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+#pragma once
+
+#include <cstddef>
+#include <vector>
+
+#include "matrix.hpp"
+#include "vector.hpp"
+
+namespace linalg {
+
+// Result of LU factorization with partial pivoting.
+//
+// The factorization satisfies PA = LU, where:
+//   P  is the permutation matrix encoded by `perm`
+//   L  is unit lower triangular  (L[i][i] == 1)
+//   U  is upper triangular
+//
+// `perm[i]` = index of the original row that ended up at position i.
+// Applying P to a vector b means: (Pb)[i] = b[perm[i]].
+//
+// `sign` is the sign of the permutation: +1 if an even number of row
+// swaps were made, -1 if odd.  Useful for computing det(A) = sign * prod(diag(U)).
+struct LUResult {
+    Matrix L;
+    Matrix U;
+    std::vector<std::size_t> perm;
+    int sign;
+};
+
+// Compute the LU factorization of A with partial pivoting.
+//
+// Throws DimensionMismatchError if A is not square.
+// Throws SingularMatrixError   if A is (numerically) singular, i.e. any
+//   pivot is smaller in magnitude than `singular_tolerance`.
+LUResult lu_factor(const Matrix& A, double singular_tolerance = 1e-12);
+
+// Solve Ax = b given a precomputed LU factorization.
+//
+// Applies the stored permutation, then forward / backward substitution.
+// Throws DimensionMismatchError if b.size() != lu.L.rows().
+Vector lu_solve(const LUResult& lu, const Vector& b);
+
+}  // namespace linalg