FChebInterpolator.hpp 34.8 KB
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#ifndef FCHEBINTERPOLATOR_HPP
#define FCHEBINTERPOLATOR_HPP


#include "./FChebMapping.hpp"
#include "./FChebTensor.hpp"
#include "./FChebRoots.hpp"

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#include "../../Utils/FBlas.hpp"
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/**
 * @author Matthias Messner (matthias.matthias@inria.fr)
 * Please read the license
 */

/**
 * @class FChebInterpolator
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 *
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 * The class @p FChebInterpolator defines the anterpolation (M2M) and
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 * interpolation (L2L) concerning operations.
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 */
template <int ORDER>
class FChebInterpolator : FNoCopyable
{
  // compile time constants and types
  enum {nnodes = TensorTraits<ORDER>::nnodes};
  typedef FChebRoots< ORDER>  BasisType;
  typedef FChebTensor<ORDER> TensorType;

  FReal T_of_roots[ORDER][ORDER];
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  FReal T[ORDER * (ORDER-1)];
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	unsigned int node_ids[nnodes][3];
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	FReal* ChildParentInterpolator[8];

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	// permutations (only needed in the tensor product interpolation case)
	unsigned int perm[3][nnodes];

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	////////////////////////////////////////////////////////////////////
	// needed for P2M
	struct IMN2MNI {
		enum {size = ORDER * (ORDER-1) * (ORDER-1)};
		unsigned int imn[size], mni[size];
		IMN2MNI() {
			unsigned int counter = 0;
			for (unsigned int i=0; i<ORDER; ++i) {
				for (unsigned int m=0; m<ORDER-1; ++m) {
					for (unsigned int n=0; n<ORDER-1; ++n) {
						imn[counter] = n*(ORDER-1)*ORDER + m*ORDER + i;
						mni[counter] = i*(ORDER-1)*(ORDER-1) + n*(ORDER-1) + m;
						counter++;
					}
				}
			}
		}
	} perm0;
	
	struct JNI2NIJ {
		enum {size = ORDER * ORDER * (ORDER-1)};
		unsigned int jni[size], nij[size];
		JNI2NIJ() {
			unsigned int counter = 0;
			for (unsigned int i=0; i<ORDER; ++i) {
				for (unsigned int j=0; j<ORDER; ++j) {
					for (unsigned int n=0; n<ORDER-1; ++n) {
						jni[counter] = i*(ORDER-1)*ORDER + n*ORDER + j;
						nij[counter] = j*ORDER*(ORDER-1) + i*(ORDER-1) + n;
						counter++;
					}
				}
			}
		}
	} perm1;

	struct KIJ2IJK {
		enum {size = ORDER * ORDER * ORDER};
		unsigned int kij[size], ijk[size];
		KIJ2IJK() {
			unsigned int counter = 0;
			for (unsigned int i=0; i<ORDER; ++i) {
				for (unsigned int j=0; j<ORDER; ++j) {
					for (unsigned int k=0; k<ORDER; ++k) {
						kij[counter] = j*ORDER*ORDER + i*ORDER + k;
						ijk[counter] = k*ORDER*ORDER + j*ORDER + i;
						counter++;
					}
				}
			}
		}
	} perm2;
	////////////////////////////////////////////////////////////////////

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	/**
	 * Initialize the child - parent - interpolator, it is basically the matrix
	 * S which is precomputed and reused for all M2M and L2L operations, ie for
	 * all non leaf inter/anterpolations.
	 */
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	void initM2MandL2L()
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	{
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		FPoint ParentRoots[nnodes], ChildRoots[nnodes];
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		const FReal ParentWidth(2.);
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		const FPoint ParentCenter(0., 0., 0.);
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		FChebTensor<ORDER>::setRoots(ParentCenter, ParentWidth, ParentRoots);

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		FPoint ChildCenter;
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		const FReal ChildWidth(1.);
		
		// loop: child cells
		for (unsigned int child=0; child<8; ++child) {

			// allocate memory
			ChildParentInterpolator[child] = new FReal [nnodes * nnodes];

