text: // Copyright (c) 2010-2021, Lawrence Livermore National Security, LLC. Produced
  // at the Lawrence Livermore National Laboratory. All Rights reserved. See files
  // LICENSE and NOTICE for details. LLNL-CODE-806117.
  //
  // This file is part of the MFEM library. For more information and source code
  // availability visit https://mfem.org.
  //
  // MFEM is free software; you can redistribute it and/or modify it under the
  // terms of the BSD-3 license. We welcome feedback and contributions, see file
  // CONTRIBUTING.md for details.
  
  // Implementation of IntegrationRule(s) classes
  
  // Acknowledgment: Some of the high-precision triangular and tetrahedral
  // quadrature rules below were obtained from the Encyclopaedia of Cubature
  // Formulas at http://nines.cs.kuleuven.be/research/ecf/ecf.html
  
  #include "fem.hpp"
  #include <cmath>
  
  #ifdef MFEM_USE_MPFR
  #include <mpfr.h>
  #endif
  
  using namespace std;
  
  namespace mfem
  {
  
  IntegrationRule::IntegrationRule(IntegrationRule &irx, IntegrationRule &iry)
  {
     int i, j, nx, ny;
  
     nx = irx.GetNPoints();
     ny = iry.GetNPoints();
     SetSize(nx * ny);
     SetPointIndices();
  
     for (j = 0; j < ny; j++)
     {
        IntegrationPoint &ipy = iry.IntPoint(j);
        for (i = 0; i < nx; i++)
        {
           IntegrationPoint &ipx = irx.IntPoint(i);
           IntegrationPoint &ip  = IntPoint(j*nx+i);
  
           ip.x = ipx.x;
           ip.y = ipy.x;
           ip.weight = ipx.weight * ipy.weight;
        }
     }
  }
  
  IntegrationRule::IntegrationRule(IntegrationRule &irx, IntegrationRule &iry,
                                   IntegrationRule &irz)
  {
     const int nx = irx.GetNPoints();
     const int ny = iry.GetNPoints();
     const int nz = irz.GetNPoints();
     SetSize(nx*ny*nz);
     SetPointIndices();
  
     for (int iz = 0; iz < nz; ++iz)
     {
        IntegrationPoint &ipz = irz.IntPoint(iz);
        for (int iy = 0; iy < ny; ++iy)
        {
           IntegrationPoint &ipy = iry.IntPoint(iy);
           for (int ix = 0; ix < nx; ++ix)
           {
              IntegrationPoint &ipx = irx.IntPoint(ix);
              IntegrationPoint &ip  = IntPoint(iz*nx*ny + iy*nx + ix);
  
              ip.x = ipx.x;
              ip.y = ipy.x;
              ip.z = ipz.x;
              ip.weight = ipx.weight*ipy.weight*ipz.weight;
           }
        }
     }
  }
  
  const Array<double> &IntegrationRule::GetWeights() const
  {
     if (weights.Size() != GetNPoints())
     {
        weights.SetSize(GetNPoints());
        for (int i = 0; i < GetNPoints(); i++)
        {
           weights[i] = IntPoint(i).weight;
        }
     }
     return weights;
  }
  
  void IntegrationRule::SetPointIndices()
  {
     for (int i = 0; i < Size(); i++)
     {
        IntPoint(i).index = i;
     }
  }
  
  void IntegrationRule::GrundmannMollerSimplexRule(int s, int n)
  {
     // for pow on older compilers
     using std::pow;
     const int d = 2*s + 1;
     Vector fact(d + n + 1);
     Array<int> beta(n), sums(n);
  
     fact(0) = 1.;
     for (int i = 1; i < fact.Size(); i++)
     {
        fact(i) = fact(i - 1)*i;
     }
  
     // number of points is \binom{n + s + 1}{n + 1}
     int np = 1, f = 1;
     for (int i = 0; i <= n; i++)
     {
        np *= (s + i + 1), f *= (i + 1);
     }
     np /= f;
     SetSize(np);
     SetPointIndices();
  
     int pt = 0;
     for (int i = 0; i <= s; i++)
     {
        double weight;
  
        weight = pow(2., -2*s)*pow(static_cast<double>(d + n - 2*i),
                                   d)/fact(i)/fact(d + n - i);
        if (i%2)
        {
           weight = -weight;
        }
  
        // loop over all beta : beta_0 + ... + beta_{n-1} <= s - i
        int k = s - i;
        beta = 0;
        sums = 0;
        while (true)
        {
           IntegrationPoint &ip = IntPoint(pt++);
           ip.weight = weight;
           ip.x = double(2*beta[0] + 1)/(d + n - 2*i);
           ip.y = double(2*beta[1] + 1)/(d + n - 2*i);
           if (n == 3)
           {
              ip.z = double(2*beta[2] + 1)/(d + n - 2*i);
           }
  
           int j = 0;
           while (sums[j] == k)
           {
              beta[j++] = 0;
              if (j == n)
              {
                 goto done_beta;
              }
           }
           beta[j]++;
           sums[j]++;
           for (j--; j >= 0; j--)
           {
              sums[j] = sums[j+1];
           }
        }
     done_beta:
        ;
     }
  }
  
  
  #ifdef MFEM_USE_MPFR
  
  // Class for computing hi-precision (HP) quadrature in 1D
  class HP_Quadrature1D
  {
  protected:
     mpfr_t pi, z, pp, p1, p2, p3, dz, w, rtol;
  
  public:
     static const mpfr_rnd_t rnd = GMP_RNDN;
     static const int default_prec = 128;
  
     // prec = MPFR precision in bits
     HP_Quadrature1D(const int prec = default_prec)
     {
        mpfr_inits2(prec, pi, z, pp, p1, p2, p3, dz, w, rtol, (mpfr_ptr) 0);
        mpfr_const_pi(pi, rnd);
        mpfr_set_si_2exp(rtol, 1, -32, rnd); // 2^(-32) < 2.33e-10
     }
  
     // set rtol = 2^exponent
     // this is a tolerance for the last correction of x_i in Newton's algorithm;
     // this gives roughly rtol^2 accuracy for the final x_i.
     void SetRelTol(const int exponent = -32)
     {
        mpfr_set_si_2exp(rtol, 1, exponent, rnd);
     }
  
     // n - number of quadrature points
     // k - index of the point to compute, 0 <= k < n
     // see also: QuadratureFunctions1D::GaussLegendre
     void ComputeGaussLegendrePoint(const int n, const int k)
     {
        MFEM_ASSERT(n > 0 && 0 <= k && k < n, "invalid n = " << n
                    << " and/or k = " << k);
  
        int i = (k < (n+1)/2) ? k+1 : n-k;
  
        // Initial guess for the x-coordinate:
        // set z = cos(pi * (i - 0.25) / (n + 0.5)) =
        //       = sin(pi * ((n+1-2*i) / (2*n+1)))
        mpfr_set_si(z, n+1-2*i, rnd);
        mpfr_div_si(z, z, 2*n+1, rnd);
        mpfr_mul(z, z, pi, rnd);
        mpfr_sin(z, z, rnd);
  
        bool done = false;
        while (1)
        {
           mpfr_set_si(p2, 1, rnd);
           mpfr_set(p1, z, rnd);
           for (int j = 2; j <= n; j++)
           {
              mpfr_set(p3, p2, rnd);
              mpfr_set(p2, p1, rnd);
              // p1 = ((2 * j - 1) * z * p2 - (j - 1) * p3) / j;
              mpfr_mul_si(p1, z, 2*j-1, rnd);
              mpfr_mul_si(p3, p3, j-1, rnd);
              mpfr_fms(p1, p1, p2, p3, rnd);
              mpfr_div_si(p1, p1, j, rnd);
           }
           // p1 is Legendre polynomial
  
           // derivative of the Legendre polynomial:
           // pp = n * (z*p1-p2) / (z*z - 1);
           mpfr_fms(pp, z, p1, p2, rnd);
           mpfr_mul_si(pp, pp, n, rnd);
           mpfr_sqr(p2, z, rnd);
           mpfr_sub_si(p2, p2, 1, rnd);
           mpfr_div(pp, pp, p2, rnd);
  
           if (done) { break; }
  
           // set delta_z: dz = p1/pp;
           mpfr_div(dz, p1, pp, rnd);
           // compute absolute tolerance: atol = rtol*(1-z)
           mpfr_t &atol = w;
           mpfr_si_sub(atol, 1, z, rnd);
           mpfr_mul(atol, atol, rtol, rnd);
           if (mpfr_cmpabs(dz, atol) <= 0)
           {
              done = true;
              // continue the computation: get pp at the new point, then exit
           }
           // update z = z - dz
           mpfr_sub(z, z, dz, rnd);
        }
  
        // map z to (0,1): z = (1 - z)/2
        mpfr_si_sub(z, 1, z, rnd);
        mpfr_div_2si(z, z, 1, rnd);
  
