Solution for Ex 1. Write the function 1

(1)

Solution for Ex 1. Write the function 1

^function [L, U] = LUnoPiv(A)

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% LU factorization without pivoting

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n = size(A,1); % size of A

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L = eye(n); % initialize

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^for ^{k = 1:n}

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1

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^for ^{i = k+1:n}

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% multipliers for gaussian elimination

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L(i,k) = A(i,k)/A(k,k);

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^for ^{j = k:n}

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% construct A

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^{A(i,j) =} ^A(i,j)

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L(i,k)*A(k,j);

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^end

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^end

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^end

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^{U = A;}

and then the following script 1

^{clear all}

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^{close all}

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^{% clc}

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epsilon = 10^(

−

15);

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^{% Define A}

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A = [13 13 0; 2 2

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epsilon 4; 1 3 17];

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^{% define b}

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xVera = [2;5;7]; b = A*xVera;

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% compute the solution

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[L, U] = LUnoPiv(A);

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y = L\b; xLUnoPiv = U\y;

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% compute the solution with pivoting

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[L, U, P] = lu(A);

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y = L\(P*b); xLUpermut = U\y;

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% compute the errors

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format short e

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ELUgauss = norm(xVera

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xLUnoPiv)/norm(xVera)

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ELUpermut = norm(xVera

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xLUpermut)/norm(xVera)

1

(2)

Solution for Ex 2. At rst write the function 1

^function [x,iter,err]=Jacobi(A,b,x0,tol,kmax)

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% Jacobi's iterative method for the solution of Ax=b

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% with tolerance 'tol' and maximum number of iteration

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^{% 'kmax'}

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D=diag(diag(A)); DI=diag(1./diag(A));

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^J=

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_DI*(A

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D);

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disp(['Spectral radius of the iteration matrix = ',...

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num2str(max(abs(eig(J))))]);

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b1=DI*b; x1=J*x0+b1; k=1;

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while(norm(x1

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x0)>tol*norm(x1) && k<=kmax)

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^x0=x1;

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x1=J*x0+b1;

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err(k)=norm(x1

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x0,inf);

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^k=k+1;

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^end

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x=x1; iter=k

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1;

Then, the following script 1

% clear all

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% close all

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% define the problem

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v = [4 1 zeros(1,4)];

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A = toeplitz(v);

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b = A*2*ones(6,1);

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% use Jacobi

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[x,iter1,err1]=Jacobi(A,b,zeros(6,1),1.e

−

10,200);

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 Solution for Ex 1. Write the function 1