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#include "lu.hpp"
#include "matrix.hpp"
#include "norms.hpp"
#include "triangular_solve.hpp"
#include "vector.hpp"
#include <cmath>
#include <cstddef>
#include <iomanip>
#include <iostream>
#include <optional>
#include <random>
#include <string>
#include <vector>
using linalg::Matrix;
using linalg::Vector;
// --- Local no-pivot LU for comparison only. ---
struct NoPivotLU {
Matrix L;
Matrix U;
bool failed = false;
std::size_t fail_step = 0;
};
NoPivotLU lu_no_pivot(const Matrix& A, double tol = 1e-14) {
const std::size_t n = A.rows();
Matrix work = A;
Matrix L = Matrix::zeros(n, n);
for (std::size_t i = 0; i < n; ++i) L(i, i) = 1.0;
Matrix U = Matrix::zeros(n, n);
for (std::size_t k = 0; k < n; ++k) {
if (std::abs(work(k, k)) <= tol) {
return NoPivotLU{std::move(L), std::move(U), true, k};
}
for (std::size_t j = k; j < n; ++j) U(k, j) = work(k, j);
for (std::size_t i = k + 1; i < n; ++i) {
L(i, k) = work(i, k) / work(k, k);
for (std::size_t j = k + 1; j < n; ++j) {
work(i, j) -= L(i, k) * work(k, j);
}
}
}
return NoPivotLU{std::move(L), std::move(U), false, 0};
}
std::optional<Vector> solve_no_pivot(const NoPivotLU& f, const Vector& b) {
if (f.failed) return std::nullopt;
try {
const Vector y = linalg::forward_substitution(f.L, b, 1e-14, /*unit_diagonal=*/true);
return linalg::backward_substitution(f.U, y);
} catch (...) {
return std::nullopt;
}
}
// --- Metrics ---
double solve_residual(const Matrix& A, const Vector& x, const Vector& b) {
return linalg::norm2(A * x - b);
}
double reconstruction_error(const Matrix& A, const linalg::LUResult& lu) {
const std::size_t n = A.rows();
Matrix PA(n, n);
for (std::size_t i = 0; i < n; ++i)
for (std::size_t j = 0; j < n; ++j)
PA(i, j) = A(lu.perm[i], j);
const Matrix LU_prod = lu.L * lu.U;
double err = 0.0;
for (std::size_t i = 0; i < n; ++i)
for (std::size_t j = 0; j < n; ++j) {
const double d = PA(i, j) - LU_prod(i, j);
err += d * d;
}
return std::sqrt(err);
}
// --- Reporting ---
void print_header(const std::string& title) {
std::cout << "\n" << std::string(60, '=') << "\n";
std::cout << " " << title << "\n";
std::cout << std::string(60, '=') << "\n";
std::cout << std::left
<< std::setw(22) << "Method"
<< std::setw(20) << "||Ax - b||"
<< std::setw(20) << "||PA - LU||"
<< "\n";
std::cout << std::string(60, '-') << "\n";
}
void report_pivoted(const Matrix& A, const Vector& b) {
try {
const linalg::LUResult lu = linalg::lu_factor(A);
const Vector x = linalg::lu_solve(lu, b);
std::cout << std::left << std::setw(22) << "Pivoted LU"
<< std::setw(20) << std::scientific << std::setprecision(3)
<< solve_residual(A, x, b)
<< std::setw(20) << reconstruction_error(A, lu)
<< "\n";
} catch (const std::exception& e) {
std::cout << std::left << std::setw(22) << "Pivoted LU"
<< "FAILED: " << e.what() << "\n";
}
}
void report_no_pivot(const Matrix& A, const Vector& b) {
const NoPivotLU f = lu_no_pivot(A);
if (f.failed) {
std::cout << std::left << std::setw(22) << "No-pivot LU"
<< "FAILED at step " << f.fail_step << " (zero pivot)\n";
return;
}
const auto x_opt = solve_no_pivot(f, b);
if (!x_opt) {
std::cout << std::left << std::setw(22) << "No-pivot LU"
<< "FAILED during solve (singular U)\n";
return;
}
const Matrix LU_prod = f.L * f.U;
double rec_err = 0.0;
for (std::size_t i = 0; i < A.rows(); ++i)
for (std::size_t j = 0; j < A.cols(); ++j) {
const double d = A(i, j) - LU_prod(i, j);
rec_err += d * d;
}
rec_err = std::sqrt(rec_err);
std::cout << std::left << std::setw(22) << "No-pivot LU"
<< std::setw(20) << std::scientific << std::setprecision(3)
<< solve_residual(A, *x_opt, b)
<< std::setw(20) << rec_err
<< "\n";
}
void run_case(const std::string& label, const Matrix& A, const Vector& b) {
print_header(label);
report_pivoted(A, b);
report_no_pivot(A, b);
}
// --- Experiment cases ---
void exp_random(std::size_t n = 8) {
std::mt19937 rng(42);
std::uniform_real_distribution<double> dist(-5.0, 5.0);
Matrix A(n, n);
for (std::size_t i = 0; i < n; ++i)
for (std::size_t j = 0; j < n; ++j)
A(i, j) = dist(rng);
Vector b(n);
for (std::size_t i = 0; i < n; ++i) b[i] = dist(rng);
run_case("Random 8x8 (well-conditioned)", A, b);
}
void exp_badly_scaled() {
const Matrix A{
{1e-14, 1.0, 2.0 },
{1.0, 3.0, 4.0 },
{2.0, 5.0, 7.0 }
};
const Vector b{1e-14 + 3.0, 8.0, 14.0}; // b = A * [1, 1, 1]
run_case("Badly scaled (row norms differ by 10^14)", A, b);
}
void exp_epsilon_pathology() {
constexpr double eps = 1e-15;
const Matrix A{{eps, 1.0}, {1.0, 2.0}};
const Vector b{1.0 + eps, 3.0};
run_case("Epsilon pathology [[1e-15,1],[1,2]] (classic)", A, b);
std::cout << " Note: exact solution is x = [1, 1]\n";
}
void exp_amplified_multiplier() {
const Matrix A{
{0.001, 1.0, 0.0, 0.0 },
{1.0, 2.0, 1.0, 0.0 },
{0.0, 1.0, 3.0, 1.0 },
{0.0, 0.0, 1.0, 4.0 }
};
const Vector b = A * Vector{1.0, 2.0, 3.0, 4.0};
run_case("Amplified multiplier (small (1,1) pivot, 4x4)", A, b);
std::cout << " Note: exact solution is x = [1, 2, 3, 4]\n";
}
void exp_permutation() {
const Matrix A{
{0.0, 0.0, 3.0},
{0.0, 2.0, 1.0},
{5.0, 1.0, 0.0}
};
const Vector b = A * Vector{1.0, -1.0, 2.0};
run_case("Multiple row swaps required (zeros in pivot positions)", A, b);
}
int main() {
std::cout << std::string(60, '*') << "\n";
std::cout << " Pivoting vs No-Pivoting LU Experiment\n";
std::cout << std::string(60, '*') << "\n";
std::cout << "Residual = ||Ax - b||_2 (solve accuracy)\n";
std::cout << "Recon err = ||PA - LU||_F (factorization accuracy)\n";
exp_random();
exp_badly_scaled();
exp_epsilon_pathology();
exp_amplified_multiplier();
exp_permutation();
return 0;
}
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