        // weight: w = 1/(4*z*(1 - z)*pp*pp)
        mpfr_sqr(w, pp, rnd);
        mpfr_mul_2si(w, w, 2, rnd);
        mpfr_mul(w, w, z, rnd);
        mpfr_si_sub(p1, 1, z, rnd); // p1 = 1-z
        mpfr_mul(w, w, p1, rnd);
        mpfr_si_div(w, 1, w, rnd);
  
        if (k >= (n+1)/2) { mpfr_swap(z, p1); }
     }
  
     // n - number of quadrature points
     // k - index of the point to compute, 0 <= k < n
     // see also: QuadratureFunctions1D::GaussLobatto
     void ComputeGaussLobattoPoint(const int n, const int k)
     {
        MFEM_ASSERT(n > 1 && 0 <= k && k < n, "invalid n = " << n
                    << " and/or k = " << k);
  
        int i = (k < (n+1)/2) ? k : n-1-k;
  
        if (i == 0)
        {
           mpfr_set_si(z, 0, rnd);
           mpfr_set_si(p1, 1, rnd);
           mpfr_set_si(w, n*(n-1), rnd);
           mpfr_si_div(w, 1, w, rnd); // weight = 1/(n*(n-1))
           return;
        }
        // initial guess is the corresponding Chebyshev point, z:
        //    z = -cos(pi * i/(n-1)) = sin(pi * (2*i-n+1)/(2*n-2))
        mpfr_set_si(z, 2*i-n+1, rnd);
        mpfr_div_si(z, z, 2*(n-1), rnd);
        mpfr_mul(z, pi, z, rnd);
        mpfr_sin(z, z, rnd);
        bool done = false;
        for (int iter = 0 ; true ; ++iter)
        {
           // build Legendre polynomials, up to P_{n}(z)
           mpfr_set_si(p1, 1, rnd);
           mpfr_set(p2, z, rnd);
  
           for (int l = 1 ; l < (n-1) ; ++l)
           {
              // P_{l+1}(x) = [ (2*l+1)*x*P_l(x) - l*P_{l-1}(x) ]/(l+1)
              mpfr_mul_si(p1, p1, l, rnd);
              mpfr_mul_si(p3, z, 2*l+1, rnd);
              mpfr_fms(p3, p3, p2, p1, rnd);
              mpfr_div_si(p3, p3, l+1, rnd);
  
              mpfr_set(p1, p2, rnd);
              mpfr_set(p2, p3, rnd);
           }
           if (done) { break; }
           // compute dz = resid/deriv = (z*p2 - p1) / (n*p2);
           mpfr_fms(dz, z, p2, p1, rnd);
           mpfr_mul_si(p3, p2, n, rnd);
           mpfr_div(dz, dz, p3, rnd);
           // update: z = z - dz
           mpfr_sub(z, z, dz, rnd);
           // compute absolute tolerance: atol = rtol*(1 + z)
           mpfr_t &atol = w;
           mpfr_add_si(atol, z, 1, rnd);
           mpfr_mul(atol, atol, rtol, rnd);
           // check for convergence
           if (mpfr_cmpabs(dz, atol) <= 0)
           {
              done = true;
              // continue the computation: get p2 at the new point, then exit
           }
           // If the iteration does not converge fast, something is wrong.
           MFEM_VERIFY(iter < 8, "n = " << n << ", i = " << i
                       << ", dz = " << mpfr_get_d(dz, rnd));
        }
        // Map to the interval [0,1] and scale the weights
        mpfr_add_si(z, z, 1, rnd);
        mpfr_div_2si(z, z, 1, rnd);
        // set the symmetric point
        mpfr_si_sub(p1, 1, z, rnd);
        // w = 1/[ n*(n-1)*[P_{n-1}(z)]^2 ]
        mpfr_sqr(w, p2, rnd);
        mpfr_mul_si(w, w, n*(n-1), rnd);
        mpfr_si_div(w, 1, w, rnd);
  
        if (k >= (n+1)/2) { mpfr_swap(z, p1); }
     }
  
     double GetPoint() const { return mpfr_get_d(z, rnd); }
     double GetSymmPoint() const { return mpfr_get_d(p1, rnd); }
     double GetWeight() const { return mpfr_get_d(w, rnd); }
  
     const mpfr_t &GetHPPoint() const { return z; }
     const mpfr_t &GetHPSymmPoint() const { return p1; }
     const mpfr_t &GetHPWeight() const { return w; }
  
     ~HP_Quadrature1D()
     {
        mpfr_clears(pi, z, pp, p1, p2, p3, dz, w, rtol, (mpfr_ptr) 0);
        mpfr_free_cache();
     }
  };
  
  #endif // MFEM_USE_MPFR
  
  
  void QuadratureFunctions1D::GaussLegendre(const int np, IntegrationRule* ir)
  {
     ir->SetSize(np);
     ir->SetPointIndices();
  
     switch (np)
     {
        case 1:
           ir->IntPoint(0).Set1w(0.5, 1.0);
           return;
        case 2:
           ir->IntPoint(0).Set1w(0.21132486540518711775, 0.5);
           ir->IntPoint(1).Set1w(0.78867513459481288225, 0.5);
           return;
        case 3:
           ir->IntPoint(0).Set1w(0.11270166537925831148, 5./18.);
           ir->IntPoint(1).Set1w(0.5, 4./9.);
           ir->IntPoint(2).Set1w(0.88729833462074168852, 5./18.);
           return;
     }
  
     const int n = np;
     const int m = (n+1)/2;
  
  #ifndef MFEM_USE_MPFR
  
     for (int i = 1; i <= m; i++)
     {
        double z = cos(M_PI * (i - 0.25) / (n + 0.5));
        double pp, p1, dz, xi = 0.;
        bool done = false;
        while (1)
        {
           double p2 = 1;
           p1 = z;
           for (int j = 2; j <= n; j++)
           {
              double p3 = p2;
              p2 = p1;
              p1 = ((2 * j - 1) * z * p2 - (j - 1) * p3) / j;
           }
           // p1 is Legendre polynomial
  
           pp = n * (z*p1-p2) / (z*z - 1);
           if (done) { break; }
  
           dz = p1/pp;
           if (fabs(dz) < 1e-16)
           {
              done = true;
              // map the new point (z-dz) to (0,1):
              xi = ((1 - z) + dz)/2; // (1 - (z - dz))/2 has bad round-off
              // continue the computation: get pp at the new point, then exit
           }
           // update: z = z - dz
           z -= dz;
        }
  
        ir->IntPoint(i-1).x = xi;
        ir->IntPoint(n-i).x = 1 - xi;
        ir->IntPoint(i-1).weight =
           ir->IntPoint(n-i).weight = 1./(4*xi*(1 - xi)*pp*pp);
     }
  
  #else // MFEM_USE_MPFR is defined
  
     HP_Quadrature1D hp_quad;
     for (int i = 1; i <= m; i++)
     {
        hp_quad.ComputeGaussLegendrePoint(n, i-1);
  
        ir->IntPoint(i-1).x = hp_quad.GetPoint();
        ir->IntPoint(n-i).x = hp_quad.GetSymmPoint();
        ir->IntPoint(i-1).weight = ir->IntPoint(n-i).weight = hp_quad.GetWeight();
     }
  
  #endif // MFEM_USE_MPFR
  
  }
  
  void QuadratureFunctions1D::GaussLobatto(const int np, IntegrationRule* ir)
  {
     /* An np point Gauss-Lobatto quadrature has (np - 2) free abscissa the other
        (2) abscissa are the interval endpoints.
  
        The interior x_i are the zeros of P'_{np-1}(x). The weights of the
        interior points on the interval [-1,1] are:
  
        w_i = 2/(np*(np-1)*[P_{np-1}(x_i)]^2)
  
        The end point weights (on [-1,1]) are: w_{end} = 2/(np*(np-1)).
  
        The interior abscissa are found via a nonlinear solve, the initial guess
        for each point is the corresponding Chebyshev point.
  
        After we find all points on the interval [-1,1], we will map and scale the
        points and weights to the MFEM natural interval [0,1].
  
