text: // Copyright Nick Thompson, 2019
  // Use, modification and distribution are subject to the
  // Boost Software License, Version 1.0.
  // (See accompanying file LICENSE_1_0.txt
  // or copy at http://www.boost.org/LICENSE_1_0.txt)
  
  #ifndef BOOST_MATH_INTERPOLATORS_CARDINAL_QUINTIC_B_SPLINE_DETAIL_HPP
  #define BOOST_MATH_INTERPOLATORS_CARDINAL_QUINTIC_B_SPLINE_DETAIL_HPP
  #include <cmath>
  #include <vector>
  #include <utility>
  #include <boost/math/special_functions/cardinal_b_spline.hpp>
  
  namespace boost{ namespace math{ namespace interpolators{ namespace detail{
  
  
  template <class Real>
  class cardinal_quintic_b_spline_detail
  {
  public:
      // If you don't know the value of the derivative at the endpoints, leave them as nans and the routine will estimate them.
      // y[0] = y(a), y[n -1] = y(b), step_size = (b - a)/(n -1).
  
      cardinal_quintic_b_spline_detail(const Real* const y,
                                       size_t n,
                                       Real t0 /* initial time, left endpoint */,
                                       Real h  /*spacing, stepsize*/,
                                       std::pair<Real, Real> left_endpoint_derivatives,
                                       std::pair<Real, Real> right_endpoint_derivatives)
      {
          static_assert(!std::is_integral<Real>::value, "The quintic B-spline interpolator only works with floating point types.");
          if (h <= 0) {
              throw std::logic_error("Spacing must be > 0.");
          }
          m_inv_h = 1/h;
          m_t0 = t0;
  
          if (n < 8) {
              throw std::logic_error("The quintic B-spline interpolator requires at least 8 points.");
          }
  
          using std::isnan;
          // This interpolator has error of order h^6, so the derivatives should be estimated with the same error.
          // See: https://en.wikipedia.org/wiki/Finite_difference_coefficient
          if (isnan(left_endpoint_derivatives.first)) {
              Real tmp = -49*y[0]/20 + 6*y[1] - 15*y[2]/2 + 20*y[3]/3 - 15*y[4]/4 + 6*y[5]/5 - y[6]/6;
              left_endpoint_derivatives.first = tmp/h;
          }
          if (isnan(right_endpoint_derivatives.first)) {
              Real tmp = 49*y[n-1]/20 - 6*y[n-2] + 15*y[n-3]/2 - 20*y[n-4]/3 + 15*y[n-5]/4 - 6*y[n-6]/5 + y[n-7]/6;
              right_endpoint_derivatives.first = tmp/h;
          }
          if(isnan(left_endpoint_derivatives.second)) {
              Real tmp = 469*y[0]/90 - 223*y[1]/10 + 879*y[2]/20 - 949*y[3]/18 + 41*y[4] - 201*y[5]/10 + 1019*y[6]/180 - 7*y[7]/10;
              left_endpoint_derivatives.second = tmp/(h*h);
          }
          if (isnan(right_endpoint_derivatives.second)) {
              Real tmp = 469*y[n-1]/90 - 223*y[n-2]/10 + 879*y[n-3]/20 - 949*y[n-4]/18 + 41*y[n-5] - 201*y[n-6]/10 + 1019*y[n-7]/180 - 7*y[n-8]/10;
              right_endpoint_derivatives.second = tmp/(h*h);
          }
  
          // This is really challenging my mental limits on by-hand row reduction.
          // I debated bringing in a dependency on a sparse linear solver, but given that that would cause much agony for users I decided against it.
  