			// set child info
			FChebTensor<ORDER>::setRelativeChildCenter(child, ChildCenter);
			FChebTensor<ORDER>::setRoots(ChildCenter, ChildWidth, ChildRoots);

			// assemble child - parent - interpolator
			assembleInterpolator(nnodes, ChildRoots, ChildParentInterpolator[child]);
		}
	}

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	/**
	 * Initialize the child - parent - interpolator, it is basically the matrix
	 * S which is precomputed and reused for all M2M and L2L operations, ie for
	 * all non leaf inter/anterpolations.
	 */
	void initTensorM2MandL2L()
	{
		FPoint ParentRoots[nnodes];
		FReal ChildCoords[3][ORDER];
		const FReal ParentWidth(2.);
		const FPoint ParentCenter(0., 0., 0.);
		FChebTensor<ORDER>::setRoots(ParentCenter, ParentWidth, ParentRoots);

		FPoint ChildCenter;
		const FReal ChildWidth(1.);
		
		// loop: child cells
		for (unsigned int child=0; child<8; ++child) {

			// set child info
			FChebTensor<ORDER>::setRelativeChildCenter(child, ChildCenter);
			FChebTensor<ORDER>::setChebyshevRoots(ChildCenter, ChildWidth, ChildCoords);

			// allocate memory
			ChildParentInterpolator[child] = new FReal [3 * ORDER*ORDER];
			assembleInterpolator(ORDER, ChildCoords[0], ChildParentInterpolator[child]);
			assembleInterpolator(ORDER, ChildCoords[1], ChildParentInterpolator[child] + 1 * ORDER*ORDER);
			assembleInterpolator(ORDER, ChildCoords[2], ChildParentInterpolator[child] + 2 * ORDER*ORDER);
		}


		// init permutations
		for (unsigned int i=0; i<ORDER; ++i) {
			for (unsigned int j=0; j<ORDER; ++j) {
				for (unsigned int k=0; k<ORDER; ++k) {
					const unsigned int index = k*ORDER*ORDER + j*ORDER + i;
					perm[0][index] = k*ORDER*ORDER + j*ORDER + i;
					perm[1][index] = i*ORDER*ORDER + k*ORDER + j;
					perm[2][index] = j*ORDER*ORDER + i*ORDER + k;
				}
			}
		}
		
	}

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public:
	/**
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	 * Constructor: Initialize the Chebyshev polynomials at the Chebyshev
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	 * roots/interpolation point
	 */
	explicit FChebInterpolator()
	{
		// initialize chebyshev polynomials of root nodes: T_o(x_j)
    for (unsigned int o=1; o<ORDER; ++o)
      for (unsigned int j=0; j<ORDER; ++j)
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        T_of_roots[o][j] = FReal(BasisType::T(o, FReal(BasisType::roots[j])));
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		// initialize chebyshev polynomials of root nodes: T_o(x_j)
    for (unsigned int o=1; o<ORDER; ++o)
      for (unsigned int j=0; j<ORDER; ++j)
        T[(o-1)*ORDER + j] = FReal(BasisType::T(o, FReal(BasisType::roots[j])));
		

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		// initialize root node ids
		TensorType::setNodeIds(node_ids);
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		// initialize interpolation operator for non M2M and L2L (non leaf
		// operations)
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		//this -> initM2MandL2L();     // non tensor-product interpolation
		this -> initTensorM2MandL2L(); // tensor-product interpolation
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	}

	
	/**
	 * Destructor: Delete dynamically allocated memory for M2M and L2L operator
	 */
	~FChebInterpolator()
	{
		for (unsigned int child=0; child<8; ++child)
			delete [] ChildParentInterpolator[child];
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	}


