Question

just 1,2,4

Problem 1 Consider the linear system of equations Ax = b, where x € R4X1, and A= 120 b = and h= 0.1. [2+d -1 0 0 1 1 -1 2+d -1 0 h2 0 -1 2 + 1 Lo 0 -1 2+d] 1. Is the above matrix diagonally dominant? Why 2. Use hand calculations to solve the linear system Ax = b with d=1 with the following methods: (a) Gaussian elimination. (b) LU decomposition. Use MATLAB (L, U] = lu (A) to compute the LU decomposition (c) Jacobi iterations. Show the first three iterations with the initial guess of x(0) 21-12-27 3. Write a MATLAB program to solve the linear system Ax = b with d=1 with the following methods: (a) Jacobi iterations. (b) Gauss-Seidel iterations. Use the convergence criterion of ||r) || < 10-6, where r(k) = Ax(k) – b is the residual for the kth iteration and r(k) || = (r+r)1/2 is the norm of residual. Plot the convergence history (residual versus iterations) of both methods in one figure and legend accordingly. Use semilogy for your plots. Explain your results. 4. Repeat part 2 with d= -1. Explain your results.

Answer 1

%&Matlab code for Gauss Siedel Method and Jacobi method clear all close all Loop for table for n iterations -1 2+d]; fprintf('Table for Jacobi and Gauss iterations\n\n') $creating A matrix d=1; h=0.1; A=(1./(h*2)).*[2+d -1 0 0; -1 2+d -1 0; 0 -1 2+d -1; 0 0 fprintf('A matrix is for d=1 and h=0.1\n') disp (vpa (A,3)) &creating B vector b=[1;0;0;0]; Gaussian Elimination [xl, B]=gauss_eliminationn(A,b); fprintf('Solution using Jacobi method is\n') disp(xl') XLU Decomposition of A [L,U]=lu(A); fprintf('LU Decomposition of A\n') fprintf('L matrix is \n') disp (vpa (L,3)) fprintf('U matrix is \n') disp (vpa (U,3)) Xinitial x and convergence x0=h2.*[1;-1;2; -2]; conv=10^-6; Jacobi method [x_j,it_j]=Jacobi_method(A,b,x0, conv); fprintf('Total iteration is $d and Solution using Jacobi method is \n', it_j) disp(x_j) Gauss Siedel method [x_g, it_9]=Gauss_method(A,b, x0, conv); fprintf('Total iteration is td and Solution using Gauss method is \n', it g) disp(x_g') texact solution x_ext=A\b; fprintf('Exact solution is\n') disp(x_ext) -1 2+d); &creating A matrix d=-1; h=0.1; A=(1./(h*2)). * [2+d -1 0 0; -1 2+d -1 0; 0 -1 2+d -1; 0 0 fprintf( 'A matrix is for d=1 and h=0.1\n') disp (vpa (A,3)) creating B vector b=[1;0;0;0];

