To do this problem, you should first go through the code for the GaussianElimina

Question

To do this problem, you should first go through the code for the GaussianElimination function, and understand it. Then write a function that solves the system of equations for the case where the equation matrix is upper-triangular. The function header is given as: function x = SolveUpper (U,b) where the input U is an upper-triangular matrix, and b is a column vector; the output x is the solution to Ux = b. (Hint: if the system begins with a matrix U which is already in an upper-triangular form, we do not need to do elimination at all; hence only the back-substitute part from the GaussianElimination function is needed.)

This is GaussianElimination function:

function x=GaussianElimination(A,b)

[m,n]=size(A);
if m~=n
error('A matrix needs to be square');
end

Ab=[A b];

for i = 1:n-1
pivot = Ab(i,i);
for k = i+1:n   
ratio=Ab(k,i)/pivot;   
Ab(k,i:n+1) = Ab(k,i:n+1) - Ab(i,i:n+1)*ratio;
end
end

x = zeros(n,1); % preallocate memory for and initialize x
x(n) = Ab(n,n+1)/Ab(n,n);
for i=n-1:-1:1
x(i) = (Ab(i,n+1) - Ab(i,i+1:n)*x(i+1:n))/Ab(i,i);
end

Explanation / Answer

function x = GaussianElimination(A,b);
n = length(b);
x = zeros(n,1);
for k=1:n-1

% forward elimination
for i=k+1:n
x_mul = A(i,k)/A(k,k);
for j=k+1:n
A(i,j) = A(i,j)-x_mul*A(k,j);
end
b(i) = b(i)-x_mul*b(k);
end
end

% back substitution
x(n) = b(n)/A(n,n);
for i=n-1:-1:1
sum = b(i);
for j=i+1:n
sum = sum-A(i,j)*x(j);
end
x(i) = sum/A(i,i);
end

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