Abstract: In this paper, efforts were made to examine and compare the algorithmic iterative solutions of conjugate gradient method as against other methods such as Gauss-Seidel and Jacobi approaches for solving systems of linear equations of the form Ax = b, where A is a real n x n symmetric and positive definite matrix. We performed algorithmic iterative steps and obtained analytical solutions of a typical 3 x 3 symmetric and positive definite matrix using the three methods described in this paper (Gauss-Seidel, Jacobi and Conjugate Gradient methods) respectively. From the results obtained, we discovered that the Conjugate Gradient method converges faster to exact solutions in fewer iterative steps than the two other methods which took much iteration, much time and kept tending to the exact solutions.
Abstract: This paper presents a comparative study of the Gauss Seidel and Newton-Raphson polar coordinates methods for power flow analysis. The effectiveness of these methods are evaluated and tested through a different IEEE bus test system on the basis of number of iteration, computational time, tolerance value and convergence.
Abstract: In an attempt to investigate the performance of single
basin solar still for climate conditions of Ludhiana a single basin
solar still was designed, fabricated and tested. The energy balance
equations for various parts of the still are solved by Gauss-Seidel
iteration method. Computer model was made and experimentally
validated. The validated computer model was used to estimate the
annual distillation yield and performance ratio of the still for
Ludhiana. The Theoretical and experimental distillation yield were
4318.79 ml and 3850 ml respectively for the typical day. The
predicted distillation yield was 12.5% higher than the experimental
yield. The annual distillation yield per square metre aperture area and
annual performance ratio for single basin solar still is 1095 litres and
0.43 respectively. The payback period for micro-stepped solar still is
2.5 years.
Abstract: Analysis of real life problems often results in linear
systems of equations for which solutions are sought. The method to
employ depends, to some extent, on the properties of the coefficient
matrix. It is not always feasible to solve linear systems of equations
by direct methods, as such the need to use an iterative method
becomes imperative. Before an iterative method can be employed
to solve a linear system of equations there must be a guaranty that
the process of solution will converge. This guaranty, which must
be determined apriori, involve the use of some criterion expressible
in terms of the entries of the coefficient matrix. It is, therefore,
logical that the convergence criterion should depend implicitly on the
algebraic structure of such a method. However, in deference to this
view is the practice of conducting convergence analysis for Gauss-
Seidel iteration on a criterion formulated based on the algebraic
structure of Jacobi iteration. To remedy this anomaly, the Gauss-
Seidel iteration was studied for its algebraic structure and contrary
to the usual assumption, it was discovered that some property of the
iteration matrix of Gauss-Seidel method is only diagonally dominant
in its first row while the other rows do not satisfy diagonal dominance.
With the aid of this structure we herein fashion out an improved
version of Gauss-Seidel iteration with the prospect of enhancing
convergence and robustness of the method. A numerical section is
included to demonstrate the validity of the theoretical results obtained
for the improved Gauss-Seidel method.
Abstract: In this paper, the modified Gauss-Seidel method with the new preconditioner for solving the linear system Ax = b, where A is a nonsingular M-matrix with unit diagonal, is considered. The convergence property and the comparison theorems of the proposed method are established. Two examples are given to show the efficiency and effectiveness of the modified Gauss-Seidel method
with the presented new preconditioner.
Abstract: We study the semiconvergence of Gauss-Seidel iterative
methods for the least squares solution of minimal norm of rank
deficient linear systems of equations. Necessary and sufficient conditions
for the semiconvergence of the Gauss-Seidel iterative method
are given. We also show that if the linear system of equations is
consistent, then the proposed methods with a zero vector as an initial
guess converge in one iteration. Some numerical results are given to
illustrate the theoretical results.
Abstract: A generalized Dirichlet to Neumann map is
one of the main aspects characterizing a recently introduced
method for analyzing linear elliptic PDEs, through which it
became possible to couple known and unknown components
of the solution on the boundary of the domain without
solving on its interior. For its numerical solution, a well conditioned
quadratically convergent sine-Collocation method
was developed, which yielded a linear system of equations
with the diagonal blocks of its associated coefficient matrix
being point diagonal. This structural property, among others,
initiated interest for the employment of iterative methods for
its solution. In this work we present a conclusive numerical
study for the behavior of classical (Jacobi and Gauss-Seidel)
and Krylov subspace (GMRES and Bi-CGSTAB) iterative
methods when they are applied for the solution of the Dirichlet
to Neumann map associated with the Laplace-s equation
on regular polygons with the same boundary conditions on
all edges.
Abstract: A numerical analysis used to simulate the effects of wavy surfaces and thermal radiation on natural convection heat transfer boundary layer flow over an inclined wavy plate has been investigated. A simple coordinate transformation is employed to transform the complex wavy surface into a flat plate. The boundary layer equations and the boundary conditions are discretized by the finite difference scheme and solved numerically using the Gauss-Seidel algorithm with relaxation coefficient. Effects of the wavy geometry, the inclination angle of the wavy plate and the thermal radiation on the velocity profiles, temperature profiles and the local Nusselt number are presented and discussed in detail.
Abstract: In this paper, a nonlinear acoustic echo cancellation
(AEC) system is proposed, whereby 3rd order Volterra filtering is
utilized along with a variable step-size Gauss-Seidel pseudo affine
projection (VSSGS-PAP) algorithm. In particular, the proposed
nonlinear AEC system is developed by considering a double-talk
situation with near-end signal variation. Simulation results
demonstrate that the proposed approach yields better nonlinear AEC
performance than conventional approaches.
Abstract: The objective of this paper is to analyse the
application of the Half-Sweep Gauss-Seidel (HSGS) method by using
the Half-sweep approximation equation based on central difference
(CD) and repeated trapezoidal (RT) formulas to solve linear fredholm
integro-differential equations of first order. The formulation and
implementation of the Full-Sweep Gauss-Seidel (FSGS) and Half-
Sweep Gauss-Seidel (HSGS) methods are also presented. The HSGS
method has been shown to rapid compared to the FSGS methods.
Some numerical tests were illustrated to show that the HSGS method
is superior to the FSGS method.
Abstract: In this paper, a new pseudo affine projection (AP)
algorithm based on Gauss-Seidel (GS) iterations is proposed for
acoustic echo cancellation (AEC). It is shown that the algorithm is
robust against near-end signal variations (including double-talk).
Abstract: The temperature distribution and the heat transfer
rates through a multi-layer door of a furnace were investigated. The
inside of the door was in contact with hot air and the other side of the
door was in contact with room air. Radiation heat transfer from the
walls of the furnace to the door and the door to the surrounding area
was included in the problem. This work is a two dimensional steady
state problem. The Churchill and Chu correlation was used to find
local convection heat transfer coefficients at the surfaces of the
furnace door. The thermophysical properties of air were the functions
of the temperatures. Polynomial curve fitting for the fluid properties
were carried out. Finite difference method was used to discretize for
conduction heat transfer within the furnace door. The Gauss-Seidel
Iteration was employed to compute the temperature distribution in
the door.
The temperature distribution in the horizontal mid plane of the
furnace door in a two dimensional problem agrees with the one
dimensional problem. The local convection heat transfer coefficients
at the inside and outside surfaces of the furnace door are exhibited.