Question

Write a MATLAB function function = pivGauss(.....) to solve linear equations using Gaussian Elimination with Partial Pivoting.

You'll need to employ Nested Loops.

Thank you !

Answer 1

The algorithm to solves a linear system Ax=b using the Gaussian elimination method with Partial pivoting is as following :

1) Initialize a permutation vector r = [1, 2,...,n] where r(i) corresponds to row i in A.
2) For k = 1,...,n-1 find the largest (in absolute value) element among a(r(k),k),a(r(k+1),k),...,a(r(n),k).
3) Assume r(j,k) is the largest element. Switch r(j) and r(k).
4) For i=1,...,k-1,k+1,...,n calculates:
zeta = a(r(i),k) / a(r(k),k)
5) For j=k,...,n calculate:
a(r(i),j)=a(r(i),j)-a(r(k),j)*zeta
b(r(i)) = b(r(i))-b(r(k))*zeta
6) Steps 1 through 6 has effectively diagonalized A.
7) Each element in the solution vector is:
x(r(i)) = b(i)/a(i,i);

MATLAB code :-

function x = GAUSS_ELIM(A, b)

%% Create permutation vector

n = size(A, 1); % Size of input matrix

r = zeros(n, 1); % Initialize permutation vector

for i = 1 : 1 : n

r(i) = i;

end

%% Apply Gaussian elimination and rearrange permutation vector

x = zeros(n, 1); % Initialize solution vector

for k = 1 : 1 : n % Go through each element in permutation vector

% Compare each element in r(k)th column for the max

max = abs(A(r(k), r(k)));

max_pos = k;

for l = k : 1 : n

if abs(A(r(l), r(k))) > max

max = abs(A(r(l), r(k)));

max_pos = l;

end

end

% Switch the kth r-vector element with max r-vector element

temp_r = r;

r(k) = temp_r(max_pos);

r(max_pos) = temp_r(k);

% Eliminate A-vector elements in r(k)th column below r(k)th row

for i = 1 : 1 : n

if i ~= k

zeta = A(r(i), k) / A(r(k), k);

for j = k : 1 : n

A(r(i), j) = A(r(i), j) - A(r(k), j) * zeta;

end

b(r(i)) = b(r(i)) - b(r(k)) * zeta;

end

end

end

% Compute the solution frpm the diagonalized A-matrix

for i = 1 : 1 : n

x(i) = b(r(i)) / A(r(i), i);

end

end