feat(phase4a+4b): Newton solver + CGAL mesh I/O
Phase 4a — newton_solver.hpp:
- newton_euclidean(): SimplicialLDLT on H (PSD); solves H·Δx = −G
- newton_spherical(): SimplicialLDLT on −H (NSD→PSD); solves (−H)·Δx = G
- Backtracking line search (halving α up to 20×) for global convergence
- NewtonResult struct: x, iterations, grad_inf_norm, converged
- 7 tests: 4 spherical (convergence, few iters, large perturbation,
field consistency) + 3 Euclidean (triangle pinned, quad pinned,
mixed pinned — all with natural-theta equilibrium at x=0)
Phase 4b — mesh_io.hpp:
- read_mesh() / write_mesh(): CGAL::IO::read/write_polygon_mesh wrappers
- load_mesh() / save_mesh(): throwing convenience versions
- Supports OFF, OBJ, PLY (format detected by file extension)
- 6 tests: OFF round-trip (tet + quad), OBJ round-trip, missing-file throw,
vertex-position preservation, save/load convenience wrappers
All 75 cgal tests pass (3 skipped as before).
Co-Authored-By: Claude Sonnet 4.5 <noreply@anthropic.com>
This commit is contained in:
Tarik Moussa
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code/include/newton_solver.hpp
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code/include/newton_solver.hpp
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#pragma once
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// newton_solver.hpp
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//
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// Phase 4a — Newton solver for the discrete conformal functionals.
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//
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// Solves G(x) = 0 where G is the gradient of the discrete conformal energy:
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// G_v = Θ_v − Σ_f α_v^f (angle-sum residual at each vertex DOF)
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//
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// Algorithm per iteration:
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// 1. Compute gradient G(x)
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// 2. Check convergence: max|G_i| < tol → done
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// 3. Compute sparse Hessian H(x)
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// 4. Factorize and solve the Newton system:
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// Euclidean: H · Δx = −G (H is PSD → SimplicialLDLT directly)
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// Spherical: (−H) · Δx = G (H is NSD → negate to get PSD matrix)
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// 5. Backtracking line search: halve α until ||G(x+α·Δx)|| < ||G(x)||
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// 6. x ← x + α·Δx, go to 1
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//
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// Requires:
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// Eigen::SimplicialLDLT (part of Eigen's sparse Cholesky module)
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#include "euclidean_hessian.hpp"
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#include "spherical_hessian.hpp"
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#include <Eigen/SparseCholesky>
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#include <Eigen/Dense>
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#include <algorithm>
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#include <cmath>
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namespace conformallab {
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// ── Result ────────────────────────────────────────────────────────────────────
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struct NewtonResult {
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std::vector<double> x; ///< DOF vector at termination
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int iterations; ///< Newton steps taken
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double grad_inf_norm;///< max |G_i| at termination
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bool converged; ///< true iff grad_inf_norm < tol
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};
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// ── Internal helpers ──────────────────────────────────────────────────────────
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namespace detail {
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// Backtracking line search: find the largest α in {1, 0.5, 0.25, …} such that
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// ||G(x + α·Δx)||₂ < ||G(x)||₂. Returns the accepted step (α may stay 1).
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template <typename GradFn>
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inline std::vector<double> line_search(
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const std::vector<double>& x,
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const Eigen::VectorXd& dx,
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double norm0,
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GradFn&& grad_fn,
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int max_halvings = 20)
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{
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const int n = static_cast<int>(x.size());
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double alpha = 1.0;
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std::vector<double> xnew(static_cast<std::size_t>(n));
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for (int ls = 0; ls < max_halvings; ++ls) {
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for (int i = 0; i < n; ++i)
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xnew[static_cast<std::size_t>(i)] = x[static_cast<std::size_t>(i)]
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+ alpha * dx[i];
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auto Gnew = grad_fn(xnew);
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double norm_new = 0.0;
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for (double v : Gnew) norm_new += v * v;
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norm_new = std::sqrt(norm_new);
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if (norm_new < norm0) return xnew;
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alpha *= 0.5;
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}
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// No improvement found — return best attempt (full step)
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for (int i = 0; i < n; ++i)
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xnew[static_cast<std::size_t>(i)] = x[static_cast<std::size_t>(i)] + dx[i];
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return xnew;
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}
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} // namespace detail
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// ── Euclidean Newton solver ────────────────────────────────────────────────────
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//
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// Minimises the Euclidean discrete conformal energy by solving G(x) = 0.
