superancillary.h

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#pragma once
 
#include <iostream>
#include <utility>
#include <optional>
#include <Eigen/Dense>
 
#include "boost/math/tools/toms748_solve.hpp"
#include "nlohmann/json.hpp"
 
namespace CoolProp {
namespace superancillary {
 
namespace detail {
 
// From https://arxiv.org/pdf/1401.5766.pdf (Algorithm #3)
template <typename Matrix>
inline void balance_matrix(const Matrix& A, Matrix& Aprime, Matrix& D) {
    const int p = 2;
    double beta = 2;  // Radix base (2?)
    int iter = 0;
    Aprime = A;
    D = Matrix::Identity(A.rows(), A.cols());
    bool converged = false;
    do {
        converged = true;
        for (Eigen::Index i = 0; i < A.rows(); ++i) {
            double c = Aprime.col(i).template lpNorm<p>();
            double r = Aprime.row(i).template lpNorm<p>();
            double s = pow(c, p) + pow(r, p);
            double f = 1;
            //if (!ValidNumber(c)){
            //    std::cout << A << std::endl;
            //    throw std::range_error("c is not a valid number in balance_matrix"); }
            //if (!ValidNumber(r)) { throw std::range_error("r is not a valid number in balance_matrix"); }
            while (c < r / beta) {
                c *= beta;
                r /= beta;
                f *= beta;
            }
            while (c >= r * beta) {
                c /= beta;
                r *= beta;
                f /= beta;
            }
            if (pow(c, p) + pow(r, p) < 0.95 * s) {
                converged = false;
                D(i, i) *= f;
                Aprime.col(i) *= f;
                Aprime.row(i) /= f;
            }
        }
        iter++;
        if (iter > 50) {
            break;
        }
    } while (!converged);
}
 
inline void companion_matrix_transposed(const Eigen::ArrayXd& coeffs, Eigen::MatrixXd& A) {
    auto N = coeffs.size() - 1;  // degree
    if (A.rows() != N) {
        throw std::invalid_argument("A.rows() != N");
    }
    A.setZero();
    // First col
    A(1, 0) = 1;
    // Last col
    A.col(N - 1) = -coeffs.head(N) / (2.0 * coeffs(N));
    A(N - 2, N - 1) += 0.5;
    // All the other cols
    for (int j = 1; j < N - 1; ++j) {
        A(j - 1, j) = 0.5;
        A(j + 1, j) = 0.5;
    }
}
inline void companion_matrix_transposed(const std::vector<double>& coeffs, Eigen::MatrixXd& A) {
    Eigen::ArrayXd coeffs_ = Eigen::Map<const Eigen::ArrayXd>(&coeffs[0], coeffs.size());
    return companion_matrix_transposed(coeffs_, A);
}
 
inline auto get_LU_matrices(std::size_t N) {
    Eigen::MatrixXd L(N + 1, N + 1);  
    Eigen::MatrixXd U(N + 1, N + 1);  
    for (std::size_t j = 0; j <= N; ++j) {
        for (std::size_t k = j; k <= N; ++k) {
            double p_j = (j == 0 || j == N) ? 2 : 1;
            double p_k = (k == 0 || k == N) ? 2 : 1;
            double cosjPikN = cos((j * EIGEN_PI * k) / N);
            L(j, k) = 2.0 / (p_j * p_k * N) * cosjPikN;
            // Exploit symmetry to fill in the symmetric elements in the matrix
            L(k, j) = L(j, k);
 
            U(j, k) = cosjPikN;
            // Exploit symmetry to fill in the symmetric elements in the matrix
            U(k, j) = U(j, k);
        }
    }
    return std::make_tuple(L, U);
}
 
inline double M_element_norm(const std::vector<double>& x, Eigen::Index M) {
    Eigen::Map<const Eigen::ArrayXd> X(&x[0], x.size());
    return X.tail(M).matrix().norm() / X.head(M).matrix().norm();
}
 
inline double M_element_norm(const Eigen::ArrayXd& x, Eigen::Index M) {
    return x.tail(M).matrix().norm() / x.head(M).matrix().norm();
}
 
template <typename Function, typename Container>
inline auto dyadic_splitting(const std::size_t N, const Function& func, const double xmin, const double xmax, const int M = 3,
                             const double tol = 1e-12, const int max_refine_passes = 8,
                             const std::optional<std::function<void(int, const Container&)>>& callback = std::nullopt) -> Container {
    using CE_t = std::decay_t<decltype(Container().front())>;
    using ArrayType = std::decay_t<decltype(CE_t().coeff())>;
 
    Eigen::MatrixXd Lmat, Umat;
    std::tie(Lmat, Umat) = detail::get_LU_matrices(N);
 
    auto builder = [&](double xmin, double xmax) -> CE_t {
        auto get_nodes_realworld = [N, xmin, xmax]() -> Eigen::ArrayXd {
            Eigen::ArrayXd nodes = (Eigen::ArrayXd::LinSpaced(N + 1, 0, static_cast<double>(N)).array() * EIGEN_PI / N).cos();
            return ((xmax - xmin) * nodes + (xmax + xmin)) * 0.5;
        };
 
        Eigen::ArrayXd x = get_nodes_realworld();
 
        // Apply the function to do the transformation
        // of the functional values at the nodes
        for (auto j = 0L; j < x.size(); ++j) {
            x(j) = func(j, static_cast<long>(x.size()), x(j));
        }
 
        // And now rebuild the expansion by left-multiplying by the L matrix
        Eigen::ArrayXd c = Lmat * x.matrix();
        //        std::cout << "c: " << c << std::endl;
        // Check if any coefficients are invalid, stop if so
        if (!c.allFinite()) {
            throw std::invalid_argument("At least one coefficient is non-finite");
        }
        if constexpr (std::is_same_v<ArrayType, std::vector<double>>) {
            return {xmin, xmax, std::vector<double>(&c[0], &c[0] + c.size())};
        } else {
            // Probably an Eigen::ArrayXd, just pass that into constructor
            return {xmin, xmax, c};
        }
    };
 
    // Start off with the full domain from xmin to xmax
    Container expansions;
    expansions.emplace_back(builder(xmin, xmax));
 
