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// We compare classical GS, modified GS, and Householder QR on:
// - reconstruction error ||A - QR||_F
// - orthogonality error ||Q^T Q - I||_F
// - wall-clock time (minimum over several trials)
#include "matrix.hpp"
#include "qr.hpp"
#include <chrono>
#include <cmath>
#include <cstddef>
#include <functional>
#include <iomanip>
#include <iostream>
#include <optional>
#include <stdexcept>
#include <string>
using linalg::Matrix;
using linalg::QRResult;
// ---------------------------------------------------------------------------
// Matrix construction
// ---------------------------------------------------------------------------
Matrix hilbert(std::size_t n) {
Matrix H(n, n);
for (std::size_t i = 0; i < n; ++i)
for (std::size_t j = 0; j < n; ++j)
H(i, j) = 1.0 / static_cast<double>(i + j + 1);
return H;
}
// ---------------------------------------------------------------------------
// Metrics
// ---------------------------------------------------------------------------
double reconstruction_error(const Matrix& A, const QRResult& qr) {
const std::size_t m = A.rows();
const std::size_t n = A.cols();
const Matrix QR = qr.Q * qr.R;
double err = 0.0;
for (std::size_t i = 0; i < m; ++i)
for (std::size_t j = 0; j < n; ++j) {
const double d = A(i, j) - QR(i, j);
err += d * d;
}
return std::sqrt(err);
}
double orthogonality_error(const QRResult& qr) {
const Matrix& Q = qr.Q;
const std::size_t n = Q.cols();
double err = 0.0;
for (std::size_t i = 0; i < n; ++i)
for (std::size_t j = 0; j < n; ++j) {
double s = 0.0;
for (std::size_t k = 0; k < Q.rows(); ++k) s += Q(k, i) * Q(k, j);
const double d = s - (i == j ? 1.0 : 0.0);
err += d * d;
}
return std::sqrt(err);
}
using Clock = std::chrono::high_resolution_clock;
using Seconds = std::chrono::duration<double>;
// Run fn() `trials` times, return minimum elapsed seconds.
template<typename Fn>
double min_time(Fn fn, int trials = 5) {
double best = 1e18;
for (int t = 0; t < trials; ++t) {
const auto t0 = Clock::now();
fn();
const auto t1 = Clock::now();
best = std::min(best, Seconds(t1 - t0).count());
}
return best;
}
using QRFn = std::function<QRResult(const Matrix&)>;
struct Result { double recon, ortho, time_s; };
std::optional<Result> measure(const Matrix& A, QRFn fn) {
try {
// Run once to get metrics.
const QRResult qr = fn(A);
const double re = reconstruction_error(A, qr);
const double oe = orthogonality_error(qr);
// Time over multiple trials.
const double t = min_time([&] { fn(A); });
return Result{re, oe, t};
} catch (const std::exception&) {
return std::nullopt;
}
}
void print_row(const std::string& method, std::optional<Result> r) {
std::cout << std::left << std::setw(16) << method;
if (!r) {
std::cout << " FAILED (linearly dependent columns)\n";
return;
}
std::cout << std::scientific << std::setprecision(2)
<< std::setw(14) << r->recon
<< std::setw(14) << r->ortho
<< std::fixed << std::setprecision(3)
<< std::setw(10) << r->time_s * 1e6 << " µs\n";
}
// ---------------------------------------------------------------------------
// main
// ---------------------------------------------------------------------------
int main() {
std::cout << std::string(70, '*') << "\n";
std::cout << " Hilbert QR Experiment — comparing GS variants and Householder\n";
std::cout << std::string(70, '*') << "\n\n";
std::cout <<
"H[i][j] = 1/(i+j+1). Condition number grows ~exponentially with n.\n"
"Orthogonality loss in classical GS tracks condition number directly.\n"
"Modified GS recovers ~half the lost digits. Householder is unaffected.\n\n";
const std::size_t sizes[] = {2, 3, 4, 5, 6, 7, 8, 10, 12};
for (std::size_t n : sizes) {
const Matrix H = hilbert(n);
std::cout << std::string(70, '-') << "\n";
std::cout << " n = " << n << "\n";
std::cout << std::string(70, '-') << "\n";
std::cout << std::left
<< std::setw(16) << "Method"
<< std::setw(14) << "||A-QR||_F"
<< std::setw(14) << "||QtQ-I||_F"
<< std::setw(10) << "Time\n";
std::cout << std::string(70, ' ') << "\n";
print_row("classical_gs",
measure(H, [](const Matrix& A) { return linalg::qr_classical_gs(A); }));
print_row("modified_gs",
measure(H, [](const Matrix& A) { return linalg::qr_modified_gs(A); }));
print_row("householder",
measure(H, [](const Matrix& A) { return linalg::qr_householder(A); }));
}
return 0;
}
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