solve method

@override List<double> solve()

Solves the Ax = b equation and returns the x vector.

Implementation

@override
List<double> solve() {
  final n = knownValues.length;
  final A = matrix.toListOfList();
  final b = knownValues.toList();

  // Swapping rows and pivoting.
  for (var p = 0; p < n; ++p) {
    // Finding a pivot.
    var max = p;
    for (var i = p + 1; i < n; i++) {
      if (A[i][p].abs() > A[max][p].abs()) {
        max = i;
      }
    }

    // Swapping rows.
    final temp = A[p];
    A[p] = A[max];
    A[max] = temp;
    final t = b[p];
    b[p] = b[max];
    b[max] = t;

    // Making sure the matrix is not singular.
    if (A[p][p].abs() <= precision) {
      throw const SystemSolverException(
        'The matrix is singular or nearly singular.',
      );
    }

    // pivot within A and b.
    for (var i = p + 1; i < n; i++) {
      final alpha = A[i][p] / A[p][p];
      b[i] -= alpha * b[p];
      for (var j = p; j < n; j++) {
        A[i][j] -= alpha * A[p][j];
      }
    }
  }

  // Back substitution.
  return SystemSolver.backSubstitution(A, b);
}