    // Now enter into refinement passes
    for (int refine_pass = 0; refine_pass < max_refine_passes; ++refine_pass) {
        bool all_converged = true;
        // Start at the right and move left because insertions will make the length increase
        for (int iexpansion = static_cast<int>(expansions.size()) - 1; iexpansion >= 0; --iexpansion) {
            auto& expan = expansions[iexpansion];
            auto err = M_element_norm(expan.coeff(), M);
            //            std::cout << "err: " <<  err << std::endl;
            if (err > tol) {
                // Splitting is required, do a dyadic split
                auto xmid = (expan.xmin() + expan.xmax()) / 2;
                CE_t newleft{builder(expan.xmin(), xmid)};
                CE_t newright{builder(xmid, expan.xmax())};
 
                expansions.at(iexpansion) = std::move(newleft);
                expansions.insert(expansions.begin() + iexpansion + 1, newright);
                all_converged = false;
            }
            //            std::cout << expansions.size() << std::endl;
        }
        if (callback) {
            callback.value()(refine_pass, expansions);
        }
        if (all_converged) {
            break;
        }
    }
    return expansions;
}
 
}  // namespace detail
 
template <typename ArrayType>
class ChebyshevExpansion
{
   private:
    double m_xmin,      
      m_xmax;           
    ArrayType m_coeff;  
   public:
    ChebyshevExpansion() {};
 
    ChebyshevExpansion(double xmin, double xmax, const ArrayType& coeff) : m_xmin(xmin), m_xmax(xmax), m_coeff(coeff) {};
 
    ChebyshevExpansion(const ChebyshevExpansion& ex) = default;
    ChebyshevExpansion& operator=(ChebyshevExpansion&& ex) = default;
    ChebyshevExpansion& operator=(const ChebyshevExpansion& ex) = default;
 
    const auto xmin() const {
        return m_xmin;
    }
 
    const auto xmax() const {
        return m_xmax;
    }
 
    const auto& coeff() const {
        return m_coeff;
    }
 
    template <typename T>
    auto eval(const T& x) const {
        // Scale to (-1, 1)
        T xscaled = (2.0 * x - (m_xmax + m_xmin)) / (m_xmax - m_xmin);
        int Norder = static_cast<int>(m_coeff.size()) - 1;
        T u_k = 0, u_kp1 = m_coeff[Norder], u_kp2 = 0;
        for (int k = Norder - 1; k > 0; k--) {  // k must be signed!
            // Do the recurrent calculation
            u_k = 2.0 * xscaled * u_kp1 - u_kp2 + m_coeff[k];
            // Update the values
            u_kp2 = u_kp1;
            u_kp1 = u_k;
        }
        T retval = m_coeff[0] + xscaled * u_kp1
                   - u_kp2;  // This seems inefficient but required to ensure that Eigen::Array are evaluated and an expression is not returned
        return retval;
    }
 
    template <typename T>
    auto eval_many(const T& x, T& y) const {
        if (x.size() != y.size()) {
            throw std::invalid_argument("x and y are not the same size");
        }
        for (auto i = 0; i < x.size(); ++i) {
            y(i) = eval(x(i));
        }
    }
 
    template <typename T>
    auto eval_manyC(const T x[], T y[], std::size_t N) const {
        for (std::size_t i = 0; i < N; ++i) {
            y[i] = eval(x[i]);
        }
    }
 
    template <typename T>
    auto eval_Eigen(const T& x, T& y) const {
        if (x.size() != y.size()) {
            throw std::invalid_argument("x and y are not the same size");
        }
        y = eval(x);
    }
 
    Eigen::ArrayXd get_nodes_realworld() const {
        Eigen::Index N = m_coeff.size() - 1;
        Eigen::ArrayXd nodes = (Eigen::ArrayXd::LinSpaced(N + 1, 0, N).array() * EIGEN_PI / N).cos();
        return ((m_xmax - m_xmin) * nodes + (m_xmax + m_xmin)) * 0.5;
    }
 
    auto solve_for_x_count(double y, double a, double b, unsigned int bits, std::size_t max_iter, double boundsytol) const {
        using namespace boost::math::tools;
        std::size_t counter = 0;
        auto f = [&](double x) {
            counter++;
            return eval(x) - y;
        };
        double fa = f(a);
        if (std::abs(fa) < boundsytol) {
            return std::make_tuple(a, std::size_t{1});
        }
        double fb = f(b);
        if (std::abs(fb) < boundsytol) {
            return std::make_tuple(b, std::size_t{2});
        }
        boost::math::uintmax_t max_iter_ = static_cast<boost::math::uintmax_t>(max_iter);
        auto [l, r] = toms748_solve(f, a, b, fa, fb, eps_tolerance<double>(bits), max_iter_);
        return std::make_tuple((r + l) / 2.0, counter);
    }
 
    auto solve_for_x(double y, double a, double b, unsigned int bits, std::size_t max_iter, double boundsytol) const {
        return std::get<0>(solve_for_x_count(y, a, b, bits, max_iter, boundsytol));
    }
 
    template <typename T, typename CountsT>
    auto solve_for_x_many(const T& y, double a, double b, unsigned int bits, std::size_t max_iter, double boundsytol, T& x, CountsT& counts) const {
        if (x.size() != y.size()) {
            throw std::invalid_argument("x and y are not the same size");
        }
        for (auto i = 0; i < x.size(); ++i) {
            std::tie(x(i), counts(i)) = solve_for_x_count(y(i), a, b, bits, max_iter, boundsytol);
        }
    }
 
    template <typename T, typename CountsT>
    auto solve_for_x_manyC(const T y[], std::size_t N, double a, double b, unsigned int bits, std::size_t max_iter, double boundsytol, T x[],
                           CountsT counts[]) const {
        for (std::size_t i = 0; i < N; ++i) {
            std::tie(x[i], counts[i]) = solve_for_x_count(y[i], a, b, bits, max_iter, boundsytol);
        }
    }
 