        References:
        [1] E. E. Lewis and W. F. Millier, "Computational Methods of Neutron
            Transport", Appendix A
        [2] the QUADRULE software by John Burkardt,
            https://people.sc.fsu.edu/~jburkardt/cpp_src/quadrule/quadrule.cpp
     */
  
     ir->SetSize(np);
     ir->SetPointIndices();
     if ( np == 1 )
     {
        ir->IntPoint(0).Set1w(0.5, 1.0);
     }
     else
     {
  
  #ifndef MFEM_USE_MPFR
  
        // endpoints and respective weights
        ir->IntPoint(0).x = 0.0;
        ir->IntPoint(np-1).x = 1.0;
        ir->IntPoint(0).weight = ir->IntPoint(np-1).weight = 1.0/(np*(np-1));
  
        // interior points and weights
        // use symmetry and compute just half of the points
        for (int i = 1 ; i <= (np-1)/2 ; ++i)
        {
           // initial guess is the corresponding Chebyshev point, x_i:
           //    x_i = -cos(\pi * (i / (np-1)))
           double x_i = std::sin(M_PI * ((double)(i)/(np-1) - 0.5));
           double z_i = 0., p_l;
           bool done = false;
           for (int iter = 0 ; true ; ++iter)
           {
              // build Legendre polynomials, up to P_{np}(x_i)
              double p_lm1 = 1.0;
              p_l = x_i;
  
              for (int l = 1 ; l < (np-1) ; ++l)
              {
                 // The Legendre polynomials can be built by recursion:
                 // x * P_l(x) = 1/(2*l+1)*[ (l+1)*P_{l+1}(x) + l*P_{l-1} ], i.e.
                 // P_{l+1}(x) = [ (2*l+1)*x*P_l(x) - l*P_{l-1} ]/(l+1)
                 double p_lp1 = ( (2*l + 1)*x_i*p_l - l*p_lm1)/(l + 1);
  
                 p_lm1 = p_l;
                 p_l = p_lp1;
              }
              if (done) { break; }
              // after this loop, p_l holds P_{np-1}(x_i)
              // resid = (x^2-1)*P'_{np-1}(x_i)
              // but use the recurrence relationship
              // (x^2 -1)P'_l(x) = l*[ x*P_l(x) - P_{l-1}(x) ]
              // thus, resid = (np-1) * (x_i*p_l - p_lm1)
  
              // The derivative of the residual is:
              // \frac{d}{d x} \left[ (x^2 -1)P'_l(x) ] \right] =
              // l * (l+1) * P_l(x), with l = np-1,
              // therefore, deriv = np * (np-1) * p_l;
  
              // compute dx = resid/deriv
              double dx = (x_i*p_l - p_lm1) / (np*p_l);
              if (std::abs(dx) < 1e-16)
              {
                 done = true;
                 // Map the point to the interval [0,1]
                 z_i = ((1.0 + x_i) - dx)/2;
                 // continue the computation: get p_l at the new point, then exit
              }
              // If the iteration does not converge fast, something is wrong.
              MFEM_VERIFY(iter < 8, "np = " << np << ", i = " << i
                          << ", dx = " << dx);
              // update x_i:
              x_i -= dx;
           }
           // Map to the interval [0,1] and scale the weights
           IntegrationPoint &ip = ir->IntPoint(i);
           ip.x = z_i;
           // w_i = (2/[ n*(n-1)*[P_{n-1}(x_i)]^2 ]) / 2
           ip.weight = (double)(1.0 / (np*(np-1)*p_l*p_l));
  
           // set the symmetric point
           IntegrationPoint &symm_ip = ir->IntPoint(np-1-i);
           symm_ip.x = 1.0 - z_i;
           symm_ip.weight = ip.weight;
        }
  
  #else // MFEM_USE_MPFR is defined
  
        HP_Quadrature1D hp_quad;
        // use symmetry and compute just half of the points
        for (int i = 0 ; i <= (np-1)/2 ; ++i)
        {
           hp_quad.ComputeGaussLobattoPoint(np, i);
           ir->IntPoint(i).x = hp_quad.GetPoint();
           ir->IntPoint(np-1-i).x = hp_quad.GetSymmPoint();
           ir->IntPoint(i).weight =
              ir->IntPoint(np-1-i).weight = hp_quad.GetWeight();
        }
  
  #endif // MFEM_USE_MPFR
  
     }
  }
  
  void QuadratureFunctions1D::OpenUniform(const int np, IntegrationRule* ir)
  {
     ir->SetSize(np);
     ir->SetPointIndices();
  
     // The Newton-Cotes quadrature is based on weights that integrate exactly the
     // interpolatory polynomial through the equally spaced quadrature points.
     for (int i = 0; i < np ; ++i)
     {
        ir->IntPoint(i).x = double(i+1) / double(np + 1);
     }
  
     CalculateUniformWeights(ir, Quadrature1D::OpenUniform);
  }
  
  void QuadratureFunctions1D::ClosedUniform(const int np,
                                            IntegrationRule* ir)
  {
     ir->SetSize(np);
     ir->SetPointIndices();
     if ( np == 1 ) // allow this case as "closed"
     {
        ir->IntPoint(0).Set1w(0.5, 1.0);
        return;
     }
  
     for (int i = 0; i < np ; ++i)
     {
        ir->IntPoint(i).x = double(i) / (np-1);
     }
  
     CalculateUniformWeights(ir, Quadrature1D::ClosedUniform);
  }
  
  void QuadratureFunctions1D::OpenHalfUniform(const int np, IntegrationRule* ir)
  {
     ir->SetSize(np);
     ir->SetPointIndices();
  
     // Open half points: the centers of np uniform intervals
     for (int i = 0; i < np ; ++i)
     {
        ir->IntPoint(i).x = double(2*i+1) / (2*np);
     }
  
     CalculateUniformWeights(ir, Quadrature1D::OpenHalfUniform);
  }
  
  void QuadratureFunctions1D::ClosedGL(const int np, IntegrationRule* ir)
  {
     ir->SetSize(np);
     ir->SetPointIndices();
     ir->IntPoint(0).x = 0.0;
     ir->IntPoint(np-1).x = 1.0;
  
     if ( np > 2 )
     {
        IntegrationRule gl_ir;
        GaussLegendre(np-1, &gl_ir);
  
        for (int i = 1; i < np-1; ++i)
        {
           ir->IntPoint(i).x = (gl_ir.IntPoint(i-1).x + gl_ir.IntPoint(i).x)/2;
        }
     }
  
     CalculateUniformWeights(ir, Quadrature1D::ClosedGL);
  }
  
  void QuadratureFunctions1D::GivePolyPoints(const int np, double *pts,
                                             const int type)
  {
     IntegrationRule ir(np);
  
     switch (type)
     {
        case Quadrature1D::GaussLegendre:
        {
           GaussLegendre(np,&ir);
           break;
        }
        case Quadrature1D::GaussLobatto:
        {
           GaussLobatto(np, &ir);
           break;
        }
        case Quadrature1D::OpenUniform:
        {
           OpenUniform(np,&ir);
           break;
        }
        case Quadrature1D::ClosedUniform:
        {
           ClosedUniform(np,&ir);
           break;
        }
        case Quadrature1D::OpenHalfUniform:
        {
           OpenHalfUniform(np, &ir);
           break;
        }
        case Quadrature1D::ClosedGL:
        {
           ClosedGL(np, &ir);
           break;
        }
        default:
        {
           MFEM_ABORT("Asking for an unknown type of 1D Quadrature points, "
                      "type = " << type);
        }
     }
  
     for (int i = 0 ; i < np ; ++i)
     {
        pts[i] = ir.IntPoint(i).x;
     }
  }
  
  void QuadratureFunctions1D::CalculateUniformWeights(IntegrationRule *ir,
                                                      const int type)
  {
     /* The Lagrange polynomials are:
             p_i = \prod_{j \neq i} {\frac{x - x_j }{x_i - x_j}}
  
        The weight associated with each abscissa is the integral of p_i over
        [0,1]. To calculate the integral of p_i, we use a Gauss-Legendre
        quadrature rule. This approach does not suffer from bad round-off/
        cancellation errors for large number of points.
     */
     const int n = ir->Size();
     switch (n)
     {
        case 1:
           ir->IntPoint(0).weight = 1.;
           return;
        case 2:
           ir->IntPoint(0).weight = .5;
           ir->IntPoint(1).weight = .5;
           return;
     }
  
  #ifndef MFEM_USE_MPFR
  
     // This algorithm should work for any set of points, not just uniform
     const IntegrationRule &glob_ir = IntRules.Get(Geometry::SEGMENT, n-1);
     const int m = glob_ir.GetNPoints();
     Vector xv(n);
     for (int j = 0; j < n; j++)
     {
        xv(j) = ir->IntPoint(j).x;
     }
     Poly_1D::Basis basis(n-1, xv.GetData()); // nodal basis, with nodes at 'xv'
     Vector w(n);
     // Integrate all nodal basis functions using 'glob_ir':
     w = 0.0;
     for (int i = 0; i < m; i++)
     {
        const IntegrationPoint &ip = glob_ir.IntPoint(i);
        basis.Eval(ip.x, xv);
        w.Add(ip.weight, xv); // w += ip.weight * xv
     }
     for (int j = 0; j < n; j++)
     {
        ir->IntPoint(j).weight = w(j);
     }
  
