feat(solvers): implement Gaussian Elimination with Pivoting

- Implements forward elimination with partial pivoting for numerical stability. - Includes checks for singular or inconsistent systems. - Replaces SymPy back-substitution with a pure MATLAB vectorized loop. - Returns the solution vector x.

Author: msmrexe

+linearSolvers/gaussianElimination.m

@@ -0,0 +1,47 @@
+% +linearSolvers/gaussianElimination.m
+function x = gaussianElimination(A, b)
+% Solves Ax = b using Gaussian Elimination with partial pivoting
+% followed by back-substitution.
+
+ [n, ~] = size(A);
+ Ab = [A, b(:)]; % Augmented matrix
+
+ % Forward Elimination with Partial Pivoting
+ for i = 1:n-1
+ % Find pivot row
+ [~, maxRowIdx] = max(abs(Ab(i:n, i)));
+ maxRowIdx = maxRowIdx + i - 1; % Adjust index relative to full matrix
+
+ % Swap rows if necessary
+ if maxRowIdx ~= i
+ Ab([i, maxRowIdx], :) = Ab([maxRowIdx, i], :);
+ end
+
+ % Check for singular matrix (zero pivot after pivoting)
+ if abs(Ab(i, i)) < eps % Use epsilon for floating point comparison
+ error('Matrix is singular or nearly singular. Cannot solve using Gaussian Elimination.');
+ end
+
+ % Eliminate column i in rows below i
+ for j = i+1:n
+ factor = Ab(j, i) / Ab(i, i);
+ Ab(j, i:n+1) = Ab(j, i:n+1) - factor * Ab(i, i:n+1);
+ end
+ end
+
+ % Check for inconsistency in the last row
+ if abs(Ab(n, n)) < eps && abs(Ab(n, n+1)) > eps
+ error('System has no solution (inconsistent).');
+ elseif abs(Ab(n, n)) < eps && abs(Ab(n, n+1)) < eps
+ error('System has infinitely many solutions.'); % Or handle appropriately
+ end
+
+ % Back Substitution (Pure MATLAB, no SymPy)
+ x = zeros(n, 1);
+ x(n) = Ab(n, n+1) / Ab(n, n);
+ for i = n-1:-1:1
+ sum_ax = Ab(i, i+1:n) * x(i+1:n);
+ x(i) = (Ab(i, n+1) - sum_ax) / Ab(i, i);
+ end
+
+end