Question

Using MATLAB please.

Recall the formula for backward substitution to solve the upper triangular system ain 011 0 012 022 bi b. alan ... 0 0 ann be which is bi - Ej=i+1 Qi; I; i= 1,...n Write a MATLAB function backward_substitution that solves an upper triangular system. Test your code using the example T 6 A 2 0 0 1 1 0 2 3 b 9 1 15

Answer 1

Matlab code for the problem

function x = backward_substitution(A,b)

% Function to solve the upper triangular system Ax = b

n = length(b); % size of the system

x = zeros(n,1); % initilizing the x vector

for i = n:-1:1 % for loop to compute x_i

sum = 0; % variable to store the sum i+1 to n

for j = i+1:n % fo loop to compute the sum Aij*xj

sum = sum + A(i,j)*x(j); % computing the sum

end

x(i) = (b(i)-sum)/A(i,i); % computing the ith x value

end % end of for loop

end % end of function

Screen Print of the code

X 2 backward_substitution.m + function x = backward_substitution(A,b) % Function to solve the upper triangular system Ax = b n = length(b); % size of the system x = zeros(n,1); % initilizing the x vector for i = n:-1:1 % for loop to compute x_i sum = 0; % variable to store the sum i+1 ton for j = i+1:n % fo loop to compute the sum Aij*xj sum = sum + Ali,j)*x(); % computing the sum end x(i) = (b(i)-sum)/A(i,i); % computing the ith x value end % end of for loop end % end of function 13 10 11 12 -

Screen Print of the output

COMMAND WINDOW >> A = [ 2, 0, 0; 1, 1, 0; 1, 2, 3]' % the A matrix given A = 2 1 1 0 1 2 3 b = [6; 7; 15] % given b vector b = 6 7 15 >> % calling the function to compute the solution >> x = backward_substitution(A,b) X = 2 -3 5 >> % validating the function using the matlab solver >> Alb ans = 2 -3 5 >> % the result is validated