  #else // MFEM_USE_MPFR is defined
  
     static const mpfr_rnd_t rnd = HP_Quadrature1D::rnd;
     HP_Quadrature1D hp_quad;
     mpfr_t l, lk, w0, wi, tmp, *weights;
     mpfr_inits2(hp_quad.default_prec, l, lk, w0, wi, tmp, (mpfr_ptr) 0);
     weights = new mpfr_t[n];
     for (int i = 0; i < n; i++)
     {
        mpfr_init2(weights[i], hp_quad.default_prec);
        mpfr_set_si(weights[i], 0, rnd);
     }
     hp_quad.SetRelTol(-48); // rtol = 2^(-48) ~ 3.5e-15
     const int p = n-1;
     const int m = p/2+1; // number of points for Gauss-Legendre quadrature
     int hinv = 0, ihoffset = 0; // x_i = (i+ihoffset/2)/hinv
     switch (type)
     {
        case Quadrature1D::ClosedUniform:
           // x_i = i/p, i=0,...,p
           hinv = p;
           ihoffset = 0;
           break;
        case Quadrature1D::OpenUniform:
           // x_i = (i+1)/(p+2), i=0,...,p
           hinv = p+2;
           ihoffset = 2;
           break;
        case Quadrature1D::OpenHalfUniform:
           // x_i = (i+1/2)/(p+1), i=0,...,p
           hinv = p+1;
           ihoffset = 1;
           break;
        default:
           MFEM_ABORT("invalid Quadrature1D type: " << type);
     }
     // set w0 = (-1)^p*(p!)/(hinv^p)
     mpfr_fac_ui(w0, p, rnd);
     mpfr_ui_pow_ui(tmp, hinv, p, rnd);
     mpfr_div(w0, w0, tmp, rnd);
     if (p%2) { mpfr_neg(w0, w0, rnd); }
  
     for (int j = 0; j < m; j++)
     {
        hp_quad.ComputeGaussLegendrePoint(m, j);
  
        // Compute l = \prod_{i=0}^p (x-x_i) and lk = l/(x-x_k), where
        // x = hp_quad.GetHPPoint(), x_i = (i+ihoffset/2)/hinv, and x_k is the
        // node closest to x, i.e. k = min(max(round(x*hinv-ihoffset/2),0),p)
        mpfr_mul_si(tmp, hp_quad.GetHPPoint(), hinv, rnd);
        mpfr_sub_d(tmp, tmp, 0.5*ihoffset, rnd);
        mpfr_round(tmp, tmp);
        int k = min(max((int)mpfr_get_si(tmp, rnd), 0), p);
        mpfr_set_si(lk, 1, rnd);
        for (int i = 0; i <= p; i++)
        {
           mpfr_set_si(tmp, 2*i+ihoffset, rnd);
           mpfr_div_si(tmp, tmp, 2*hinv, rnd);
           mpfr_sub(tmp, hp_quad.GetHPPoint(), tmp, rnd);
           if (i != k)
           {
              mpfr_mul(lk, lk, tmp, rnd);
           }
           else
           {
              mpfr_set(l, tmp, rnd);
           }
        }
        mpfr_mul(l, l, lk, rnd);
        mpfr_set(wi, w0, rnd);
        for (int i = 0; true; i++)
        {
           if (i != k)
           {
              // tmp = l/(wi*(x - x_i))
              mpfr_set_si(tmp, 2*i+ihoffset, rnd);
              mpfr_div_si(tmp, tmp, 2*hinv, rnd);
              mpfr_sub(tmp, hp_quad.GetHPPoint(), tmp, rnd);
              mpfr_mul(tmp, tmp, wi, rnd);
              mpfr_div(tmp, l, tmp, rnd);
           }
           else
           {
              // tmp = lk/wi
              mpfr_div(tmp, lk, wi, rnd);
           }
           // weights[i] += hp_quad.weight*tmp
           mpfr_mul(tmp, tmp, hp_quad.GetHPWeight(), rnd);
           mpfr_add(weights[i], weights[i], tmp, rnd);
  
           if (i == p) { break; }
  
           // update wi *= (i+1)/(i-p)
           mpfr_mul_si(wi, wi, i+1, rnd);
           mpfr_div_si(wi, wi, i-p, rnd);
        }
     }
     for (int i = 0; i < n; i++)
     {
        ir->IntPoint(i).weight = mpfr_get_d(weights[i], rnd);
        mpfr_clear(weights[i]);
     }
     delete [] weights;
     mpfr_clears(l, lk, w0, wi, tmp, (mpfr_ptr) 0);
  
  #endif // MFEM_USE_MPFR
  
  }
  
  
  int Quadrature1D::CheckClosed(int type)
  {
     switch (type)
     {
        case GaussLobatto:
        case ClosedUniform:
           return type;
        default:
           return Invalid;
     }
  }
  
  int Quadrature1D::CheckOpen(int type)
  {
     switch (type)
     {
        case GaussLegendre:
        case GaussLobatto:
        case OpenUniform:
        case ClosedUniform:
        case OpenHalfUniform:
           return type; // all types can work as open
        default:
           return Invalid;
     }
  }
  
  
  IntegrationRules IntRules(0, Quadrature1D::GaussLegendre);
  
  IntegrationRules RefinedIntRules(1, Quadrature1D::GaussLegendre);
  
  IntegrationRules::IntegrationRules(int Ref, int type_):
     quad_type(type_)
  {
     refined = Ref;
  
     if (refined < 0) { own_rules = 0; return; }
  
     own_rules = 1;
  
     const MemoryType h_mt = MemoryType::HOST;
     PointIntRules.SetSize(2, h_mt);
     PointIntRules = NULL;
  
     SegmentIntRules.SetSize(32, h_mt);
     SegmentIntRules = NULL;
  
     // TriangleIntegrationRule() assumes that this size is >= 26
     TriangleIntRules.SetSize(32, h_mt);
     TriangleIntRules = NULL;
  
     SquareIntRules.SetSize(32, h_mt);
     SquareIntRules = NULL;
  
     // TetrahedronIntegrationRule() assumes that this size is >= 10
     TetrahedronIntRules.SetSize(32, h_mt);
     TetrahedronIntRules = NULL;
  
     PyramidIntRules.SetSize(32, h_mt);
     PyramidIntRules = NULL;
  
     PrismIntRules.SetSize(32, h_mt);
     PrismIntRules = NULL;
  
     CubeIntRules.SetSize(32, h_mt);
     CubeIntRules = NULL;
  }
  
  const IntegrationRule &IntegrationRules::Get(int GeomType, int Order)
  {
     Array<IntegrationRule *> *ir_array;
  
     switch (GeomType)
     {
        case Geometry::POINT:       ir_array = &PointIntRules; Order = 0; break;
        case Geometry::SEGMENT:     ir_array = &SegmentIntRules; break;
        case Geometry::TRIANGLE:    ir_array = &TriangleIntRules; break;
        case Geometry::SQUARE:      ir_array = &SquareIntRules; break;
        case Geometry::TETRAHEDRON: ir_array = &TetrahedronIntRules; break;
        case Geometry::CUBE:        ir_array = &CubeIntRules; break;
        case Geometry::PRISM:       ir_array = &PrismIntRules; break;
        case Geometry::PYRAMID:     ir_array = &PyramidIntRules; break;
        default:
           mfem_error("IntegrationRules::Get(...) : Unknown geometry type!");
           ir_array = NULL;
     }
  
     if (Order < 0)
     {
        Order = 0;
     }
  
     if (!HaveIntRule(*ir_array, Order))
     {
  #ifdef MFEM_USE_LEGACY_OPENMP
        #pragma omp critical
  #endif
        {
           if (!HaveIntRule(*ir_array, Order))
           {
              IntegrationRule *ir = GenerateIntegrationRule(GeomType, Order);
              int RealOrder = Order;
              while (RealOrder+1 < ir_array->Size() &&
              /*  */ (*ir_array)[RealOrder+1] == ir)
              {
                 RealOrder++;
              }
              ir->SetOrder(RealOrder);
           }
        }
     }
  
     return *(*ir_array)[Order];
  }
  
  void IntegrationRules::Set(int GeomType, int Order, IntegrationRule &IntRule)
  {
     Array<IntegrationRule *> *ir_array;
  
     switch (GeomType)
     {
        case Geometry::POINT:       ir_array = &PointIntRules; break;
        case Geometry::SEGMENT:     ir_array = &SegmentIntRules; break;
        case Geometry::TRIANGLE:    ir_array = &TriangleIntRules; break;
        case Geometry::SQUARE:      ir_array = &SquareIntRules; break;
        case Geometry::TETRAHEDRON: ir_array = &TetrahedronIntRules; break;
        case Geometry::CUBE:        ir_array = &CubeIntRules; break;
        case Geometry::PRISM:       ir_array = &PrismIntRules; break;
        case Geometry::PYRAMID:     ir_array = &PyramidIntRules; break;
        default:
           mfem_error("IntegrationRules::Set(...) : Unknown geometry type!");
           ir_array = NULL;
     }
  
     if (HaveIntRule(*ir_array, Order))
     {
        MFEM_ABORT("Overwriting set rules is not supported!");
     }
  
