Gaussian elimination
Gaussian elimination is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
Problem: Solve Ax=b using Gaussian elimination then backwards substitution. A being an n by n matrix. Also, x and b are n by 1 vectors. To improve accuracy, please use partial pivoting and scaling.
MATLAB
<lang MATLAB> function [ x ] = GaussElim( A, b)
% Ensures A is n by n sz = size(A); if sz(1)~=sz(2)
fprintf('A is not n by n\n'); clear x; return;
end
n = sz(1);
% Ensures b is n x 1. if n~=sz(1)
fprintf('b is not 1 by n.\n'); return
end
x = zeros(n,1); aug = [A b]; tempmatrix = aug;
for i=2:sz(1)
% Find maximum of row and divide by the maximum tempmatrix(1,:) = tempmatrix(1,:)/max(tempmatrix(1,:)); % Finds the maximum in column temp = find(abs(tempmatrix) - max(abs(tempmatrix(:,1)))); if length(temp)>2 for j=1:length(temp)-1 if j~=temp(j) maxi = j; %maxi = column number of maximum break; end end else % length(temp)==2 maxi=1; end % Row swap if maxi is not 1 if maxi~=1 temp = tempmatrix(maxi,:); tempmatrix(maxi,:) = tempmatrix(1,:); tempmatrix(1,:) = temp; end % Row reducing for j=2:length(tempmatrix)-1 tempmatrix(j,:) = tempmatrix(j,:)-tempmatrix(j,1)/tempmatrix(1,1)*tempmatrix(1,:); if tempmatrix(j,j)==0 || isnan(tempmatrix(j,j)) || abs(tempmatrix(j,j))==Inf fprintf('Error: Matrix is singular.\n'); clear x; return end end aug(i-1:end,i-1:end) = tempmatrix; % Decrease matrix size tempmatrix = tempmatrix(2:end,2:end);
end
% Backwards Substitution x(end) = aug(end,end)/aug(end,end-1); for i=n-1:-1:1
x(i) = (aug(i,end)-dot(aug(i,1:end-1),x))/aug(i,i);
end
end </lang>