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// The Hessian H is PSD; Eigen::SimplicialLDLT is used directly.
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inline NewtonResult newton_euclidean(
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ConformalMesh& mesh,
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std::vector<double> x0,
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const EuclideanMaps& m,
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double tol = 1e-8,
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int max_iter = 200)
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{
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std::vector<double> x = x0;
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const int n = static_cast<int>(x.size());
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NewtonResult res;
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res.converged = false;
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res.iterations = 0;
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res.grad_inf_norm = 0.0;
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Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> solver;
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for (int iter = 0; iter < max_iter; ++iter) {
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// ── Gradient ──────────────────────────────────────────────────────────
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auto G_std = euclidean_gradient(mesh, x, m);
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Eigen::Map<const Eigen::VectorXd> G(G_std.data(), n);
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double inf_norm = G.cwiseAbs().maxCoeff();
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if (inf_norm < tol) {
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res.converged = true;
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res.grad_inf_norm = inf_norm;
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res.iterations = iter;
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res.x = x;
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return res;
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}
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// ── Hessian + factorisation ───────────────────────────────────────────
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auto H = euclidean_hessian(mesh, x, m);
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solver.compute(H);
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if (solver.info() != Eigen::Success) break;
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// ── Newton step: solve H·Δx = −G ────────────────────────────────────
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Eigen::VectorXd dx = solver.solve(-G);
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if (solver.info() != Eigen::Success) break;
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// ── Backtracking line search ──────────────────────────────────────────
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double norm0 = G.norm();
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x = detail::line_search(x, dx, norm0,
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[&](const std::vector<double>& xnew) {
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return euclidean_gradient(mesh, xnew, m);
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});
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res.iterations = iter + 1;
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}
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// Report final gradient norm
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auto G_final = euclidean_gradient(mesh, x, m);
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double inf_final = 0.0;
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for (double v : G_final) inf_final = std::max(inf_final, std::abs(v));
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res.grad_inf_norm = inf_final;
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res.x = x;
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return res;
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}
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// ── Spherical Newton solver ───────────────────────────────────────────────────
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//
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// Solves G(x) = 0 for the spherical discrete conformal functional.
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// The Hessian H is NSD at the solution; −H is PSD, so we factorise −H and
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// solve (−H)·Δx = G ⟺ H·Δx = −G.
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inline NewtonResult newton_spherical(
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ConformalMesh& mesh,
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std::vector<double> x0,
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const SphericalMaps& m,
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double tol = 1e-8,
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int max_iter = 200)
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{
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std::vector<double> x = x0;
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const int n = static_cast<int>(x.size());
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NewtonResult res;
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res.converged = false;
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res.iterations = 0;
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res.grad_inf_norm = 0.0;
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Eigen::SimplicialLDLT<Eigen::SparseMatrix<double>> solver;
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for (int iter = 0; iter < max_iter; ++iter) {
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// ── Gradient ──────────────────────────────────────────────────────────
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auto G_std = spherical_gradient(mesh, x, m);
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Eigen::Map<const Eigen::VectorXd> G(G_std.data(), n);
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double inf_norm = G.cwiseAbs().maxCoeff();
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if (inf_norm < tol) {
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res.converged = true;
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res.grad_inf_norm = inf_norm;
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res.iterations = iter;
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res.x = x;
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return res;
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}
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// ── Hessian: negate to get PSD matrix ────────────────────────────────
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auto H = spherical_hessian(mesh, x, m);
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auto negH = Eigen::SparseMatrix<double>(-H);
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solver.compute(negH);
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if (solver.info() != Eigen::Success) break;
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// ── Newton step: solve (−H)·Δx = G ─────────────────────────────────
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Eigen::VectorXd dx = solver.solve(G);
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if (solver.info() != Eigen::Success) break;
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// ── Backtracking line search ──────────────────────────────────────────
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double norm0 = G.norm();
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x = detail::line_search(x, dx, norm0,
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[&](const std::vector<double>& xnew) {
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return spherical_gradient(mesh, xnew, m);
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});
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res.iterations = iter + 1;
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}
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auto G_final = spherical_gradient(mesh, x, m);
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double inf_final = 0.0;
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for (double v : G_final) inf_final = std::max(inf_final, std::abs(v));
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res.grad_inf_norm = inf_final;
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res.x = x;
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return res;
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}
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} // namespace conformallab
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