     AllocIntRule(*ir_array, Order);
  
     (*ir_array)[Order] = &IntRule;
  }
  
  void IntegrationRules::DeleteIntRuleArray(Array<IntegrationRule *> &ir_array)
  {
     int i;
     IntegrationRule *ir = NULL;
  
     // Many of the intrules have multiple contiguous copies in the ir_array
     // so we have to be careful to not delete them twice.
     for (i = 0; i < ir_array.Size(); i++)
     {
        if (ir_array[i] != NULL && ir_array[i] != ir)
        {
           ir = ir_array[i];
           delete ir;
        }
     }
  }
  
  IntegrationRules::~IntegrationRules()
  {
     if (!own_rules) { return; }
  
     DeleteIntRuleArray(PointIntRules);
     DeleteIntRuleArray(SegmentIntRules);
     DeleteIntRuleArray(TriangleIntRules);
     DeleteIntRuleArray(SquareIntRules);
     DeleteIntRuleArray(TetrahedronIntRules);
     DeleteIntRuleArray(CubeIntRules);
     DeleteIntRuleArray(PrismIntRules);
     DeleteIntRuleArray(PyramidIntRules);
  }
  
  
  IntegrationRule *IntegrationRules::GenerateIntegrationRule(int GeomType,
                                                             int Order)
  {
     switch (GeomType)
     {
        case Geometry::POINT:
           return PointIntegrationRule(Order);
        case Geometry::SEGMENT:
           return SegmentIntegrationRule(Order);
        case Geometry::TRIANGLE:
           return TriangleIntegrationRule(Order);
        case Geometry::SQUARE:
           return SquareIntegrationRule(Order);
        case Geometry::TETRAHEDRON:
           return TetrahedronIntegrationRule(Order);
        case Geometry::CUBE:
           return CubeIntegrationRule(Order);
        case Geometry::PRISM:
           return PrismIntegrationRule(Order);
        case Geometry::PYRAMID:
           return PyramidIntegrationRule(Order);
        default:
           mfem_error("IntegrationRules::Set(...) : Unknown geometry type!");
           return NULL;
     }
  }
  
  
  // Integration rules for a point
  IntegrationRule *IntegrationRules::PointIntegrationRule(int Order)
  {
     if (Order > 1)
     {
        mfem_error("Point Integration Rule of Order > 1 not defined");
        return NULL;
     }
  
     IntegrationRule *ir = new IntegrationRule(1);
     ir->IntPoint(0).x = .0;
     ir->IntPoint(0).weight = 1.;
  
     PointIntRules[1] = PointIntRules[0] = ir;
  
     return ir;
  }
  
  // Integration rules for line segment [0,1]
  IntegrationRule *IntegrationRules::SegmentIntegrationRule(int Order)
  {
     int RealOrder = GetSegmentRealOrder(Order); // RealOrder >= Order
     // Order is one of {RealOrder-1,RealOrder}
     AllocIntRule(SegmentIntRules, RealOrder);
  
     IntegrationRule tmp, *ir;
     ir = refined ? &tmp : new IntegrationRule;
  
     int n = 0;
     // n is the number of points to achieve the exact integral of a
     // degree Order polynomial
     switch (quad_type)
     {
        case Quadrature1D::GaussLegendre:
        {
           // Gauss-Legendre is exact for 2*n-1
           n = Order/2 + 1;
           QuadratureFunctions1D::GaussLegendre(n, ir);
           break;
        }
        case Quadrature1D::GaussLobatto:
        {
           // Gauss-Lobatto is exact for 2*n-3
           n = Order/2 + 2;
           QuadratureFunctions1D::GaussLobatto(n, ir);
           break;
        }
        case Quadrature1D::OpenUniform:
        {
           // Open Newton Cotes is exact for n-(n+1)%2 = n-1+n%2
           n = Order | 1; // n is always odd
           QuadratureFunctions1D::OpenUniform(n, ir);
           break;
        }
        case Quadrature1D::ClosedUniform:
        {
           // Closed Newton Cotes is exact for n-(n+1)%2 = n-1+n%2
           n = Order | 1; // n is always odd
           QuadratureFunctions1D::ClosedUniform(n, ir);
           break;
        }
        case Quadrature1D::OpenHalfUniform:
        {
           // Open half Newton Cotes is exact for n-(n+1)%2 = n-1+n%2
           n = Order | 1; // n is always odd
           QuadratureFunctions1D::OpenHalfUniform(n, ir);
           break;
        }
        default:
        {
           MFEM_ABORT("unknown Quadrature1D type: " << quad_type);
        }
     }
     if (refined)
     {
        // Effectively passing memory management to SegmentIntegrationRules
        ir = new IntegrationRule(2*n);
        for (int j = 0; j < n; j++)
        {
           ir->IntPoint(j).x = tmp.IntPoint(j).x/2.0;
           ir->IntPoint(j).weight = tmp.IntPoint(j).weight/2.0;
           ir->IntPoint(j+n).x = 0.5 + tmp.IntPoint(j).x/2.0;
           ir->IntPoint(j+n).weight = tmp.IntPoint(j).weight/2.0;
        }
     }
     SegmentIntRules[RealOrder-1] = SegmentIntRules[RealOrder] = ir;
     return ir;
  }
  
  // Integration rules for reference triangle {[0,0],[1,0],[0,1]}
  IntegrationRule *IntegrationRules::TriangleIntegrationRule(int Order)
  {
     IntegrationRule *ir = NULL;
     // Note: Set TriangleIntRules[*] to ir only *after* ir is fully constructed.
     // This is needed in multithreaded environment.
  
     // assuming that orders <= 25 are pre-allocated
     switch (Order)
     {
        case 0:  // 1 point - 0 degree
        case 1:
           ir = new IntegrationRule(1);
           ir->AddTriMidPoint(0, 0.5);
           TriangleIntRules[0] = TriangleIntRules[1] = ir;
           return ir;
  
        case 2:  // 3 point - 2 degree
           ir = new IntegrationRule(3);
           ir->AddTriPoints3(0, 1./6., 1./6.);
           TriangleIntRules[2] = ir;
           // interior points
           return ir;
  
        case 3:  // 4 point - 3 degree (has one negative weight)
           ir = new IntegrationRule(4);
           ir->AddTriMidPoint(0, -0.28125); // -9./32.
           ir->AddTriPoints3(1, 0.2, 25./96.);
           TriangleIntRules[3] = ir;
           return ir;
  
        case 4:  // 6 point - 4 degree
           ir = new IntegrationRule(6);
           ir->AddTriPoints3(0, 0.091576213509770743460, 0.054975871827660933819);
           ir->AddTriPoints3(3, 0.44594849091596488632, 0.11169079483900573285);
           TriangleIntRules[4] = ir;
           return ir;
  
        case 5:  // 7 point - 5 degree
           ir = new IntegrationRule(7);
           ir->AddTriMidPoint(0, 0.1125);
           ir->AddTriPoints3(1, 0.10128650732345633880, 0.062969590272413576298);
           ir->AddTriPoints3(4, 0.47014206410511508977, 0.066197076394253090369);
           TriangleIntRules[5] = ir;
           return ir;
  
        case 6:  // 12 point - 6 degree
           ir = new IntegrationRule(12);
           ir->AddTriPoints3(0, 0.063089014491502228340, 0.025422453185103408460);
           ir->AddTriPoints3(3, 0.24928674517091042129, 0.058393137863189683013);
           ir->AddTriPoints6(6, 0.053145049844816947353, 0.31035245103378440542,
                             0.041425537809186787597);
           TriangleIntRules[6] = ir;
           return ir;
  
        case 7:  // 12 point - degree 7
           ir = new IntegrationRule(12);
           ir->AddTriPoints3R(0, 0.062382265094402118174, 0.067517867073916085443,
                              0.026517028157436251429);
           ir->AddTriPoints3R(3, 0.055225456656926611737, 0.32150249385198182267,
                              0.043881408714446055037);
           // slightly better with explicit 3rd area coordinate
           ir->AddTriPoints3R(6, 0.034324302945097146470, 0.66094919618673565761,
                              0.30472650086816719592, 0.028775042784981585738);
           ir->AddTriPoints3R(9, 0.51584233435359177926, 0.27771616697639178257,
                              0.20644149867001643817, 0.067493187009802774463);
           TriangleIntRules[7] = ir;
           return ir;
  
        case 8:  // 16 point - 8 degree
           ir = new IntegrationRule(16);
           ir->AddTriMidPoint(0, 0.0721578038388935841255455552445323);
           ir->AddTriPoints3(1, 0.170569307751760206622293501491464,
                             0.0516086852673591251408957751460645);
           ir->AddTriPoints3(4, 0.0505472283170309754584235505965989,
                             0.0162292488115990401554629641708902);
           ir->AddTriPoints3(7, 0.459292588292723156028815514494169,
                             0.0475458171336423123969480521942921);
           ir->AddTriPoints6(10, 0.008394777409957605337213834539296,
                             0.263112829634638113421785786284643,
                             0.0136151570872174971324223450369544);
           TriangleIntRules[8] = ir;
           return ir;
  