    ArrayType do_derivs(std::size_t Nderiv) const {
        // See Mason and Handscomb, p. 34, Eq. 2.52
        // and example in https ://github.com/numpy/numpy/blob/master/numpy/polynomial/chebyshev.py#L868-L964
        ArrayType c = m_coeff;
        for (std::size_t deriv_counter = 0; deriv_counter < Nderiv; ++deriv_counter) {
            std::size_t N = c.size() - 1,  
              Nd = N - 1;                  
            ArrayType cd(N);
            for (std::size_t r = 0; r <= Nd; ++r) {
                cd[r] = 0;
                for (std::size_t k = r + 1; k <= N; ++k) {
                    // Terms where k-r is odd have values, otherwise, they are zero
                    if ((k - r) % 2 == 1) {
                        cd[r] += 2 * k * c[k];
                    }
                }
                // The first term with r = 0 is divided by 2 (the single prime in Mason and Handscomb, p. 34, Eq. 2.52)
                if (r == 0) {
                    cd[r] /= 2;
                }
                // Rescale the values if the range is not [-1,1].  Arrives from the derivative of d(xreal)/d(x_{-1,1})
                cd[r] /= (m_xmax - m_xmin) / 2.0;
            }
            if (Nderiv == 1) {
                return cd;
            } else {
                c = cd;
            }
        }
        return c;
    }
};
 
static_assert(std::is_move_assignable_v<ChebyshevExpansion<std::vector<double>>>);
//static_assert(std::is_copy_assignable_v<ChebyshevExpansion<std::vector<double>>>);
 
struct MonotonicExpansionMatch
{
    std::size_t idx;  
    double ymin,      
      ymax,           
      xmin,           
      xmax;           
    bool contains_y(double y) const {
        return y >= ymin && y <= ymax;
    }
};
 
struct IntervalMatch
{
    std::vector<MonotonicExpansionMatch> expansioninfo;  
    double xmin,                                         
      xmax,                                              
      ymin,                                              
      ymax;                                              
    bool contains_y(double y) const {
        return y >= ymin && y <= ymax;
    }
};
 
template <typename ArrayType = Eigen::ArrayXd>
struct ChebyshevApproximation1D
{
   private:
    const double thresh_imag = 1e-15;                         
    std::vector<ChebyshevExpansion<ArrayType>> m_expansions;  
    std::vector<double> m_x_at_extrema;                       
    std::vector<IntervalMatch> m_monotonic_intervals;         
 
    Eigen::ArrayXd head(const Eigen::ArrayXd& c, Eigen::Index N) const {
        return c.head(N);
    }
    std::vector<double> head(const std::vector<double>& c, Eigen::Index N) const {
        return std::vector<double>(c.begin(), c.begin() + N);
    }
 
    std::vector<double> determine_extrema(const std::decay_t<decltype(m_expansions)>& expansions, double thresh_im) const {
        std::vector<double> x_at_extrema;
        Eigen::MatrixXd companion_matrix, cprime, D;
        for (auto& expan : expansions) {
            // Get the coefficients of the derivative's expansion (for x in [-1, 1])
            auto cd = expan.do_derivs(1);
            // First, right-trim the coefficients that are equal to zero
            int ilastnonzero = static_cast<int>(cd.size()) - 1;
            for (int i = static_cast<int>(cd.size()) - 1; i >= 0; --i) {
                if (std::abs(cd[i]) != 0) {
                    ilastnonzero = i;
                    break;
                }
            }
            if (ilastnonzero != static_cast<int>(cd.size() - 1)) {
                cd = head(cd, ilastnonzero);
            }
            // Then do eigenvalue rootfinding after balancing
            // Define working buffers here to avoid allocations all over the place
            if (companion_matrix.rows() != static_cast<int>(cd.size() - 1)) {
                companion_matrix.resize(cd.size() - 1, cd.size() - 1);
                companion_matrix.setZero();
                D.resizeLike(companion_matrix);
                D.setZero();
                cprime.resizeLike(companion_matrix);
                cprime.setZero();
            }
            detail::companion_matrix_transposed(cd, companion_matrix);
            cprime = companion_matrix;
 
            detail::balance_matrix(companion_matrix, cprime, D);
            for (auto& root : companion_matrix.eigenvalues()) {
                // re_n11 is in [-1,1], need to rescale back to real units
                auto re_n11 = root.real(), im = root.imag();
                if (std::abs(im) < thresh_im) {
                    if (re_n11 >= -1 && re_n11 <= 1) {
                        x_at_extrema.push_back(((expan.xmax() - expan.xmin()) * re_n11 + (expan.xmax() + expan.xmin())) / 2.0);
                    }
                }
            }
        }
        std::sort(x_at_extrema.begin(), x_at_extrema.end());
        return x_at_extrema;
    }
 
    std::vector<IntervalMatch> build_monotonic_intervals(const std::vector<double>& x_at_extrema) const {
        std::vector<IntervalMatch> intervals;
 
        auto sort = [](double& x, double& y) {
            if (x > y) {
                std::swap(x, y);
            }
        };
 
        if (false) {  //x_at_extrema.empty()){
            auto xmin = m_expansions.front().xmin();
            auto xmax = m_expansions.back().xmax();
            auto ymin = eval(xmin), ymax = eval(xmax);
            sort(ymin, ymax);
            MonotonicExpansionMatch mem;
            mem.idx = 0;
            mem.xmin = xmin;
            mem.xmax = xmax;
            mem.ymin = ymin;
            mem.ymax = ymax;
            IntervalMatch im;
            im.expansioninfo = {mem};
            im.xmin = mem.xmin;
            im.xmax = mem.xmax;
            im.ymin = mem.ymin;
            im.ymax = mem.ymax;
            intervals.push_back(im);
        } else {
            auto newx = x_at_extrema;
            newx.insert(newx.begin(), m_expansions.front().xmin());
            newx.insert(newx.end(), m_expansions.back().xmax());
 
            auto interval_intersection = [](const auto& t1, const auto& t2) {
                auto a = std::max(t1.xmin(), t2.xmin);
                auto b = std::min(t1.xmax(), t2.xmax);
                return std::make_tuple(a, b);
            };
 