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	/**
	 * Assembles the interpolator \f$S_\ell\f$ of size \f$N\times
	 * \ell^3\f$. Here local points is meant as points whose global coordinates
	 * have already been mapped to the reference interval [-1,1].
	 *
	 * @param[in] NumberOfLocalPoints
	 * @param[in] LocalPoints
	 * @param[out] Interpolator
	 */
	void assembleInterpolator(const unsigned int NumberOfLocalPoints,
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				  const FPoint *const LocalPoints,
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				  FReal *const Interpolator) const
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	{
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		// values of chebyshev polynomials of source particle: T_o(x_i)
		FReal T_of_x[ORDER][3];
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		// loop: local points (mapped in [-1,1])
		for (unsigned int m=0; m<NumberOfLocalPoints; ++m) {
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			// evaluate chebyshev polynomials at local points
			for (unsigned int o=1; o<ORDER; ++o) {
				T_of_x[o][0] = BasisType::T(o, LocalPoints[m].getX());
				T_of_x[o][1] = BasisType::T(o, LocalPoints[m].getY());
				T_of_x[o][2] = BasisType::T(o, LocalPoints[m].getZ());
			}
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			// assemble interpolator
			for (unsigned int n=0; n<nnodes; ++n) {
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				//Interpolator[n*nnodes + m] = FReal(1.);
				Interpolator[n*NumberOfLocalPoints + m] = FReal(1.);
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				for (unsigned int d=0; d<3; ++d) {
					const unsigned int j = node_ids[n][d];
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					FReal S_d = FReal(1.) / ORDER;
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					for (unsigned int o=1; o<ORDER; ++o)
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					 	S_d += FReal(2.) / ORDER * T_of_x[o][d] * T_of_roots[o][j];
					//Interpolator[n*nnodes + m] *= S_d;
					Interpolator[n*NumberOfLocalPoints + m] *= S_d;
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				}
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			}
			
		}
		
	}

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	void assembleInterpolator(const unsigned int M, const FReal *const x, FReal *const S) const
	{
		// values of chebyshev polynomials of source particle: T_o(x_i)
		FReal T_of_x[ORDER];

		// loop: local points (mapped in [-1,1])
		for (unsigned int m=0; m<M; ++m) {
			// evaluate chebyshev polynomials at local points
			for (unsigned int o=1; o<ORDER; ++o)
				T_of_x[o] = BasisType::T(o, x[m]);
			
			for (unsigned int n=0; n<ORDER; ++n) {
				S[n*M + m] = FReal(1.) / ORDER;
				for (unsigned int o=1; o<ORDER; ++o)
					S[n*M + m] += FReal(2.) / ORDER * T_of_x[o] * T_of_roots[o][n];
			}
			
		}
		
	}
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	/**
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	 * Particle to moment: application of \f$S_\ell(y,\bar y_n)\f$
	 * (anterpolation, it is the transposed interpolation)
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	 */
	template <class ContainerClass>
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	void applyP2M(const FPoint& center,
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								const FReal width,
								FReal *const multipoleExpansion,
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								const ContainerClass *const sourceParticles) const;
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	/**
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	 * Local to particle operation: application of \f$S_\ell(x,\bar x_m)\f$ (interpolation)
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	 */
	template <class ContainerClass>
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	void applyL2P(const FPoint& center,
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								const FReal width,
								const FReal *const localExpansion,
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								ContainerClass *const localParticles) const;
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	/**
	 * Local to particle operation: application of \f$\nabla_x S_\ell(x,\bar x_m)\f$ (interpolation)
	 */
	template <class ContainerClass>
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	void applyL2PGradient(const FPoint& center,
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												const FReal width,
												const FReal *const localExpansion,
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												ContainerClass *const localParticles) const;
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	/**
	 * Local to particle operation: application of \f$S_\ell(x,\bar x_m)\f$ and
	 * \f$\nabla_x S_\ell(x,\bar x_m)\f$ (interpolation)
	 */
	template <class ContainerClass>
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	void applyL2PTotal(const FPoint& center,
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										 const FReal width,
										 const FReal *const localExpansion,
										 ContainerClass *const localParticles) const;
	