        case 9:  // 19 point - 9 degree
           ir = new IntegrationRule(19);
           ir->AddTriMidPoint(0, 0.0485678981413994169096209912536443);
           ir->AddTriPoints3b(1, 0.020634961602524744433,
                              0.0156673501135695352684274156436046);
           ir->AddTriPoints3b(4, 0.12582081701412672546,
                              0.0389137705023871396583696781497019);
           ir->AddTriPoints3(7, 0.188203535619032730240961280467335,
                             0.0398238694636051265164458871320226);
           ir->AddTriPoints3(10, 0.0447295133944527098651065899662763,
                             0.0127888378293490156308393992794999);
           ir->AddTriPoints6(13, 0.0368384120547362836348175987833851,
                             0.2219629891607656956751025276931919,
                             0.0216417696886446886446886446886446);
           TriangleIntRules[9] = ir;
           return ir;
  
        case 10:  // 25 point - 10 degree
           ir = new IntegrationRule(25);
           ir->AddTriMidPoint(0, 0.0454089951913767900476432975500142);
           ir->AddTriPoints3b(1, 0.028844733232685245264984935583748,
                              0.0183629788782333523585030359456832);
           ir->AddTriPoints3(4, 0.109481575485037054795458631340522,
                             0.0226605297177639673913028223692986);
           ir->AddTriPoints6(7, 0.141707219414879954756683250476361,
                             0.307939838764120950165155022930631,
                             0.0363789584227100543021575883096803);
           ir->AddTriPoints6(13, 0.025003534762686386073988481007746,
                             0.246672560639902693917276465411176,
                             0.0141636212655287424183685307910495);
           ir->AddTriPoints6(19, 0.0095408154002994575801528096228873,
                             0.0668032510122002657735402127620247,
                             4.71083348186641172996373548344341E-03);
           TriangleIntRules[10] = ir;
           return ir;
  
        case 11: // 28 point -- 11 degree
           ir = new IntegrationRule(28);
           ir->AddTriPoints6(0, 0.0,
                             0.141129718717363295960826061941652,
                             3.68119189165027713212944752369032E-03);
           ir->AddTriMidPoint(6, 0.0439886505811161193990465846607278);
           ir->AddTriPoints3(7, 0.0259891409282873952600324854988407,
                             4.37215577686801152475821439991262E-03);
           ir->AddTriPoints3(10, 0.0942875026479224956305697762754049,
                             0.0190407859969674687575121697178070);
           ir->AddTriPoints3b(13, 0.010726449965572372516734795387128,
                              9.42772402806564602923839129555767E-03);
           ir->AddTriPoints3(16, 0.207343382614511333452934024112966,
                             0.0360798487723697630620149942932315);
           ir->AddTriPoints3b(19, 0.122184388599015809877869236727746,
                              0.0346645693527679499208828254519072);
           ir->AddTriPoints6(22, 0.0448416775891304433090523914688007,
                             0.2772206675282791551488214673424523,
                             0.0205281577146442833208261574536469);
           TriangleIntRules[11] = ir;
           return ir;
  
        case 12: // 33 point - 12 degree
           ir = new IntegrationRule(33);
           ir->AddTriPoints3b(0, 2.35652204523900E-02, 1.28655332202275E-02);
           ir->AddTriPoints3b(3, 1.20551215411079E-01, 2.18462722690190E-02);
           ir->AddTriPoints3(6, 2.71210385012116E-01, 3.14291121089425E-02);
           ir->AddTriPoints3(9, 1.27576145541586E-01, 1.73980564653545E-02);
           ir->AddTriPoints3(12, 2.13173504532100E-02, 3.08313052577950E-03);
           ir->AddTriPoints6(15, 1.15343494534698E-01, 2.75713269685514E-01,
                             2.01857788831905E-02);
           ir->AddTriPoints6(21, 2.28383322222570E-02, 2.81325580989940E-01,
                             1.11783866011515E-02);
           ir->AddTriPoints6(27, 2.57340505483300E-02, 1.16251915907597E-01,
                             8.65811555432950E-03);
           TriangleIntRules[12] = ir;
           return ir;
  
        case 13: // 37 point - 13 degree
           ir = new IntegrationRule(37);
           ir->AddTriPoints3b(0, 0.0,
                              2.67845189554543044455908674650066E-03);
           ir->AddTriMidPoint(3, 0.0293480398063595158995969648597808);
           ir->AddTriPoints3(4, 0.0246071886432302181878499494124643,
                             3.92538414805004016372590903990464E-03);
           ir->AddTriPoints3b(7, 0.159382493797610632566158925635800,
                              0.0253344765879434817105476355306468);
           ir->AddTriPoints3(10, 0.227900255506160619646298948153592,
                             0.0250401630452545330803738542916538);
           ir->AddTriPoints3(13, 0.116213058883517905247155321839271,
                             0.0158235572961491595176634480481793);
           ir->AddTriPoints3b(16, 0.046794039901841694097491569577008,
                              0.0157462815379843978450278590138683);
           ir->AddTriPoints6(19, 0.0227978945382486125477207592747430,
                             0.1254265183163409177176192369310890,
                             7.90126610763037567956187298486575E-03);
           ir->AddTriPoints6(25, 0.0162757709910885409437036075960413,
                             0.2909269114422506044621801030055257,
                             7.99081889046420266145965132482933E-03);
           ir->AddTriPoints6(31, 0.0897330604516053590796290561145196,
                             0.2723110556841851025078181617634414,
                             0.0182757511120486476280967518782978);
           TriangleIntRules[13] = ir;
           return ir;
  
        case 14: // 42 point - 14 degree
           ir = new IntegrationRule(42);
           ir->AddTriPoints3b(0, 2.20721792756430E-02, 1.09417906847145E-02);
           ir->AddTriPoints3b(3, 1.64710561319092E-01, 1.63941767720625E-02);
           ir->AddTriPoints3(6, 2.73477528308839E-01, 2.58870522536460E-02);
           ir->AddTriPoints3(9, 1.77205532412543E-01, 2.10812943684965E-02);
           ir->AddTriPoints3(12, 6.17998830908730E-02, 7.21684983488850E-03);
           ir->AddTriPoints3(15, 1.93909612487010E-02, 2.46170180120000E-03);
           ir->AddTriPoints6(18, 5.71247574036480E-02, 1.72266687821356E-01,
                             1.23328766062820E-02);
           ir->AddTriPoints6(24, 9.29162493569720E-02, 3.36861459796345E-01,
                             1.92857553935305E-02);
           ir->AddTriPoints6(30, 1.46469500556540E-02, 2.98372882136258E-01,
                             7.21815405676700E-03);
           ir->AddTriPoints6(36, 1.26833093287200E-03, 1.18974497696957E-01,
                             2.50511441925050E-03);
           TriangleIntRules[14] = ir;
           return ir;
  
        case 15: // 54 point - 15 degree
           ir = new IntegrationRule(54);
           ir->AddTriPoints3b(0, 0.0834384072617499333, 0.016330909424402645);
           ir->AddTriPoints3b(3, 0.192779070841738867, 0.01370640901568218);
           ir->AddTriPoints3(6, 0.293197167913025367, 0.01325501829935165);
           ir->AddTriPoints3(9, 0.146467786942772933, 0.014607981068243055);
           ir->AddTriPoints3(12, 0.0563628676656034333, 0.005292304033121995);
           ir->AddTriPoints3(15, 0.0165751268583703333, 0.0018073215320460175);
           ir->AddTriPoints6(18, 0.0099122033092248, 0.239534554154794445,
                             0.004263874050854718);
           ir->AddTriPoints6(24, 0.015803770630228, 0.404878807318339958,
                             0.006958088258345965);
           ir->AddTriPoints6(30, 0.00514360881697066667, 0.0950021131130448885,
                             0.0021459664703674175);
           ir->AddTriPoints6(36, 0.0489223257529888, 0.149753107322273969,
                             0.008117664640887445);
           ir->AddTriPoints6(42, 0.0687687486325192, 0.286919612441334979,
                             0.012803670460631195);
           ir->AddTriPoints6(48, 0.1684044181246992, 0.281835668099084562,
                             0.016544097765822835);
           TriangleIntRules[15] = ir;
           return ir;
  