            for (auto j = 0; j < static_cast<int>(newx.size() - 1); ++j) {
                double xmin = newx[j], xmax = newx[j + 1];
                IntervalMatch im;
                // Loop over the expansions that contain one of the values of x
                // that intersect the interval defined by [xmin, xmax]
                for (auto i = 0UL; i < m_expansions.size(); ++i) {
                    struct A
                    {
                        double xmin, xmax;
                    };
                    auto [a, b] = interval_intersection(m_expansions[i], A{xmin, xmax});
                    if (a < b) {
                        const auto& e = m_expansions[i];
                        MonotonicExpansionMatch mem;
                        mem.idx = i;
                        mem.xmin = a;
                        mem.xmax = b;
                        double ymin = e.eval(a), ymax = e.eval(b);
                        sort(ymin, ymax);
                        mem.ymin = ymin;
                        mem.ymax = ymax;
                        im.expansioninfo.push_back(mem);
                    }
                }
                // These are the limits for the entire interval
                im.xmin = xmin;
                im.xmax = xmax;
                double yminoverall = eval(xmin), ymaxoverall = eval(xmax);
                sort(yminoverall, ymaxoverall);
                im.ymin = yminoverall;
                im.ymax = ymaxoverall;
                intervals.push_back(im);
            }
        }
        return intervals;
    }
    double m_xmin,  
      m_xmax;       
 
   public:
    // Move constructor given a vector of expansions
    ChebyshevApproximation1D(std::vector<ChebyshevExpansion<ArrayType>>&& expansions)
      : m_expansions(std::move(expansions)),
        m_x_at_extrema(determine_extrema(m_expansions, thresh_imag)),
        m_monotonic_intervals(build_monotonic_intervals(m_x_at_extrema)),
        m_xmin(get_expansions().front().xmin()),
        m_xmax(get_expansions().back().xmax()) {}
 
    ChebyshevApproximation1D(const ChebyshevApproximation1D& other)
      : m_expansions(other.m_expansions),
        m_x_at_extrema(other.m_x_at_extrema),
        m_monotonic_intervals(other.m_monotonic_intervals),
        m_xmin(other.xmin()),
        m_xmax(other.xmax()) {}
 
    ChebyshevApproximation1D& operator=(ChebyshevApproximation1D&& other) = default;
    //    ChebyshevApproximation1D& operator=(const ChebyshevApproximation1D& other) = default;
    ChebyshevApproximation1D& operator=(ChebyshevApproximation1D other) {
        std::swap(m_expansions, other.m_expansions);
        std::swap(m_x_at_extrema, other.m_x_at_extrema);
        std::swap(m_monotonic_intervals, other.m_monotonic_intervals);
        std::swap(m_xmin, other.m_xmin);
        std::swap(m_xmax, other.m_xmax);
        return *this;
    }
 
    const auto& get_expansions() const {
        return m_expansions;
    }
 
    const auto& get_x_at_extrema() const {
        return m_x_at_extrema;
    }
 
    const auto& get_monotonic_intervals() const {
        return m_monotonic_intervals;
    }
 
    const auto xmin() const {
        return m_xmin;
    }
 
    const auto xmax() const {
        return m_xmax;
    }
 
    bool is_monotonic() const {
        return m_monotonic_intervals.size() == 1;
    }
 
    auto get_index(double x) const {
 
        // https://proquest.safaribooksonline.com/9780321637413
        // https://web.stanford.edu/class/archive/cs/cs107/cs107.1202/lab1/
        auto midpoint_Knuth = [](int x, int y) { return (x & y) + ((x ^ y) >> 1); };
 
        int iL = 0U, iR = static_cast<int>(m_expansions.size()) - 1, iM;
        while (iR - iL > 1) {
            iM = midpoint_Knuth(iL, iR);
            if (x >= m_expansions[iM].xmin()) {
                iL = iM;
            } else {
                iR = iM;
            }
        }
        return (x < m_expansions[iL].xmax()) ? iL : iR;
    };
 
    double eval(double x) const {
        return m_expansions[get_index(x)].eval(x);
    }
 
    template <typename Container>
    const auto eval_many(const Container& x, Container& y) const {
        if (x.size() != y.size()) {
            throw std::invalid_argument("x and y are not the same size");
        }
        for (auto i = 0U; i < x.size(); ++i) {
            y(i) = eval(x(i));
        }
    }
 
    template <typename Container>
    const auto eval_manyC(const Container x[], Container y[], std::size_t N) const {
        for (auto i = 0U; i < N; ++i) {
            y[i] = eval(x[i]);
        }
    }
 
    const std::vector<IntervalMatch> get_intervals_containing_y(double y) const {
        std::vector<IntervalMatch> matches;
        for (auto& interval : m_monotonic_intervals) {
            if (y >= interval.ymin && y <= interval.ymax) {
                matches.push_back(interval);
            }
        }
        return matches;
    }
 
    const auto get_x_for_y(double y, unsigned int bits, std::size_t max_iter, double boundsftol) const {
        std::vector<std::pair<double, int>> solns;
        for (const auto& interval : m_monotonic_intervals) {
            // Each interval is required to be monotonic internally, so if the value of
            // y is within the y values at the endpoints it is a candidate
            // for rootfinding, otherwise continue
            if (interval.contains_y(y)) {
                // Now look at the expansions that intersect the interval
                for (const auto& ei : interval.expansioninfo) {
                    // Again, since the portions of the expansions are required
                    // to be monotonic, if it is contained then a solution must exist
                    if (ei.contains_y(y)) {
                        const ChebyshevExpansion<ArrayType>& e = m_expansions[ei.idx];
                        auto [xvalue, num_steps] = e.solve_for_x_count(y, ei.xmin, ei.xmax, bits, max_iter, boundsftol);
                        solns.emplace_back(xvalue, static_cast<int>(num_steps));
                    }
                }
            }
        }
        return solns;
    }
 
    template <typename YContainer, typename CountContainer>
    const auto count_x_for_y_many(const YContainer& y, unsigned int bits, std::size_t max_iter, double boundsftol, CountContainer& x) const {
        if (x.size() != y.size()) {
            throw std::invalid_argument("x and y are not the same size");
        }
        for (auto i = 0U; i < x.size(); ++i) {
            x(i) = get_x_for_y(y(i), bits, max_iter, boundsftol).size();
        }
    }
 
    template <typename YContainer, typename CountContainer>
    const auto count_x_for_y_manyC(const YContainer y[], size_t N, unsigned int bits, std::size_t max_iter, double boundsftol,
                                   CountContainer x[]) const {
        for (auto i = 0U; i < N; ++i) {
            x[i] = get_x_for_y(y[i], bits, max_iter, boundsftol).size();
        }
    }
};
 
static_assert(std::is_copy_constructible_v<ChebyshevApproximation1D<std::vector<double>>>);
static_assert(std::is_copy_assignable_v<ChebyshevApproximation1D<std::vector<double>>>);
 
struct SuperAncillaryTwoPhaseSolution
{
    double T,             
      q;                  
    std::size_t counter;  
};
 
template <typename ArrayType = Eigen::ArrayXd>
class SuperAncillary
{
   private:
    ChebyshevApproximation1D<ArrayType> m_rhoL,  
      m_rhoV,                                    
      m_p;                                       
 
