gauss_seidel.cpp

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#include "linalg/solvers/gauss_seidel.hpp"
#include <cmath>
#include <stdexcept>
 
namespace num {
 
// Gauss-Seidel  -- sequential update order, parallel residual when omp backend.
//
// Note: standard Gauss-Seidel has sequential data dependencies (x[i] depends
// on x[0..i-1] updated in the same sweep).  We keep the sequential update
// order here and parallelise only the residual computation.  For a truly
// parallel relaxation scheme, use the Jacobi solver with Backend::omp.
SolverResult gauss_seidel(const Matrix &A, const Vector &b, Vector &x, real tol,
                          idx max_iter, Backend backend) {
  constexpr real zero_diag_tol = 1e-15;
  idx n = b.size();
  if (A.rows() != n || A.cols() != n || x.size() != n)
    throw std::invalid_argument("Dimension mismatch in Gauss-Seidel solver");
 
  SolverResult result{0, 0.0, false};
 
  for (idx iter = 0; iter < max_iter; ++iter) {
    // Sequential update  -- maintain GS convergence properties
    for (idx i = 0; i < n; ++i) {
      if (std::abs(A(i, i)) < zero_diag_tol)
        throw std::runtime_error("Zero diagonal in Gauss-Seidel at row " +
                                 std::to_string(i));
      real sigma = 0.0;
      for (idx j = 0; j < n; ++j)
        if (j != i)
          sigma += A(i, j) * x[j];
      x[i] = (b[i] - sigma) / A(i, i);
    }
 
    // Residual ||b - Ax||
    real res = 0.0;
#ifdef NUMERICS_HAS_OMP
#pragma omp parallel for reduction(+ : res)                                    \
    schedule(static) if (backend == Backend::omp)
#endif
    for (idx i = 0; i < n; ++i) {
      real ri = b[i];
      for (idx j = 0; j < n; ++j)
        ri -= A(i, j) * x[j];
      res += ri * ri;
    }
    result.residual = std::sqrt(res);
    result.iterations = iter + 1;
 
    if (result.residual < tol) {
      result.converged = true;
      break;
    }
  }
  return result;
}
 
} // namespace num