        case 16:  // 61 point - 17 degree (used for 16 as well)
        case 17:
           ir = new IntegrationRule(61);
           ir->AddTriMidPoint(0,  1.67185996454015E-02);
           ir->AddTriPoints3b(1,  5.65891888645200E-03, 2.54670772025350E-03);
           ir->AddTriPoints3b(4,  3.56473547507510E-02, 7.33543226381900E-03);
           ir->AddTriPoints3b(7,  9.95200619584370E-02, 1.21754391768360E-02);
           ir->AddTriPoints3b(10, 1.99467521245206E-01, 1.55537754344845E-02);
           ir->AddTriPoints3 (13, 2.52141267970953E-01, 1.56285556093100E-02);
           ir->AddTriPoints3 (16, 1.62047004658461E-01, 1.24078271698325E-02);
           ir->AddTriPoints3 (19, 7.58758822607460E-02, 7.02803653527850E-03);
           ir->AddTriPoints3 (22, 1.56547269678220E-02, 1.59733808688950E-03);
           ir->AddTriPoints6 (25, 1.01869288269190E-02, 3.34319867363658E-01,
                              4.05982765949650E-03);
           ir->AddTriPoints6 (31, 1.35440871671036E-01, 2.92221537796944E-01,
                              1.34028711415815E-02);
           ir->AddTriPoints6 (37, 5.44239242905830E-02, 3.19574885423190E-01,
                              9.22999660541100E-03);
           ir->AddTriPoints6 (43, 1.28685608336370E-02, 1.90704224192292E-01,
                              4.23843426716400E-03);
           ir->AddTriPoints6 (49, 6.71657824135240E-02, 1.80483211648746E-01,
                              9.14639838501250E-03);
           ir->AddTriPoints6 (55, 1.46631822248280E-02, 8.07113136795640E-02,
                              3.33281600208250E-03);
           TriangleIntRules[16] = TriangleIntRules[17] = ir;
           return ir;
  
        case 18: // 73 point - 19 degree (used for 18 as well)
        case 19:
           ir = new IntegrationRule(73);
           ir->AddTriMidPoint(0,  0.0164531656944595);
           ir->AddTriPoints3b(1,  0.020780025853987, 0.005165365945636);
           ir->AddTriPoints3b(4,  0.090926214604215, 0.011193623631508);
           ir->AddTriPoints3b(7,  0.197166638701138, 0.015133062934734);
           ir->AddTriPoints3 (10, 0.255551654403098, 0.015245483901099);
           ir->AddTriPoints3 (13, 0.17707794215213,  0.0120796063708205);
           ir->AddTriPoints3 (16, 0.110061053227952, 0.0080254017934005);
           ir->AddTriPoints3 (19, 0.05552862425184,  0.004042290130892);
           ir->AddTriPoints3 (22, 0.012621863777229, 0.0010396810137425);
           ir->AddTriPoints6 (25, 0.003611417848412, 0.395754787356943,
                              0.0019424384524905);
           ir->AddTriPoints6 (31, 0.13446675453078, 0.307929983880436,
                              0.012787080306011);
           ir->AddTriPoints6 (37, 0.014446025776115, 0.26456694840652,
                              0.004440451786669);
           ir->AddTriPoints6 (43, 0.046933578838178, 0.358539352205951,
                              0.0080622733808655);
           ir->AddTriPoints6 (49, 0.002861120350567, 0.157807405968595,
                              0.0012459709087455);
           ir->AddTriPoints6 (55, 0.075050596975911, 0.223861424097916,
                              0.0091214200594755);
           ir->AddTriPoints6 (61, 0.03464707481676, 0.142421601113383,
                              0.0051292818680995);
           ir->AddTriPoints6 (67, 0.065494628082938, 0.010161119296278,
                              0.001899964427651);
           TriangleIntRules[18] = TriangleIntRules[19] = ir;
           return ir;
  
        case 20: // 85 point - 20 degree
           ir = new IntegrationRule(85);
           ir->AddTriMidPoint(0, 0.01380521349884976);
           ir->AddTriPoints3b(1, 0.001500649324429,     0.00088951477366337);
           ir->AddTriPoints3b(4, 0.0941397519389508667, 0.010056199056980585);
           ir->AddTriPoints3b(7, 0.2044721240895264,    0.013408923629665785);
           ir->AddTriPoints3(10, 0.264500202532787333,  0.012261566900751005);
           ir->AddTriPoints3(13, 0.211018964092076767,  0.008197289205347695);
           ir->AddTriPoints3(16, 0.107735607171271333,  0.0073979536993248);
           ir->AddTriPoints3(19, 0.0390690878378026667, 0.0022896411388521255);
           ir->AddTriPoints3(22, 0.0111743797293296333, 0.0008259132577881085);
           ir->AddTriPoints6(25, 0.00534961818733726667, 0.0635496659083522206,
                             0.001174585454287792);
           ir->AddTriPoints6(31, 0.00795481706619893333, 0.157106918940706982,
                             0.0022329628770908965);
           ir->AddTriPoints6(37, 0.0104223982812638,     0.395642114364374018,
                             0.003049783403953986);
           ir->AddTriPoints6(43, 0.0109644147961233333,  0.273167570712910522,
                             0.0034455406635941015);
           ir->AddTriPoints6(49, 0.0385667120854623333,  0.101785382485017108,
                             0.0039987375362390815);
           ir->AddTriPoints6(55, 0.0355805078172182,     0.446658549176413815,
                             0.003693067142668012);
           ir->AddTriPoints6(61, 0.0496708163627641333,  0.199010794149503095,
                             0.00639966593932413);
           ir->AddTriPoints6(67, 0.0585197250843317333,  0.3242611836922827,
                             0.008629035587848275);
           ir->AddTriPoints6(73, 0.121497787004394267,   0.208531363210132855,
                             0.009336472951467735);
           ir->AddTriPoints6(79, 0.140710844943938733,   0.323170566536257485,
                             0.01140911202919763);
           TriangleIntRules[20] = ir;
           return ir;
  
        case 21: // 126 point - 25 degree (used also for degrees from 21 to 24)
        case 22:
        case 23:
        case 24:
        case 25:
           ir = new IntegrationRule(126);
           ir->AddTriPoints3b(0, 0.0279464830731742,   0.0040027909400102085);
           ir->AddTriPoints3b(3, 0.131178601327651467, 0.00797353841619525);
           ir->AddTriPoints3b(6, 0.220221729512072267, 0.006554570615397765);
           ir->AddTriPoints3 (9, 0.298443234019804467,   0.00979150048281781);
           ir->AddTriPoints3(12, 0.2340441723373718,     0.008235442720768635);
           ir->AddTriPoints3(15, 0.151468334609017567,   0.00427363953704605);
           ir->AddTriPoints3(18, 0.112733893545993667,   0.004080942928613246);
           ir->AddTriPoints3(21, 0.0777156920915263,     0.0030605732699918895);
           ir->AddTriPoints3(24, 0.034893093614297,      0.0014542491324683325);
           ir->AddTriPoints3(27, 0.00725818462093236667, 0.00034613762283099815);
           ir->AddTriPoints6(30,  0.0012923527044422,     0.227214452153364077,
                             0.0006241445996386985);
           ir->AddTriPoints6(36,  0.0053997012721162,     0.435010554853571706,
                             0.001702376454401511);
           ir->AddTriPoints6(42,  0.006384003033975,      0.320309599272204437,
                             0.0016798271630320255);
           ir->AddTriPoints6(48,  0.00502821150199306667, 0.0917503222800051889,
                             0.000858078269748377);
           ir->AddTriPoints6(54,  0.00682675862178186667, 0.0380108358587243835,
                             0.000740428158357803);
           ir->AddTriPoints6(60,  0.0100161996399295333,  0.157425218485311668,
                             0.0017556563053643425);
           ir->AddTriPoints6(66,  0.02575781317339,       0.239889659778533193,
                             0.003696775074853242);
           ir->AddTriPoints6(72,  0.0302278981199158,     0.361943118126060531,
                             0.003991543738688279);
           ir->AddTriPoints6(78,  0.0305049901071620667,  0.0835519609548285602,
                             0.0021779813065790205);
           ir->AddTriPoints6(84,  0.0459565473625693333,  0.148443220732418205,
                             0.003682528350708916);
           ir->AddTriPoints6(90,  0.0674428005402775333,  0.283739708727534955,
                             0.005481786423209775);
           ir->AddTriPoints6(96,  0.0700450914159106,     0.406899375118787573,
                             0.00587498087177056);
           ir->AddTriPoints6(102, 0.0839115246401166,     0.194113987024892542,
                             0.005007800356899285);
           ir->AddTriPoints6(108, 0.120375535677152667,   0.32413434700070316,
                             0.00665482039381434);
           ir->AddTriPoints6(114, 0.148066899157366667,   0.229277483555980969,
                             0.00707722325261307);
           ir->AddTriPoints6(120, 0.191771865867325067,   0.325618122595983752,
                             0.007440689780584005);
           TriangleIntRules[21] =
              TriangleIntRules[22] =
                 TriangleIntRules[23] =
                    TriangleIntRules[24] =
                       TriangleIntRules[25] = ir;
           return ir;
  
        default:
           // Grundmann-Moller rules
           int i = (Order / 2) * 2 + 1;   // Get closest odd # >= Order
           AllocIntRule(TriangleIntRules, i);
           ir = new IntegrationRule;
           ir->GrundmannMollerSimplexRule(i/2,2);
           TriangleIntRules[i-1] = TriangleIntRules[i] = ir;
           return ir;
     }
  }
  