    // These ones *may* be present
    std::optional<ChebyshevApproximation1D<ArrayType>> m_hL,  
      m_hV,                                                   
      m_sL,                                                   
      m_sV,                                                   
      m_uL,                                                   
      m_uV,                                                   
      m_invlnp;                                               
 
    double m_Tmin;         
    double m_Tcrit_num;    
    double m_rhocrit_num;  
    double m_pmin;         
    double m_pmax;         
 
    auto loader(const nlohmann::json& j, const std::string& key) {
        std::vector<ChebyshevExpansion<ArrayType>> buf;
        // If you want to use Eigen...
        // auto toeig = [](const nlohmann::json &j) -> Eigen::ArrayXd{
        //     auto vec = j.get<std::vector<double>>();
        //     return Eigen::Map<const Eigen::ArrayXd>(&vec[0], vec.size());
        // };
        // for (auto& block : j[key]){
        //     buf.emplace_back(block.at("xmin"), block.at("xmax"), toeig(block.at("coef")));
        // }
        for (auto& block : j[key]) {
            buf.emplace_back(block.at("xmin"), block.at("xmax"), block.at("coef"));
        }
        return buf;
    }
 
    auto make_invlnp(Eigen::Index Ndegree) {
 
        auto pmin = m_p.eval(m_p.xmin());
        auto pmax = m_p.eval(m_p.xmax());
        // auto N = m_p.get_expansions().front().coeff().size()-1;
        const double EPSILON = std::numeric_limits<double>::epsilon();
 
        auto func = [&](long i, long Npts, double lnp) -> double {
            double p = exp(lnp);
            auto solns = m_p.get_x_for_y(p, 64, 100U, 1e-8);
 
            if (solns.size() != 1) {
                if ((i == 0 || i == Npts - 1) && (p > pmin * (1 - EPSILON * 1000) && p < pmin * (1 + EPSILON * 1000))) {
                    return m_p.get_monotonic_intervals()[0].xmin;
                }
                if ((i == 0 || i == Npts - 1) && (p > pmax * (1 - EPSILON * 1000) && p < pmax * (1 + EPSILON * 1000))) {
                    return m_p.get_monotonic_intervals()[0].xmax;
                }
                std::stringstream ss;
                ss << std::setprecision(20) << "Must have one solution for T(p), got " << solns.size() << " for " << p << " Pa; limits are [" << pmin
                   << +" Pa , " << pmax << " Pa]; i is " << i;
                throw std::invalid_argument(ss.str());
            }
            auto [T, iters] = solns[0];
            return T;
        };
 
        using CE_t = std::vector<ChebyshevExpansion<ArrayType>>;
        return detail::dyadic_splitting<decltype(func), CE_t>(Ndegree, func, log(pmin), log(pmax), 3, 1e-12, 26);
    }
 
    // using PropertyPairs = properties::PropertyPairs; ///< A convenience alias to save some typing
 
   public:
    SuperAncillary(const nlohmann::json& j)
      : m_rhoL(std::move(loader(j, "jexpansions_rhoL"))),
        m_rhoV(std::move(loader(j, "jexpansions_rhoV"))),
        m_p(std::move(loader(j, "jexpansions_p"))),
        m_Tmin(m_p.xmin()),
        m_Tcrit_num(j.at("meta").at("Tcrittrue / K")),
        m_rhocrit_num(j.at("meta").at("rhocrittrue / mol/m^3")),
        m_pmin(m_p.eval(m_p.xmin())),
        m_pmax(m_p.eval(m_p.xmax())) {};
 
    SuperAncillary(const std::string& s) : SuperAncillary(nlohmann::json::parse(s)) {};
 
    const auto& get_approx1d(char k, short Q) const {
        auto get_or_throw = [&](const auto& v) -> const auto& {
            if (v) {
                return v.value();
            } else {
                throw std::invalid_argument("unable to get the variable " + std::string(1, k) + ", make sure it has been added to superancillary");
            }
        };
        switch (k) {
            case 'P':
                return m_p;
            case 'D':
                return (Q == 0) ? m_rhoL : m_rhoV;
            case 'H':
                return (Q == 0) ? get_or_throw(m_hL) : get_or_throw(m_hV);
            case 'S':
                return (Q == 0) ? get_or_throw(m_sL) : get_or_throw(m_sV);
            case 'U':
                return (Q == 0) ? get_or_throw(m_uL) : get_or_throw(m_uV);
            default:
                throw std::invalid_argument("Bad key of '" + std::string(1, k) + "'");
        }
    }
    const auto& get_invlnp() {
        // Lazily construct on the first access
        if (!m_invlnp) {
            // Degree of expansion is the same as
            auto Ndegree = m_p.get_expansions()[0].coeff().size() - 1;
            m_invlnp = make_invlnp(Ndegree);
        }
        return m_invlnp;
    }
    const double get_pmin() const {
        return m_pmin;
    }
    const double get_pmax() const {
        return m_pmax;
    }
    const double get_Tmin() const {
        return m_Tmin;
    }
    const double get_Tcrit_num() const {
        return m_Tcrit_num;
    }
    const double get_rhocrit_num() const {
        return m_rhocrit_num;
    }
 
    void add_variable(char var, const std::function<double(double, double)>& caller) {
        Eigen::MatrixXd Lmat, Umat;
        std::tie(Lmat, Umat) = detail::get_LU_matrices(12);
 
