Add QR decomposition

2 files changed, 167 insertions, 10 deletions

diff --git a/experiments/hilbert_qr.cpp b/experiments/hilbert_qr.cpp
new file mode 100644
index 0000000..9209a30
--- /dev/null
+++ b/experiments/hilbert_qr.cpp
@@ -0,0 +1,167 @@
+// Experiment: QR methods on Hilbert matrices
+//
+// The Hilbert matrix H[i][j] = 1/(i+j+1) is the canonical ill-conditioned
+// dense matrix.  Its condition number grows roughly as (3.5 * e)^n / sqrt(n),
+// reaching ~10^13 at n=10 and ~10^18 at n=14.
+//
+// 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);
+}
+
+// ---------------------------------------------------------------------------
+// Timing
+// ---------------------------------------------------------------------------
+
+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;
+}
+
+// ---------------------------------------------------------------------------
+// Run one method on one size, return {recon, ortho, time} or nullopt on failure
+// ---------------------------------------------------------------------------
+
+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;
+    }
+}
+
+// ---------------------------------------------------------------------------
+// Pretty printing
+// ---------------------------------------------------------------------------
+
+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;
+}
diff --git a/experiments/pivoting_vs_no_pivoting.cpp b/experiments/pivoting_vs_no_pivoting.cpp
index ce2b9cc..0e71dc1 100644
--- a/experiments/pivoting_vs_no_pivoting.cpp
+++ b/experiments/pivoting_vs_no_pivoting.cpp
@@ -246,15 +246,5 @@ int main() {
     exp_amplified_multiplier();
     exp_permutation();
 
-    std::cout << "\n" << std::string(60, '=') << "\n";
-    std::cout << "  Conclusion\n";
-    std::cout << std::string(60, '=') << "\n";
-    std::cout <<
-        "Partial pivoting keeps multipliers bounded by 1 in magnitude.\n"
-        "Without it, a near-zero pivot inflates multipliers and destroys\n"
-        "accuracy via catastrophic cancellation.  The 'epsilon pathology'\n"
-        "case is the textbook example: a pivot of 1e-15 makes no-pivot LU\n"
-        "compute x ≈ [0, 0.5] instead of the exact [1, 1].\n\n";
-
     return 0;
 }