  // Integration rules for unit square
  IntegrationRule *IntegrationRules::SquareIntegrationRule(int Order)
  {
     int RealOrder = GetSegmentRealOrder(Order);
     // Order is one of {RealOrder-1,RealOrder}
     if (!HaveIntRule(SegmentIntRules, RealOrder))
     {
        SegmentIntegrationRule(RealOrder);
     }
     AllocIntRule(SquareIntRules, RealOrder); // RealOrder >= Order
     SquareIntRules[RealOrder-1] =
        SquareIntRules[RealOrder] =
           new IntegrationRule(*SegmentIntRules[RealOrder],
                               *SegmentIntRules[RealOrder]);
     return SquareIntRules[Order];
  }
  
  /** Integration rules for reference tetrahedron
      {[0,0,0],[1,0,0],[0,1,0],[0,0,1]}          */
  IntegrationRule *IntegrationRules::TetrahedronIntegrationRule(int Order)
  {
     IntegrationRule *ir;
     // Note: Set TetrahedronIntRules[*] to ir only *after* ir is fully
     // constructed. This is needed in multithreaded environment.
  
     // assuming that orders <= 9 are pre-allocated
     switch (Order)
     {
        case 0:  // 1 point - degree 1
        case 1:
           ir = new IntegrationRule(1);
           ir->AddTetMidPoint(0, 1./6.);
           TetrahedronIntRules[0] = TetrahedronIntRules[1] = ir;
           return ir;
  
        case 2:  // 4 points - degree 2
           ir = new IntegrationRule(4);
           // ir->AddTetPoints4(0, 0.13819660112501051518, 1./24.);
           ir->AddTetPoints4b(0, 0.58541019662496845446, 1./24.);
           TetrahedronIntRules[2] = ir;
           return ir;
  
        case 3:  // 5 points - degree 3 (negative weight)
           ir = new IntegrationRule(5);
           ir->AddTetMidPoint(0, -2./15.);
           ir->AddTetPoints4b(1, 0.5, 0.075);
           TetrahedronIntRules[3] = ir;
           return ir;
  
        case 4:  // 11 points - degree 4 (negative weight)
           ir = new IntegrationRule(11);
           ir->AddTetPoints4(0, 1./14., 343./45000.);
           ir->AddTetMidPoint(4, -74./5625.);
           ir->AddTetPoints6(5, 0.10059642383320079500, 28./1125.);
           TetrahedronIntRules[4] = ir;
           return ir;
  
        case 5:  // 14 points - degree 5
           ir = new IntegrationRule(14);
           ir->AddTetPoints6(0, 0.045503704125649649492,
                             7.0910034628469110730E-03);
           ir->AddTetPoints4(6, 0.092735250310891226402, 0.012248840519393658257);
           ir->AddTetPoints4b(10, 0.067342242210098170608,
                              0.018781320953002641800);
           TetrahedronIntRules[5] = ir;
           return ir;
  
        case 6:  // 24 points - degree 6
           ir = new IntegrationRule(24);
           ir->AddTetPoints4(0, 0.21460287125915202929,
                             6.6537917096945820166E-03);
           ir->AddTetPoints4(4, 0.040673958534611353116,
                             1.6795351758867738247E-03);
           ir->AddTetPoints4b(8, 0.032986329573173468968,
                              9.2261969239424536825E-03);
           ir->AddTetPoints12(12, 0.063661001875017525299, 0.26967233145831580803,
                              8.0357142857142857143E-03);
           TetrahedronIntRules[6] = ir;
           return ir;
  
        case 7:  // 31 points - degree 7 (negative weight)
           ir = new IntegrationRule(31);
           ir->AddTetPoints6(0, 0.0, 9.7001763668430335097E-04);
           ir->AddTetMidPoint(6, 0.018264223466108820291);
           ir->AddTetPoints4(7, 0.078213192330318064374, 0.010599941524413686916);
           ir->AddTetPoints4(11, 0.12184321666390517465,
                             -0.062517740114331851691);
           ir->AddTetPoints4b(15, 2.3825066607381275412E-03,
                              4.8914252630734993858E-03);
           ir->AddTetPoints12(19, 0.1, 0.2, 0.027557319223985890653);
           TetrahedronIntRules[7] = ir;
           return ir;
  
        case 8:  // 43 points - degree 8 (negative weight)
           ir = new IntegrationRule(43);
           ir->AddTetPoints4(0, 5.7819505051979972532E-03,
                             1.6983410909288737984E-04);
           ir->AddTetPoints4(4, 0.082103588310546723091,
                             1.9670333131339009876E-03);
           ir->AddTetPoints12(8, 0.036607749553197423679, 0.19048604193463345570,
                              2.1405191411620925965E-03);
           ir->AddTetPoints6(20, 0.050532740018894224426,
                             4.5796838244672818007E-03);
           ir->AddTetPoints12(26, 0.22906653611681113960, 0.035639582788534043717,
                              5.7044858086819185068E-03);
           ir->AddTetPoints4(38, 0.20682993161067320408, 0.014250305822866901248);
           ir->AddTetMidPoint(42, -0.020500188658639915841);
           TetrahedronIntRules[8] = ir;
           return ir;
  
        case 9: // orders 9 and higher -- Grundmann-Moller rules
           ir = new IntegrationRule;
           ir->GrundmannMollerSimplexRule(4,3);
           TetrahedronIntRules[9] = ir;
           return ir;
  
        default: // Grundmann-Moller rules
           int i = (Order / 2) * 2 + 1;   // Get closest odd # >= Order
           AllocIntRule(TetrahedronIntRules, i);
           ir = new IntegrationRule;
           ir->GrundmannMollerSimplexRule(i/2,3);
           TetrahedronIntRules[i-1] = TetrahedronIntRules[i] = ir;
           return ir;
     }
  }
  
  // Integration rules for reference pyramid
  IntegrationRule *IntegrationRules::PyramidIntegrationRule(int Order)
  {
     // This is a simple integration rule adapted from an integration
     // rule for a cube which seems to be adequate for now. When we
     // implement high order finite elements for pyramids we should
     // revisit this and see if we can improve upon it.
     const IntegrationRule &irc = Get(Geometry::CUBE, Order);
     int npts = irc.GetNPoints();
     AllocIntRule(PyramidIntRules, Order);
     PyramidIntRules[Order] = new IntegrationRule(npts);
  
     for (int k=0; k<npts; k++)
     {
        const IntegrationPoint & ipc = irc.IntPoint(k);
        IntegrationPoint & ipp = PyramidIntRules[Order]->IntPoint(k);
        ipp.x = ipc.x * (1.0 - ipc.z);
        ipp.y = ipc.y * (1.0 - ipc.z);
        ipp.z = ipc.z;
        ipp.weight = ipc.weight / 3.0;
     }
     return PyramidIntRules[Order];
  }
  
  // Integration rules for reference prism
  IntegrationRule *IntegrationRules::PrismIntegrationRule(int Order)
  {
     const IntegrationRule & irt = Get(Geometry::TRIANGLE, Order);
     const IntegrationRule & irs = Get(Geometry::SEGMENT, Order);
     int nt = irt.GetNPoints();
     int ns = irs.GetNPoints();
     AllocIntRule(PrismIntRules, Order);
     PrismIntRules[Order] = new IntegrationRule(nt * ns);
  
     for (int ks=0; ks<ns; ks++)
     {
        const IntegrationPoint & ips = irs.IntPoint(ks);
        for (int kt=0; kt<nt; kt++)
        {
           int kp = ks * nt + kt;
           const IntegrationPoint & ipt = irt.IntPoint(kt);
           IntegrationPoint & ipp = PrismIntRules[Order]->IntPoint(kp);
           ipp.x = ipt.x;
           ipp.y = ipt.y;
           ipp.z = ips.x;
           ipp.weight = ipt.weight * ips.weight;
        }
     }
     return PrismIntRules[Order];
  }
  
  // Integration rules for reference cube
  IntegrationRule *IntegrationRules::CubeIntegrationRule(int Order)
  {
     int RealOrder = GetSegmentRealOrder(Order);
     if (!HaveIntRule(SegmentIntRules, RealOrder))
     {
        SegmentIntegrationRule(RealOrder);
     }
     AllocIntRule(CubeIntRules, RealOrder);
     CubeIntRules[RealOrder-1] =
        CubeIntRules[RealOrder] =
           new IntegrationRule(*SegmentIntRules[RealOrder],
                               *SegmentIntRules[RealOrder],
                               *SegmentIntRules[RealOrder]);
     return CubeIntRules[Order];
  }
  
  }
  