        auto builder = [&](char var, auto& variantL, auto& variantV) {
            std::vector<ChebyshevExpansion<ArrayType>> newexpL, newexpV;
            const auto& expsL = get_approx1d('D', 0);
            const auto& expsV = get_approx1d('D', 1);
            if (expsL.get_expansions().size() != expsV.get_expansions().size()) {
                throw std::invalid_argument("L&V are not the same size");
            }
            for (auto i = 0U; i < expsL.get_expansions().size(); ++i) {
                const auto& expL = expsL.get_expansions()[i];
                const auto& expV = expsV.get_expansions()[i];
                const auto& T = expL.get_nodes_realworld();
                // Get the functional values at the Chebyshev nodes
                Eigen::ArrayXd funcL = Umat * expL.coeff().matrix();
                Eigen::ArrayXd funcV = Umat * expV.coeff().matrix();
                // Apply the function inplace to do the transformation
                // of the functional values at the nodes
                for (auto j = 0; j < funcL.size(); ++j) {
                    funcL(j) = caller(T(j), funcL(j));
                    funcV(j) = caller(T(j), funcV(j));
                }
                // And now rebuild the expansions by left-multiplying by the L matrix
                newexpL.emplace_back(expL.xmin(), expL.xmax(), (Lmat * funcL.matrix()).eval());
                newexpV.emplace_back(expV.xmin(), expV.xmax(), (Lmat * funcV.matrix()).eval());
            }
 
            variantL.emplace(std::move(newexpL));
            variantV.emplace(std::move(newexpV));
        };
 
        switch (var) {
            case 'H':
                builder(var, m_hL, m_hV);
                break;
            case 'S':
                builder(var, m_sL, m_sV);
                break;
            case 'U':
                builder(var, m_uL, m_uV);
                break;
            default:
                throw std::invalid_argument("nope");
        }
    }
 
    double eval_sat(double T, char k, short Q) const {
        if (Q == 1 || Q == 0) {
            return get_approx1d(k, Q).eval(T);
        } else {
            throw std::invalid_argument("invalid_value for Q");
        }
    }
 
    template <typename Container>
    void eval_sat_many(const Container& T, char k, short Q, Container& y) const {
        if (T.size() != y.size()) {
            throw std::invalid_argument("T and y are not the same size");
        }
        const auto& approx = get_approx1d(k, Q);
        for (auto i = 0; i < T.size(); ++i) {
            y(i) = approx.eval(T(i));
        }
    }
 
    template <typename Container>
    void eval_sat_manyC(const Container T[], std::size_t N, char k, short Q, Container y[]) const {
        // if (T.size() != y.size()){ throw std::invalid_argument("T and y are not the same size"); }
        const auto& approx = get_approx1d(k, Q);
        for (std::size_t i = 0; i < N; ++i) {
            y[i] = approx.eval(T[i]);
        }
    }
 
    auto solve_for_T(double propval, char k, bool Q, unsigned int bits = 64, unsigned int max_iter = 100, double boundsftol = 1e-13) const {
        return get_approx1d(k, Q).get_x_for_y(propval, bits, max_iter, boundsftol);
    }
 
    auto get_vaporquality(double T, double propval, char k) const {
        if (k == 'D') {
            // Need to special-case here because v = q*v_V + (1-q)*v_V but it is NOT(!!!!) the case that rho = q*rho_V + (1-q)*rho_L
            double v_L = 1 / get_approx1d('D', 0).eval(T);
            double v_V = 1 / get_approx1d('D', 1).eval(T);
            return (1 / propval - v_L) / (v_V - v_L);
        } else {
            double L = get_approx1d(k, 0).eval(T);
            double V = get_approx1d(k, 1).eval(T);
            return (propval - L) / (V - L);
        }
    }
 
    auto get_T_from_p(double p) {
        return get_invlnp().value().eval(log(p));
    }
 
    auto get_yval(double T, double q, char k) const {
 
        if (k == 'D') {
            // Need to special-case here because v = q*v_V + (1-q)*v_V but it is NOT(!!!!) the case that rho = q*rho_V + (1-q)*rho_L
            double v_L = 1 / get_approx1d('D', 0).eval(T);
            double v_V = 1 / get_approx1d('D', 1).eval(T);
            double v = q * v_V + (1 - q) * v_L;
            return 1 / v;  // rho = 1/v
        } else {
            double L = get_approx1d(k, 0).eval(T);
            double V = get_approx1d(k, 1).eval(T);
            return q * V + (1 - q) * L;
        }
    }
 
    template <typename Container>
    void get_yval_many(const Container& T, char k, const Container& q, Container& y) const {
        if (T.size() != y.size() || T.size() != q.size()) {
            throw std::invalid_argument("T, q, and y are not all the same size");
        }
 
        const auto& L = get_approx1d(k, 0);
        const auto& V = get_approx1d(k, 1);
 
        if (k == 'D') {
            for (auto i = 0; i < T.size(); ++i) {
                // Need to special-case here because v = q*v_V + (1-q)*v_V but it is NOT(!!!!) the case that rho = q*rho_V + (1-q)*rho_L
                double v_L = 1.0 / L.eval(T(i));
                double v_V = 1.0 / V.eval(T(i));
                double v = q(i) * v_V + (1 - q(i)) * v_L;
                y(i) = 1 / v;
            }
        } else {
            for (auto i = 0; i < T.size(); ++i) {
                y(i) = q(i) * V.eval(T(i)) + (1 - q(i)) * L.eval(T(i));
            }
        }
    }
    auto get_all_intersections(const char k, const double val, unsigned int bits, std::size_t max_iter, double boundsftol) const {
        const auto& L = get_approx1d(k, 0);
        const auto& V = get_approx1d(k, 1);
        auto TsatL = L.get_x_for_y(val, bits, max_iter, boundsftol);
        const auto TsatV = V.get_x_for_y(val, bits, max_iter, boundsftol);
        for (auto&& el : TsatV) {
            TsatL.push_back(el);
        }
        //        TsatL.insert(TsatL.end(),
        //                     std::make_move_iterator(TsatV.begin() + TsatV.size()),
        //                     std::make_move_iterator(TsatV.end()));
        return TsatL;
    }
 
    std::optional<SuperAncillaryTwoPhaseSolution> iterate_for_Tq_XY(double Tmin, double Tmax, char ch1, double val1, char ch2, double val2,
                                                                    unsigned int bits, std::size_t max_iter, double boundsftol) const {
 
        std::size_t counter = 0;
        auto f = [&](double T_) {
            counter++;
            double q_fromv1 = get_vaporquality(T_, val1, ch1);
            double resid = get_yval(T_, q_fromv1, ch2) - val2;
            return resid;
        };
        using namespace boost::math::tools;
        double fTmin = f(Tmin), fTmax = f(Tmax);
 