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	/*
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	void applyM2M(const unsigned int ChildIndex,
								const FReal *const ChildExpansion,
								FReal *const ParentExpansion) const
	{
		FBlas::gemtva(nnodes, nnodes, FReal(1.),
									ChildParentInterpolator[ChildIndex],
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									const_cast<FReal*>(ChildExpansion), ParentExpansion);
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	}
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	void applyL2L(const unsigned int ChildIndex,
								const FReal *const ParentExpansion,
								FReal *const ChildExpansion) const
	{
		FBlas::gemva(nnodes, nnodes, FReal(1.),
								 ChildParentInterpolator[ChildIndex],
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								 const_cast<FReal*>(ParentExpansion), ChildExpansion);
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	}
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	*/
	

	
	void applyM2M(const unsigned int ChildIndex,
								const FReal *const ChildExpansion,
								FReal *const ParentExpansion) const
	{
		FReal Exp[nnodes], PermExp[nnodes];
		FBlas::gemtm(ORDER, ORDER, ORDER*ORDER, FReal(1.),
								 ChildParentInterpolator[ChildIndex], ORDER,
								 const_cast<FReal*>(ChildExpansion), ORDER, PermExp, ORDER);
		
		for (unsigned int n=0; n<nnodes; ++n)	Exp[n] = PermExp[perm[1][n]];
		FBlas::gemtm(ORDER, ORDER, ORDER*ORDER, FReal(1.),
								 ChildParentInterpolator[ChildIndex] + 2 * ORDER*ORDER, ORDER,
								 Exp, ORDER, PermExp, ORDER);

		for (unsigned int n=0; n<nnodes; ++n)	Exp[perm[1][n]] = PermExp[perm[2][n]];
		FBlas::gemtm(ORDER, ORDER, ORDER*ORDER, FReal(1.),
								 ChildParentInterpolator[ChildIndex] + 1 * ORDER*ORDER, ORDER,
								 Exp, ORDER, PermExp, ORDER);

		for (unsigned int n=0; n<nnodes; ++n)	ParentExpansion[perm[2][n]] += PermExp[n];
	}


	void applyL2L(const unsigned int ChildIndex,
								const FReal *const ParentExpansion,
								FReal *const ChildExpansion) const
	{
		FReal Exp[nnodes], PermExp[nnodes];
		FBlas::gemm(ORDER, ORDER, ORDER*ORDER, FReal(1.),
								ChildParentInterpolator[ChildIndex], ORDER,
								const_cast<FReal*>(ParentExpansion), ORDER, PermExp, ORDER);
		
		for (unsigned int n=0; n<nnodes; ++n)	Exp[n] = PermExp[perm[1][n]];
		FBlas::gemm(ORDER, ORDER, ORDER*ORDER, FReal(1.),
								ChildParentInterpolator[ChildIndex] + 2 * ORDER*ORDER, ORDER,
								Exp, ORDER, PermExp, ORDER);
		
		for (unsigned int n=0; n<nnodes; ++n)	Exp[perm[1][n]] = PermExp[perm[2][n]];
		FBlas::gemm(ORDER, ORDER, ORDER*ORDER, FReal(1.),
								ChildParentInterpolator[ChildIndex] + 1 * ORDER*ORDER, ORDER,
								Exp, ORDER, PermExp, ORDER);

		for (unsigned int n=0; n<nnodes; ++n)	ChildExpansion[perm[2][n]] += PermExp[n];
	}
	
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};
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/**
 * Particle to moment: application of \f$S_\ell(y,\bar y_n)\f$
 * (anterpolation, it is the transposed interpolation)
 */
template <int ORDER>
template <class ContainerClass>
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inline void FChebInterpolator<ORDER>::applyP2M(const FPoint& center,
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																							 const FReal width,
																							 FReal *const multipoleExpansion,
																							 const ContainerClass *const sourceParticles) const
{
	// set all multipole expansions to zero
	FBlas::setzero(nnodes, multipoleExpansion);

	// allocate stuff
	const map_glob_loc map(center, width);
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	FPoint localPosition;
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	FReal W1 = FReal(0.);
	FReal W2[3][ ORDER-1];
	FReal W4[3][(ORDER-1)*(ORDER-1)];
	FReal W8[   (ORDER-1)*(ORDER-1)*(ORDER-1)];
	for(unsigned int i=0; i<(ORDER-1); ++i) W2[0][i] = W2[1][i] = W2[2][i] = FReal(0.);
	for(unsigned int i=0; i<(ORDER-1)*(ORDER-1); ++i)	W4[0][i] = W4[1][i] = W4[2][i] = FReal(0.);
	for(unsigned int i=0; i<(ORDER-1)*(ORDER-1)*(ORDER-1); ++i)	W8[i] = FReal(0.);
	
