Question

matlab

1. Given the system of equations 9 + x2 +x3 +x4 = 75 xi +8x2 x3x54 X1+X1 +7X3 + X4 = 43 xi+x2 +x6x434 Write a code to find the solution of linear equations using a) Gauss elimination method b) Gauss-Seidel iterative method c) Jacobi's iterative method d) Compare the number of iterations required for b) and c) to the exact solution Assume an initial guess of the solution as (X1, X2, X3, X4) = (0,0,0,0).

Answer 1

Matlab code for solving clear all close all Matrix form of given linear equation b= [ 75:54:43,34]; displaying A and b matrix fprintf( Matrix A of given linear equation.In) disp (A) fprintf( 'Matrix b vector.\n disp(b) initial guess x0-0:0:0:01 %error convergence conv 10*-12; [x]-gauss_eliminationn (A,b) fprintf( "Solution Matrix for Gauss Elimination is x=\n ') disp(x) x,it]-Jacobi method (A,b, x0, conv); fprintfSolution Matrix for Jacobi method is x-n) disp(x) fprintf( 'Total number of iterations for Jacobi method is %d"\n' , it) [x, it]-causs-method (A, b , x0, conv) ; fprintf(nSolution Matrix for Gauss disp (x) Siedel is x-n' fprint f ("Total number of iterations for Gauss Siedel method is %d. n',it) %Function for Gauss Elimination function [x]-gauss_eliminationn(A, b) %A is the coefficient matrix and b 1s the result matrix for Ax-b is the solution matrix texample A-[15 6 8 11;6 65 3:8 5 7 6;11 3 6 91; b[40 20;26;29 so by x-gauss_eliminationn(A,b),we get x-1111]T below is the algorithm for Gaussian elimination A= [ A , b);8A įs sz_A-size (A); the matrix with A and b %Loop for Gaussian elimination matrix formation for A matrix for i-1: ss A-1 Al-A(i:end,i) b-find (A1--max (A1)); beb+ ( i-1 ) ; A2-A (b,:);

m-max (A1); for iri+1: ss A a#A(j,i)/m; end end Final A matrix after Gaussian elimination B-A; sz=size(A) ; x ( ss )=A( end, end) /A (end, end-1 ) ; %loop for backward substitution finding x for ii ss-1:-1:1 sum-0; for jj-iitl:ss of Gaussian elimination matrix for sum=sum+A(ii,jj)*x(jj); end end x=x ; end %Function for Jacobi method function [x,it ]-Jacobi method (A, b, x0,conv) Jacobi method ea-liit-0 while ea >=conv it-it+l; for i= 1:length(b) s-0; for j-l:length(b) if i-j s-s+A( i, j ) *x013 ) ; end end end if it-1 ea-10; else ea-norm(xl -x0)i end end end %Function for Gauss Siedel method

Eunction [x,it]-Gauss_method (A, b, x0,conv) Gauss Siedel method ea=1 ; it=0; while ea>=conv it=it+1 ; for i-1:length(b) s 0; for j-1:length (b) if i-j end if it-1 ea-10; else eanorm(x1 -x0)i end x-x0 end 돔88%옹옹옹옹옹옹옹옹옹옹%8888 End of Code %88888888錢888888응 Matrix A of given linear equation. Matrix b vector. 75 54 Solution Matrix for Gauss Elimination is x- Solution Matrix for Jacobi method is x= 7.00000000000009 5.0000000000001 4. 00000000000011 3. 00000000000012

Total number of iterations for Jacobi method is 35 Solution Matrix for Gauss Siedel is x- 4.99999999999999 Total number of iterations for Gauss Siedel method is 14. Published with MATLAB® R2018a

%%Matlab code for solving
clear all
close all

%Matrix form of given linear equation
A=[9 1 1 1; 1 8 1 1; 1 1 7 1; 1 1 1 6];
b=[75;54;43;34];

%displaying A and b matrix
fprintf('Matrix A of given linear equation.\n')
disp(A)
fprintf('Matrix b vector.\n')
disp(b)
%initial guess
x0=[0;0;0;0];

%error convergence
conv=10^-12;

[x]=gauss_eliminationn(A,b);
fprintf('Solution Matrix for Gauss Elimination is x=\n ')
disp(x)

[x,it]=Jacobi_method(A,b,x0,conv);
fprintf('Solution Matrix for Jacobi method is x=\n ')
disp(x)
fprintf('Total number of iterations for Jacobi method is %d.\n',it)

[x,it]=Gauss_method(A,b,x0,conv);
fprintf('\nSolution Matrix for Gauss Siedel is x=\n ')
disp(x)
fprintf('Total number of iterations for Gauss Siedel method is %d.\n',it)

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%Function for Gauss Elimination
function [x]=gauss_eliminationn(A,b)
%A is the coefficient matrix and b is the result matrix for Ax=b
%x is the solution matrix
%example A=[15 6 8 11;6 6 5 3;8 5 7 6;11 3 6 9]; b=[40;20;26;29];
%so by x=gauss_eliminationn(A,b), we get x=[1 1 1 1]T;
%below is the algorithm for Gaussian elimination
    A=[A,b];%A is the matrix with A and b
    sz_A=size(A);
    ss_A=sz_A(1,1);
    %Loop for Gaussian elimination matrix formation for A matrix
    for i=1:ss_A-1
        A1=A(i:end,i);
        b=find(A1==max(A1));
        b=b+(i-1);
        A2=A(b,:);
        A(b,:)=A(i,:);
        A(i,:)=A2;
        m=max(A1);
        for j=i+1:ss_A      
            a=A(j,i)/m;
            A(j,:)=A(j,:)-a*A(i,:);
        end
    end
    %Final A matrix after Gaussian elimination
    B=A;
    sz=size(A);
    ss=sz(1,1);
    x(ss)=A(end,end)/A(end,end-1);
    %loop for backward substitution of Gaussian elimination matrix for finding x
    for ii=ss-1:-1:1
        sum=0;
        for jj=ii+1:ss
            sum=sum+A(ii,jj)*x(jj);
        end
        x(ii)=(A(ii,end)-sum)/A(ii,ii);
    end
    x=x';
end

%Function for Jacobi method
function [x,it]=Jacobi_method(A,b,x0,conv)
    %Jacobi method
    ea=1;it=0;
    while ea>=conv
        it=it+1;
        for i=1:length(b)
            s=0;
            for j=1:length(b)
                if i~=j
                    s=s+A(i,j)*x0(j);
                end

            end
             x1(i)=(b(i)-s)/A(i,i);
        end

        if it==1
            ea=10;
        else
            ea = norm(x1 - x0);
        end
        x0=x1;
    end
    x=x0';
end

%Function for Gauss Siedel method
function [x,it]=Gauss_method(A,b,x0,conv)
    %Gauss Siedel method
    ea=1;it=0;
    while ea>=conv
        it=it+1;
        for i=1:length(b)
            s=0;
            for j=1:length(b)
                if i~=j
                    s=s+A(i,j)*x0(j);
                end

            end
             x0(i)=(b(i)-s)/A(i,i);
        end

        if it==1
            ea=10;
        else
            ea = norm(x1 - x0);
        end
        x1=x0;

    end
    x=x0;
end


%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%