          std::vector<Real> rhs(n+4);
          rhs[0] = 20*y[0] - 12*h*left_endpoint_derivatives.first +  2*h*h*left_endpoint_derivatives.second;
          rhs[1] = 60*y[0] - 12*h*left_endpoint_derivatives.first;
          for (size_t i = 2; i < n + 2; ++i) {
              rhs[i] = 120*y[i-2];
          }
          rhs[n+2] = 60*y[n-1] + 12*h*right_endpoint_derivatives.first;
          rhs[n+3] = 20*y[n-1] + 12*h*right_endpoint_derivatives.first +  2*h*h*right_endpoint_derivatives.second;
  
          std::vector<Real> diagonal(n+4, 66);
          diagonal[0] = 1;
          diagonal[1] = 18;
          diagonal[n+2] = 18;
          diagonal[n+3] = 1;
  
          std::vector<Real> first_superdiagonal(n+4, 26);
          first_superdiagonal[0] = 10;
          first_superdiagonal[1] = 33;
          first_superdiagonal[n+2] = 1;
          // There is one less superdiagonal than diagonal; make sure that if we read it, it shows up as a bug:
          first_superdiagonal[n+3] = std::numeric_limits<Real>::quiet_NaN();
  
          std::vector<Real> second_superdiagonal(n+4, 1);
          second_superdiagonal[0] = 9;
          second_superdiagonal[1] = 8;
          second_superdiagonal[n+2] = std::numeric_limits<Real>::quiet_NaN();
          second_superdiagonal[n+3] = std::numeric_limits<Real>::quiet_NaN();
  
          std::vector<Real> first_subdiagonal(n+4, 26);
          first_subdiagonal[0] = std::numeric_limits<Real>::quiet_NaN();
          first_subdiagonal[1] = 1;
          first_subdiagonal[n+2] = 33;
          first_subdiagonal[n+3] = 10;
  
          std::vector<Real> second_subdiagonal(n+4, 1);
          second_subdiagonal[0] = std::numeric_limits<Real>::quiet_NaN();
          second_subdiagonal[1] = std::numeric_limits<Real>::quiet_NaN();
          second_subdiagonal[n+2] = 8;
          second_subdiagonal[n+3] = 9;
  
  
          for (size_t i = 0; i < n+2; ++i) {
              Real di = diagonal[i];
              diagonal[i] = 1;
              first_superdiagonal[i] /= di;
              second_superdiagonal[i] /= di;
              rhs[i] /= di;
  
              // Eliminate first subdiagonal:
              Real nfsub = -first_subdiagonal[i+1];
              // Superfluous:
              first_subdiagonal[i+1] /= nfsub;
              // Not superfluous:
              diagonal[i+1] /= nfsub;
              first_superdiagonal[i+1] /= nfsub;
              second_superdiagonal[i+1] /= nfsub;
              rhs[i+1] /= nfsub;
  
              diagonal[i+1] += first_superdiagonal[i];
              first_superdiagonal[i+1] += second_superdiagonal[i];
              rhs[i+1] += rhs[i];
              // Superfluous, but clarifying:
              first_subdiagonal[i+1] = 0;
  
              // Eliminate second subdiagonal:
              Real nssub = -second_subdiagonal[i+2];
              first_subdiagonal[i+2] /= nssub;
              diagonal[i+2] /= nssub;
              first_superdiagonal[i+2] /= nssub;
              second_superdiagonal[i+2] /= nssub;
              rhs[i+2] /= nssub;
  
              first_subdiagonal[i+2] += first_superdiagonal[i];
              diagonal[i+2] += second_superdiagonal[i];
              rhs[i+2] += rhs[i];
              // Superfluous, but clarifying:
              second_subdiagonal[i+2] = 0;
          }
  
          // Eliminate last subdiagonal:
          Real dnp2 = diagonal[n+2];
          diagonal[n+2] = 1;
          first_superdiagonal[n+2] /= dnp2;
          rhs[n+2] /= dnp2;
          Real nfsubnp3 = -first_subdiagonal[n+3];
          diagonal[n+3] /= nfsubnp3;
          rhs[n+3] /= nfsubnp3;
  
          diagonal[n+3] += first_superdiagonal[n+2];
          rhs[n+3] += rhs[n+2];
  
          m_alpha.resize(n + 4, std::numeric_limits<Real>::quiet_NaN());
  
          m_alpha[n+3] = rhs[n+3]/diagonal[n+3];
          m_alpha[n+2] = rhs[n+2] - first_superdiagonal[n+2]*m_alpha[n+3];
          for (int64_t i = int64_t(n+1); i >= 0; --i) {
              m_alpha[i] = rhs[i] - first_superdiagonal[i]*m_alpha[i+1] - second_superdiagonal[i]*m_alpha[i+2];
          }
  