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	// loop over source particles
	typename ContainerClass::ConstBasicIterator iter(*sourceParticles);
	while(iter.hasNotFinished()){
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		// map global position to [-1,1]
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		map(iter.data().getPosition(), localPosition); // 15 flops
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		FReal T_of_x[3][ORDER];
		T_of_x[0][0] = FReal(1.); T_of_x[0][1] = localPosition.getX();
		T_of_x[1][0] = FReal(1.); T_of_x[1][1] = localPosition.getY();
		T_of_x[2][0] = FReal(1.); T_of_x[2][1] = localPosition.getZ();
		const FReal x2 = FReal(2.) * T_of_x[0][1]; // 1 flop
		const FReal y2 = FReal(2.) * T_of_x[1][1]; // 1 flop
		const FReal z2 = FReal(2.) * T_of_x[2][1]; // 1 flop
		for (unsigned int j=2; j<ORDER; ++j) {
			T_of_x[0][j] = x2 * T_of_x[0][j-1] - T_of_x[0][j-2]; // 2 flops
			T_of_x[1][j] = y2 * T_of_x[1][j-1] - T_of_x[1][j-2]; // 2 flops
			T_of_x[2][j] = z2 * T_of_x[2][j-1] - T_of_x[2][j-2]; // 2 flops
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		}
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		const FReal weight = iter.data().getPhysicalValue();
		W1 += weight; // 1 flop
		for (unsigned int i=1; i<ORDER; ++i) {
			const FReal wx = weight * T_of_x[0][i]; // 1 flop
			const FReal wy = weight * T_of_x[1][i]; // 1 flop
			const FReal wz = weight * T_of_x[2][i]; // 1 flop
			W2[0][i-1] += wx; // 1 flop
			W2[1][i-1] += wy; // 1 flop
			W2[2][i-1] += wz; // 1 flop
			for (unsigned int j=1; j<ORDER; ++j) {
				const FReal wxy = wx * T_of_x[1][j]; // 1 flop
				const FReal wxz = wx * T_of_x[2][j]; // 1 flop
				const FReal wyz = wy * T_of_x[2][j]; // 1 flop
				W4[0][(j-1)*(ORDER-1) + (i-1)] += wxy; // 1 flop
				W4[1][(j-1)*(ORDER-1) + (i-1)] += wxz; // 1 flop
				W4[2][(j-1)*(ORDER-1) + (i-1)] += wyz; // 1 flop
				for (unsigned int k=1; k<ORDER; ++k) {
					const FReal wxyz = wxy * T_of_x[2][k]; // 1 flop
					W8[(k-1)*(ORDER-1)*(ORDER-1) + (j-1)*(ORDER-1) + (i-1)] += wxyz; // 1 flop
				} // flops: (ORDER-1) * 2
			} // flops: (ORDER-1) * (6 + (ORDER-1) * 2) 
		} // flops: (ORDER-1) * (6 + (ORDER-1) * (6 + (ORDER-1) * 2))
		
		
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		// increment source iterator
		iter.gotoNext();
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	} // flops: N * (18 + (ORDER-2) * 6 + (ORDER-1) * (6 + (ORDER-1) * (6 + (ORDER-1) * 2)))

	////////////////////////////////////////////////////////////////////

	// loop over interpolation points
	FReal F2[3][ORDER];
	FReal F4[3][ORDER*ORDER];
	FReal F8[   ORDER*ORDER*ORDER];
	{
		// compute W2: 3 * ORDER*(2*(ORDER-1)-1) flops
		FBlas::gemv(ORDER, ORDER-1, FReal(1.), const_cast<FReal*>(T), W2[0], F2[0]);
		FBlas::gemv(ORDER, ORDER-1, FReal(1.), const_cast<FReal*>(T), W2[1], F2[1]);
		FBlas::gemv(ORDER, ORDER-1, FReal(1.), const_cast<FReal*>(T), W2[2], F2[2]);