Binitial x and convergence x0=h 2 .*[1;-1;2;-2); conv=10^-6; Jacobi method [x_j, it_j]=Jacobi_method(A,b, x0, conv); fprintf('Total iteration is td and Solution using Jacobi method is \n', it j) disp(x_j) Gauss Siedel method (x_9, it_g]=Gauss_method(A,b, x0, conv); fprintf('Total iteration is $d and Solution using Gauss method is \n', it g) disp(x_g') texact solution x_ext=A\b; fprintf('Exact solution is\n') disp(x_ext) Function for Jacobi method function [x, it]=Jacobi_method (A,b, x0, conv) Jacobi method ea=1;it=0;x1=zeros(length(b),1); 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()); end end x2(i)=(b(i)-s)/Ali, i); end x0=x2; if it==1000 break end if it==1 ea=10; else ea = norm(A*x0' - b, 'inf'); end xl=x0; end x=x0'; end Function for Gauss Siedel method function (x,it] =Gauss_method(A,b,x0,conv) Jacobi method ea=1;it=0;xl-zeros(length(b),1); 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()); end end x0(i)=(b(i)-s)/Ali, i); end if it==1000 break end if it==1 ea=10; else ea = norm(A*x0 - b, 'inf'); end xl=x0; end x=x0'; end 88Matlab function for Gaussian elimination for solving tridiagonal matrix function (x,B]=gauss_eliminationn(A,b) PA is the coefficient matrix and b is the result matrix for Ax=b 8x is the solution matrix texample 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 Al=A( i:end, i); b=find (Al==max(A1)); b=b+ (1-1); A2-Ab,:); Alb, :)=A(1,:); Ali, :)=A2; m=max(Al); for j=i+1:ss_A a=A j , i)/m; A(), :)=A(), :)-a+Ali,:); end end Final A matrix after Gaussian elimination B=A; sz=size(A); ss=S2(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)/Alii,ii); end end $888888888888888888888 End of Code $8888888888888888888888 Table for Jacobi and Gauss iterations A matrix is for d=1 and h=0.1 [ 300.0, -100.0, 0, 01 [ -100.0, 300.0, -100.0, [ 0, -100.0, 300.0, -100.0) 0, -100.0, 300.0] 0] Solution using Jacobi method is 0.00381818181818182 0.00145454545454545 0.000545454545454546 0.000181818181818182 LU Decomposition of A L matrix is [ 1.0, 0, 0, (-0.333, 1.0, 0, 0, -0.375, 1.0, 0, -0.381, 0] 0] 0] 1.0) U matrix is [ 300.0, -100.0, 0, 0] o, 267.0, -100.0, 0] 0, 0, 262.0, -100.0) 0, 262.0] Total iteration is 26 and Solution using Jacobi method is 0.00381818292974188 0.00145454360558462 0.000545456343996511 0.00018181703909 7542 Total iteration is 13 and Solution using Gauss method is 0.00381817960 796939 0.0014545435 25 7417 0.000545453505164532 0.00018181 7835054844 Exact solution is 0.00381818181818182 0.00145454545454546

0.000545454545454546 0.000181818181818182 A matrix is for d=1 and h=0.1 [ 100.0, -100.0, 0, 01 [ -100.0, 100.0, -100.0, 01 0, -100.0, 100.0, -100.0] 0, 0, -100.0, 100.0] Total iteration is 1000 and Solution using Jacobi method is 1.3039967306081 2e+207 -1.1379692539836e+207 2.10991103134268e+207 -7.03303677114229e +206 Total iteration is 738 and Solution using Gauss method is 4.2638944745576e+305 1.11630206588 34 7e +306 Inf Inf Exact solution is 0.01 -0.01 -0.01 Published with MATLAB® R2018a
%creating A matrix
    d=1; h=0.1;
    A=(1./(h^2)).*[2+d -1 0 0; -1 2+d -1 0; 0 -1 2+d -1; 0 0 -1 2+d];
    fprintf('A matrix is for d=1 and h=0.1\n')
    disp(vpa(A,3))
    %creating B vector
    b=[1;0;0;0];
  
    %Gaussian Elimination
    [x1,B]=gauss_eliminationn(A,b);
    fprintf('Solution using Jacobi method is\n')
    disp(x1')
  
    %LU Decomposition of A
    [L,U]=lu(A);
    fprintf('LU Decomposition of A\n')
    fprintf('L matrix is \n')
    disp(vpa(L,3))
    fprintf('U matrix is \n')
    disp(vpa(U,3))
    %initial x and convergence
    x0=h^2.*[1;-1;2;-2];
    conv=10^-6;
    %Jacobi method
    [x_j,it_j]=Jacobi_method(A,b,x0,conv);
    fprintf('Total iteration is %d and Solution using Jacobi method is\n',it_j)
    disp(x_j)
    %Gauss Siedel method
    [x_g,it_g]=Gauss_method(A,b,x0,conv);
    fprintf('Total iteration is %d and Solution using Gauss method is\n',it_g)
    disp(x_g')
    %exact solution
    x_ext=A\b;
    fprintf('Exact solution is\n')
    disp(x_ext)