Convergence of Gauss-Seidel Method\302\2512006 Kevin Martin, Autar Kaw, Jamie Trahan University of South FloridaUnited States of Americakaw@eng.usf.eduNOTE: This worksheet demonstrates the convergence of Gauss-Seidel method, an iterative technique used in solving a system of simultaneous linear equations.restart; IntroductionGauss-Seidel method is an advantageous approach to solving a system of simultaneous linear equations because it allows the user to control round-off error that is inherent in elimination methods such as Gaussian Elimination. However, this method is not without its pitfalls. Gauss-Seidel method is an iterative technique whose solution may or may not converge. Convergence is ensured only if the coefficient matrix, [A]nxn, is diagonally dominant, otherwise the method may or may not converge. A diagonally dominant square matrix [A] is defined by the following: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print();for all i, 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();for at least one i.Fortunately, many physical systems that result in simultaneous linear equations have diagonally dominant coefficient matrices, or with the exchange of a few equations, the coefficient matrix can become diagonally dominant. To learn more about diagonally dominant matrices as well as how to perform the Gauss-Seidel method, click here.The following simulation illustrates the convergence of the Gauss-Seidel method. 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Section 1: Input DataThe following are the input parameters for the simulation. The user may interact with the simulation by changing values only in this section. Once entered, Maple will produce plots that demonstrate the convergence of each solution Xi as a function of the iteration number.Input parameters:n = number of equations[A] = nxn coefficient matrix[RHS] = nx1 right hand side array[X] = nx1 initial guess of the solution vectormaxit = maximum number of iterationsrestart;n:=4;
A:=Matrix([[12,7,3,1],[1,5,1,2],[2,7,-11,1],[9,2,1,13]]);
RHS:=[22,7,-2,3];
X:=[1,2,1,1];
maxit:=8;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Section 2: Gauss-Seidel ProcedureGauss-Seidel method utilizes the 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print();to compute an approximate value for a solution vector [X]. The following procedure uses Gauss-Seidel method to calculate the value of the solution for the above system of equations using maxit iterations. It will then store each approximate solution, Xi, from each iteration in a matrix with maxit columns. Thereafter, Maple will plot the solutions as a function of the iteration number.Parameter names:A = nxn coefficient matrixRHS = nx1 right hand side arrayn = number of equationsXinitial = nx1 initial guess solution vectormaxit = maximum number of iterationsgauss_seidel:=proc(A,RHS,n,Xinitial,maxit)
local Xold,Xnew,Xstore,Xprev,i,j,k,summ,epsa,epsmax;
#epsa is the array that stores the absolute relative approximate error at the end of each iteration.
epsa:=Array(1..n);
#Xnew is the solution vector after each iteration is conducted.
Xnew:=Array(1..n);
#epsmax is the greatest relative approximate error of all values in the solution vector that are generated in the given iteration.
epsmax:=Array(1..maxit);
#Xstore is a matrix that stores the solution vector after each iteration.
Xstore:=Matrix(1..n,1..maxit);
#Defining the initial guess values of the solution vector.
Xprev:=Xinitial:
#conducting maxit iterations.
for k from 1 by 1 to maxit do
epsmax[k]:=0.0;
for i from 1 by 1 to n do
#Initializing the series sum to zero.
summ:=0.0;
for j from 1 by 1 to n do
#Only adding i<>j terms.
if (i<>j) then
#Generating the summation term in Equation (3.1).
summ:=summ+A[i,j]*Xprev[j];
end if:
end do:
#Using Equation (3.1) to calculate the new [X] solution vector.
Xnew[i]:=(RHS[i]-summ)/A[i,i];
#Calculating the absolute relative approximate error.
epsa[i]:=abs((Xnew[i]-Xprev[i])/Xnew[i])*100.0;
#Finding the maximum epsa value.
if epsmax[k]<=epsa[i] then
epsmax[k]:=epsa[i];
end if;
#Updating the previous guess.
Xprev[i]:=Xnew[i];
#Storing each value of X for each iteration.
Xstore[i,k]:=Xnew[i]
end do;
end do;
return(Xstore,epsmax);
end proc:Section 3: Resultssoln:=gauss_seidel(A,RHS,n,X,maxit):The following matrix stores the value of the solution for Xi in the ith row after each given iteration.Xstore:=soln[1];The following matrix stores the maximum absolute relative approximate error after each given iteration.epsmax:=soln[2];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Section 4: Convergence GraphsThe following graphs plot the value of Xi as a function of the iteration number to demonstrate the convergence of each solution.with(plots):for i from 1 by 1 to n do
ttl:=cat(`Value of X`,i,` as a function of the iteration number`):
lbl:=cat("X",i):
data:=[seq([k,Xstore[i,k]],k=1..maxit)];
pointplot(data,connect=true,color=blue,axes=boxed,title=ttl,axes=BOXED,labels=["iteration number",lbl],thickness=3);
end do;The following graph plots the maximum absolute relative approximate error as a function of the iteration number.data:=[seq([k,epsmax[k]],k=1..maxit)];
pointplot(data,connect=true,color=green,axes=boxed,title="Value of maximum absolute relative approximate error as a function of iteration number",titlefont=[HELVETICA,28], axes=BOXED,labels=["iteration number","epsmax"],thickness=2);
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References[1] Autar Kaw, Holistic Numerical Methods Institute, http://numericalmethods.eng.usf.edu/mws, SeeHow does Gauss-Seidel method work?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ConclusionMaple helped us to study the convergence of Gauss-Seidel Method.Question: Solve a set of equations for which the coefficient matrix is not diagonally dominant. For example, 5x + 6y + 7z = 18 6x + 3y + 9z = 18 7x + 9y + 10z = 26Choose an initial solution vector guess of [2, 5, 7]. Does the solution converge? Now choose [0.99, 0.995, 0.997] as the initial guess of the solution vector. Does the solution converge now?Legal Notice: The copyright for this application is owned by the author(s). Neither Maplesoft nor the author are responsible for any errors contained within and are not liable for any damages resulting from the use of this material. This application is intended for non-commercial, non-profit use only. Contact the author for permission if you wish to use this application in for-profit activities.