ◽  
Vol 45 (3) ◽  
pp. 1305-1318 ◽  
Author(s):  
S. Abdelmessih ◽  
G. Sinclair
Keyword(s):  
High Speed ◽  
Numerical Technique ◽  
Linear Equations ◽  
The Other ◽  
Incident Plane Wave ◽  
Induced Current ◽  
Scattering Problems ◽  
Incident Plane ◽  
Simultaneous Linear Equations ◽  
Sharp Edges

Certain field components become infinite at sharp edges, causing difficulties in computation when solving scattering problems for polygonal cylinders by numerical methods. Since the field behavior near an edge is known, it is possible to devise appropriate functions to represent the singularities at the edges. By suitably representing the induced current and using an integral equation for it, it is possible to obtain a set of simultaneous linear equations which can be solved by a high-speed digital computer. In this way, an explicit analytical expression for the induced current including the singularities from which the other field components can be derived, is obtained. Numerical results are shown for square and triangular cylinders of various sizes, for an incident plane wave polarized parallel to the axis of the cylinders.

An efficient approach to the seismogram synthesis for a basin structure using propagation invariants

1996 ◽  
Vol 86 (2) ◽  
pp. 379-388 ◽  
Author(s):  
H. Takenaka ◽  
M. Ohori ◽  
K. Koketsu ◽  
B. L. N. Kennett
Keyword(s):  
Integral Equation ◽  
Inverse Fourier Transform ◽  
Linear Equations ◽  
Computation Time ◽  
Reduction Technique ◽  
Final Solution ◽  
Space Domain ◽  
New Approach ◽  
Simultaneous Linear Equations ◽  
Single Integral

Abstract The Aki-Larner method is one of the cheapest methods for synthetic seismograms in irregularly layered media. In this article, we propose a new approach for a two-dimensional SH problem, solved originally by Aki and Larner (1970). This new approach is not only based on the Rayleigh ansatz used in the original Aki-Larner method but also uses further information on wave fields, i.e., the propagation invariants. We reduce two coupled integral equations formulated in the original Aki-Larner method to a single integral equation. Applying the trapezoidal rule for numerical integration and collocation matching, this integral equation is discretized to yield a set of simultaneous linear equations. Throughout the derivation of these linear equations, we do not assume the periodicity of the interface, unlike the original Aki-Larner method. But the final solution in the space domain implicitly includes it due to use of the same discretization of the horizontal wavenumber as the discrete wavenumber technique for the inverse Fourier transform from the wavenumber domain to the space domain. The scheme presented in this article is more efficient than the original Aki-Larner method. The computation time and memory required for our scheme are nearly half and one-fourth of those for the original Aki-Larner method. We demonstrate that the band-reduction technique, approximation by considering only coupling between nearby wavenumbers, can accelerate the efficiency of our scheme, although it may degrade the accuracy.

scholarly journals A method of accelerating stationary iterative methods for solving linear systems

1988 ◽  
Vol 30 (1) ◽  
pp. 1-23
Author(s):  
G. K. Robinson
Keyword(s):  
Quadratic Forms ◽  
Simple Technique ◽  
Linear Equations ◽  
Full Rank ◽  
Systems Of Equations ◽  
Linear Combinations ◽  
Simultaneous Linear Equations ◽  
Large Systems ◽  
Stationary Iterative Methods ◽  
Refined Version

AbstractThe speed of convergence of stationary iterative techniques for solving simultaneous linear equations may be increased by using a method similar to conjugate gradients but which does not require the stationary iterative technique to be symmetrisable. The method of refinement is to find linear combinations of iterates from a stationary technique which minimise a quadratic form. This basic method may be used in several ways to construct refined versions of the simple technique. In particular, quadratic forms of much less than full rank may be used. It is suggested that the method is likely to be competitive with other techniques when the number of linear equations is very large and little is known about the properties of the system of equations. A refined version of the Gauss-Seidel technique was found to converge satisfactorily for two large systems of equations arising in the estimation of genetic merit of dairy cattle.

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