        // First we check if we are converged enough (TOMS748 does not stop based on function value)
        double T;
 
        // A little lambda to make it easier to return
        // in different logical branches
        auto returner = [&]() {
            SuperAncillaryTwoPhaseSolution soln;
            soln.T = T;
            soln.q = get_vaporquality(T, val1, ch1);
            soln.counter = counter;
            return soln;
        };
 
        if (std::abs(fTmin) < boundsftol) {
            T = Tmin;
            return returner();
        }
        if (std::abs(fTmax) < boundsftol) {
            T = Tmax;
            return returner();
        }
        if (fTmin * fTmax > 0) {
            // No sign change, this means that the inputs are not within the two-phase region
            // and thus no iteration is needed
            return std::nullopt;
        }
        // Neither bound meets the convergence criterion, we need to iterate on temperature
        try {
            boost::math::uintmax_t max_iter_ = static_cast<boost::math::uintmax_t>(max_iter);
            auto [l, r] = toms748_solve(f, Tmin, Tmax, fTmin, fTmax, eps_tolerance<double>(bits), max_iter_);
            T = (r + l) / 2.0;
            return returner();
        } catch (...) {
            std::cout << "fTmin,fTmax: " << fTmin << "," << fTmax << std::endl;
            throw;
        }
    }
 
    std::optional<SuperAncillaryTwoPhaseSolution> solve_for_Tq_DX(const double rho, const double propval, const char k, unsigned int bits,
                                                                  std::size_t max_iter, double boundsftol) const {
 
        const auto& Lrho = get_approx1d('D', 0);
        const auto& Vrho = get_approx1d('D', 1);
        auto Tsat = get_all_intersections(k, propval, bits, max_iter, boundsftol);
        std::size_t rhosat_soln_count = Tsat.size();
 
        std::tuple<double, double> Tsearchrange;
        if (rhosat_soln_count == 1) {
            // Search the complete range from Tmin to the intersection point where rhosat(T) = rho
            // obtained just above
            Tsearchrange = std::make_tuple(Lrho.xmin * 0.999, std::get<0>(Tsat[0]));
        } else if (rhosat_soln_count == 2) {
            double y1 = std::get<0>(Tsat[0]), y2 = std::get<0>(Tsat[1]);
            if (y2 < y1) {
                std::swap(y2, y2);
            }  // Required to be sorted in increasing value
            Tsearchrange = std::make_tuple(y1, y2);
        } else {
            throw std::invalid_argument("cannot have number of solutions other than 1 or 2; got " + std::to_string(rhosat_soln_count) + " solutions");
        }
        auto [a, b] = Tsearchrange;
        return iterate_for_Tq_XY(a, b, 'D', rho, k, propval, bits, max_iter, boundsftol);
    }
 
    template <typename Container>
    void solve_for_Tq_DX_many(const Container& rho, const Container& propval, const char k, unsigned int bits, std::size_t max_iter,
                              double boundsftol, Container& T, Container& q, Container& count) {
        if (std::set<std::size_t>({rho.size(), propval.size(), T.size(), q.size(), count.size()}).size() != 1) {
            throw std::invalid_argument("rho, propval, T, q, count are not all the same size");
        }
        for (auto i = 0U; i < T.size(); ++i) {
            auto osoln = solve_for_Tq_DX(rho(i), propval(i), k, bits, max_iter, boundsftol);
            if (osoln) {
                const auto& soln = osoln.value();
                T(i) = soln.T;
                q(i) = soln.q;
                count(i) = soln.counter;
            } else {
                T(i) = -1;
                q(i) = -1;
                count(i) = -1;
            }
        }
    }
 
    //     /**
    //     The high-level function used to carry out a solution. It handles all the different permutations of variables and delegates to lower-level functions to actually od the calculations
 
    //      \param pair The enumerated pair of thermodynamic variables being provided
    //      \param val1 The first value
    //      \param val2 The second value
    //      */
    //     auto flash(PropertyPairs pair, double val1, double val2) const -> std::optional<SuperAncillaryTwoPhaseSolution>{
    //         double T, q;
    //         std::size_t counter = 0;
    //         double boundsftol = 1e-12;
 
    //         auto returner = [&](){
    //             SuperAncillaryTwoPhaseSolution soln;
    //             soln.T = T;
    //             soln.q = q;
    //             soln.counter = counter;
    //             return soln;
    //         };
 
    //         auto get_T_other = [&](){
    //             switch (pair){
    //                 case PropertyPairs::DT: return std::make_tuple(val2, val1, 'D');
    //                 case PropertyPairs::HT: return std::make_tuple(val2, val1, 'H');
    //                 case PropertyPairs::ST: return std::make_tuple(val2, val1, 'S');
    //                 case PropertyPairs::TU: return std::make_tuple(val1, val2, 'U');
    //                 default:
    //                     throw std::invalid_argument("not valid");
    //             };
    //         };
    //         auto get_p_other = [&](){
    //             switch (pair){
    //                 case PropertyPairs::DP: return std::make_tuple(val2, val1, 'D');
    //                 case PropertyPairs::HP: return std::make_tuple(val2, val1, 'H');
    //                 case PropertyPairs::PS: return std::make_tuple(val1, val2, 'S');
    //                 case PropertyPairs::PU: return std::make_tuple(val1, val2, 'U');
    //                 default:
    //                     throw std::invalid_argument("not valid");
    //             };
    //         };
    //         auto get_rho_other = [&](){
    //             switch (pair){
    //                 case PropertyPairs::DH: return std::make_tuple(val1, val2, 'H');
    //                 case PropertyPairs::DS: return std::make_tuple(val1, val2, 'S');
    //                 case PropertyPairs::DU: return std::make_tuple(val1, val2, 'U');
    //                 default:
    //                     throw std::invalid_argument("not valid");
    //             };
    //         };
 