      }
  
      Real operator()(Real t) const {
          using std::ceil;
          using std::floor;
          using boost::math::cardinal_b_spline;
          // tf = t0 + (n-1)*h
          // alpha.size() = n+4
          if (t < m_t0 || t > m_t0 + (m_alpha.size()-5)/m_inv_h) {
              const char* err_msg = "Tried to evaluate the cardinal quintic b-spline outside the domain of of interpolation; extrapolation does not work.";
              throw std::domain_error(err_msg);
          }
          Real x = (t-m_t0)*m_inv_h;
          // Support of B_5 is [-3, 3]. So -3 < x - j + 2 < 3, so x-1 < j < x+5.
          // TODO: Zero pad m_alpha so that only the domain check is necessary.
          int64_t j_min = std::max(int64_t(0), int64_t(ceil(x-1)));
          int64_t j_max = std::min(int64_t(m_alpha.size() - 1), int64_t(floor(x+5)) );
          Real s = 0;
          for (int64_t j = j_min; j <= j_max; ++j) {
              // TODO: Use Cox 1972 to generate all integer translates of B5 simultaneously.
              s += m_alpha[j]*cardinal_b_spline<5, Real>(x - j + 2);
          }
          return s;
      }
  
      Real prime(Real t) const {
          using std::ceil;
          using std::floor;
          using boost::math::cardinal_b_spline_prime;
          if (t < m_t0 || t > m_t0 + (m_alpha.size()-5)/m_inv_h) {
              const char* err_msg = "Tried to evaluate the cardinal quintic b-spline outside the domain of of interpolation; extrapolation does not work.";
              throw std::domain_error(err_msg);
          }
          Real x = (t-m_t0)*m_inv_h;
          // Support of B_5 is [-3, 3]. So -3 < x - j + 2 < 3, so x-1 < j < x+5
          int64_t j_min = std::max(int64_t(0), int64_t(ceil(x-1)));
          int64_t j_max = std::min(int64_t(m_alpha.size() - 1), int64_t(floor(x+5)) );
          Real s = 0;
          for (int64_t j = j_min; j <= j_max; ++j) {
              s += m_alpha[j]*cardinal_b_spline_prime<5, Real>(x - j + 2);
          }
          return s*m_inv_h;
  
      }
  
      Real double_prime(Real t) const {
          using std::ceil;
          using std::floor;
          using boost::math::cardinal_b_spline_double_prime;
          if (t < m_t0 || t > m_t0 + (m_alpha.size()-5)/m_inv_h) {
              const char* err_msg = "Tried to evaluate the cardinal quintic b-spline outside the domain of of interpolation; extrapolation does not work.";
              throw std::domain_error(err_msg);
          }
          Real x = (t-m_t0)*m_inv_h;
          // Support of B_5 is [-3, 3]. So -3 < x - j + 2 < 3, so x-1 < j < x+5
          int64_t j_min = std::max(int64_t(0), int64_t(ceil(x-1)));
          int64_t j_max = std::min(int64_t(m_alpha.size() - 1), int64_t(floor(x+5)) );
          Real s = 0;
          for (int64_t j = j_min; j <= j_max; ++j) {
              s += m_alpha[j]*cardinal_b_spline_double_prime<5, Real>(x - j + 2);
          }
          return s*m_inv_h*m_inv_h;
      }
  
  
      Real t_max() const {
          return m_t0 + (m_alpha.size()-5)/m_inv_h;
      }
  
  private:
      std::vector<Real> m_alpha;
      Real m_inv_h;
      Real m_t0;
  };
  
  }}}}
  #endif
  