8. (Back and Forward Substitution: Matlab Program) Write two programs, one that

Question

8. (Back and Forward Substitution: Matlab Program) Write two programs, one that performs back substitution on an upper triangular matrix and another that performs forward substitution on a lower triangular matrix (you may assume that the diagonal entries are all 1). Both files should begin function [x] forwardsub(L, b) n-length(b); (your code here) b and A is lower triangular. Test your code on the following In the above, Lx systems: 1 0 0 1 2-11 0 0 2 Remember, in Matlab you can solve matrix equations as follows (assuming you have defined the matrix A and the rhs vector b): >>Alb Print and hand-in the text file containing your program.

Explanation / Answer

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Save the following code as backwardSubstitution

function ans1 = backSubstitution(Upper,b)

n = size(Upper,1);

ans1 = zeros(n,1);

for j=n:-1:1

if (Upper(j,j)==0)

error('Matrix is singular!');

end;

ans1(j) = b(j)/Upper(j,j);

b(1:j-1) = b(1:j-1)-Upper(1:j-1,j)*ans1(j);

end

Save the following code as forwardSubstitution.m

function ans1 = forwardSubstitution(Lower,b)

n = size(Lower,1);

ans1 = zeros(n,1);

for j=1:n

if (Lower(j,j)==0)

error('Matrix is singular!');

end;

ans1(j) = b(j)/Lower(j,j);

b(j+1:n) = b(j+1:n)-Lower(j+1:n,j)*ans1(j);

end

Save it as test.m

clear all;

lower_mat_A = [[1, 0, 0]; [2, 1, 0] ;[3, 4, 1]];

lower_mat_B = [[-1]; [0]; [1]];

disp('Back Substitution :');

disp(backSubstitution(lower_mat_A, lower_mat_B));

upper_mat_A = [[1, 2, -1]; [0, 3, -1] ;[0, 0, 2]];

upper_mat_B = [[-1]; [0]; [1]];

disp('Forward Substitution :');

disp(forwardSubstitution(upper_mat_A,upper_mat_B));

Output :

Back Substitution :
-1
0
1

Forward Substitution :
-1.0000
0
0.5000

Thank you. Feel free to ask anything. Please upvote, if you like the answer. Thank you again.

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