    //         // Given the arguments, unpack them into (xchar, xval, ychar, yval) where x is the
    //         // variable of convenience that will be used to do the interval intersection and then
    //         // the iteration will be in the other variable
    //         auto parse_HSU = [&](){
    //             switch (pair){
    //                 case PropertyPairs::HS: return std::make_tuple('S', val2, 'H', val1);
    //                 case PropertyPairs::SU: return std::make_tuple('S', val1, 'U', val2);
    //                 case PropertyPairs::HU: return std::make_tuple('H', val1, 'U', val2);
    //                 default:
    //                     throw std::invalid_argument("not valid");
    //             };
    //         };
 
    //         switch(pair){
    //             // Case 0, PT is always single-phase, by definition
    //             case PropertyPairs::PT: throw std::invalid_argument("PT inputs are trivial");
 
    //             // Case 1, T is a given variable
    //             case PropertyPairs::DT:
    //             case PropertyPairs::HT:
    //             case PropertyPairs::ST:
    //             case PropertyPairs::TU:
    //             {
    //                 auto [Tval, other, otherchar] = get_T_other();
    //                 q = get_vaporquality(Tval, other, otherchar);
    //                 T = Tval;
    //                 if (0 <= q && q <= 1){
    //                     return returner();
    //                 }
    //                 break;
    //             }
    //             // Case 2, p is a given variable, p gets converted to T, and then q calculated
    //             case PropertyPairs::DP:
    //             case PropertyPairs::HP:
    //             case PropertyPairs::PS:
    //             case PropertyPairs::PU:
    //             {
    //                 auto [p, other, otherchar] = get_p_other();
    //                 T = get_T_from_p(p);
    //                 q = get_vaporquality(T, other, otherchar); // Based on the T and the other variable
    //                 if (0 <= q && q <= 1){
    //                     return returner();
    //                 }
    //                 break;
    //             }
    //             // Case 3, rho is a given variable, special case that because it is relatively simple
    //             // and only water and heavy water have more than one density solution along the saturation curve
    //             case PropertyPairs::DH:
    //             case PropertyPairs::DS:
    //             case PropertyPairs::DU:
    //             {
    //                 auto [rho, otherval, otherchar] = get_rho_other();
    //                 auto optsoln = solve_for_Tq_DX(rho, otherval, otherchar, 64, 100, boundsftol);
    //                 if (optsoln){
    //                     const auto& soln = optsoln.value();
    //                     T = soln.T;
    //                     q = soln.q;
    //                     counter = soln.counter;
    //                     return returner();
    //                 }
    //                 break;
    //             }
    //             // Case 4, handle all the cases where complicated interval checks are
    //             // necessary
    //             case PropertyPairs::HS:
    //             case PropertyPairs::HU:
    //             case PropertyPairs::SU:
    //             {
    //                 auto [xchar, xval, ychar, yval] = parse_HSU();
    //                 auto Tsat = get_all_intersections(xchar, xval, 64, 100, boundsftol);
    //                 auto sort_predicate = [](const auto& x, const auto& y) {
    //                     return std::get<0>(x) < std::get<0>(y);
    //                 };
    //                 std::sort(Tsat.begin(), Tsat.end(), sort_predicate);
    // //                std::cout << Tsat.size() << " T crossings for " << xval << std::endl;
    // //                for (auto & el : Tsat){
    // //                    std::cout << std::get<0>(el) << std::endl;
    // //                }
    //                 if (Tsat.size() == 1){
    //                     // A single crossing, in which the other temperature bound for iteration
    //                     // is assumed to be the minimum temperature of the superancillary
    //                     double Tmin = get_approx1d('D', 0).xmin*0.9999;
    //                     double Tmax = std::get<0>(Tsat[0]);
    //                     auto o = iterate_for_Tq_XY(Tmin, Tmax, xchar, xval, ychar, yval, 64, 100, boundsftol);
    //                     if (o){ return o.value(); }
    //                 }
    //                 else if (Tsat.size() % 2 == 0){ // Even number of crossings
    //                     // neighboring intersections should bound the solution, or if not, the point is single-phase
    //                     for (auto i = 0; i < Tsat.size(); i += 2){
    //                         auto o = iterate_for_Tq_XY(std::get<0>(Tsat[i]), std::get<0>(Tsat[i+1]), xchar, xval, ychar, yval, 64, 100, boundsftol);
    //                         if (o){ return o.value(); }
    //                     }
    //                 }
    //                 else{
    //                     throw std::invalid_argument("Don't know what to do about this number of T crossings ("+std::to_string(Tsat.size())+") yet");
    //                 }
    //                 break;
    //             }
    //             default:
    //                 throw std::invalid_argument("These inputs are not yet supported in flash");
    //         }
    //         return std::nullopt;
    //     }
 
    //     /// A vectorized version of the flash function used in the Python interface for profiling
    //     template <typename Container>
    //     void flash_many(const PropertyPairs ppair, const Container& val1, const Container& val2, Container& T, Container& q, Container& count){
    //         if (std::set<std::size_t>({val1.size(), val2.size(), T.size(), q.size(), count.size()}).size() != 1){
    //             throw std::invalid_argument("val1, val2, T, q, count are not all the same size");
    //         }
    //         for (auto i = 0U; i < T.size(); ++i){
    //             try{
    //                 auto osoln = flash(ppair, val1(i), val2(i));
    //                 if (osoln){
    //                     const auto& soln = osoln.value();
    //                     T(i) = soln.T;
    //                     q(i) = soln.q;
    //                     count(i) = soln.counter;
    //                 }
    //                 else{
    //                     T(i) = -1;
    //                     q(i) = -1;
    //                     count(i) = -1;
    //                 }
    //             }
    //             catch(std::exception&e){
    // //                std::cout << e.what() << " for " << val1(i) << "," << val2(i) << std::endl;
    //                 T(i) = -2;
    //                 q(i) = -2;
    //                 count(i) = -2;
    //             }
 
    //         }
    //     }
};
 
static_assert(std::is_copy_constructible_v<SuperAncillary<std::vector<double>>>);
static_assert(std::is_copy_assignable_v<SuperAncillary<std::vector<double>>>);
 
}  // namespace superancillary
}  // namespace CoolProp