		// compute W4: 3 * [ORDER*(ORDER-1)*(2*(ORDER-1)-1) + ORDER*ORDER*(2*(ORDER-1)-1)]
		FReal C[ORDER * (ORDER-1)];
		FBlas::gemmt(ORDER, ORDER-1, ORDER-1, FReal(1.), const_cast<FReal*>(T), ORDER, W4[0], ORDER-1, C,     ORDER);
		FBlas::gemmt(ORDER, ORDER-1, ORDER,   FReal(1.), const_cast<FReal*>(T), ORDER, C,     ORDER,   F4[0], ORDER);
		FBlas::gemmt(ORDER, ORDER-1, ORDER-1, FReal(1.), const_cast<FReal*>(T), ORDER, W4[1], ORDER-1, C,     ORDER);
		FBlas::gemmt(ORDER, ORDER-1, ORDER,   FReal(1.), const_cast<FReal*>(T), ORDER, C,     ORDER,   F4[1], ORDER);
		FBlas::gemmt(ORDER, ORDER-1, ORDER-1, FReal(1.), const_cast<FReal*>(T), ORDER, W4[2], ORDER-1, C,     ORDER);
		FBlas::gemmt(ORDER, ORDER-1, ORDER,   FReal(1.), const_cast<FReal*>(T), ORDER, C,     ORDER,   F4[2], ORDER);
	
		// compute W8: 3 * (2*(ORDER-1)-1) * [ORDER*(ORDER-1)*(ORDER-1) + ORDER*ORDER*(ORDER-1) + ORDER*ORDER*ORDER]
		FReal D[ORDER * (ORDER-1) * (ORDER-1)];
		FBlas::gemm(ORDER, ORDER-1, (ORDER-1)*(ORDER-1), FReal(1.),	const_cast<FReal*>(T), ORDER, W8, ORDER-1, D, ORDER);
		FReal E[(ORDER-1) * (ORDER-1) * ORDER];
		for (unsigned int s=0; s<perm0.size; ++s)	E[perm0.mni[s]] = D[perm0.imn[s]];
		FReal F[ORDER * (ORDER-1) * ORDER];
		FBlas::gemm(ORDER, ORDER-1, ORDER*(ORDER-1), FReal(1.), const_cast<FReal*>(T), ORDER, E, ORDER-1, F, ORDER);
		FReal G[(ORDER-1) * ORDER * ORDER];
		for (unsigned int s=0; s<perm1.size; ++s)	G[perm1.nij[s]] = F[perm1.jni[s]];
		FReal H[ORDER * ORDER * ORDER];
		FBlas::gemm(ORDER, ORDER-1, ORDER*ORDER, FReal(1.), const_cast<FReal*>(T), ORDER, G, ORDER-1, H, ORDER);
		for (unsigned int s=0; s<perm2.size; ++s)	F8[perm2.ijk[s]] = H[perm2.kij[s]];
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	}
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	// assemble multipole expansions
	for (unsigned int i=0; i<ORDER; ++i) {
		for (unsigned int j=0; j<ORDER; ++j) {
			for (unsigned int k=0; k<ORDER; ++k) {
				const unsigned int idx = k*ORDER*ORDER + j*ORDER + i;
				multipoleExpansion[idx] = (W1 + 
																	 FReal(2.) * (F2[0][i] + F2[1][j] + F2[2][k]) +
																	 FReal(4.) * (F4[0][j*ORDER+i] + F4[1][k*ORDER+i] + F4[2][k*ORDER+j]) +
																	 FReal(8.) *  F8[idx]) / nnodes; // 11 * ORDER*ORDER*ORDER flops
			}
		}
	}

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}


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///**
// * Particle to moment: application of \f$S_\ell(y,\bar y_n)\f$
// * (anterpolation, it is the transposed interpolation)
// */
//template <int ORDER>
//template <class ContainerClass>
//inline void FChebInterpolator<ORDER>::applyP2M(const FPoint& center,
//																							 const FReal width,
//																							 FReal *const multipoleExpansion,
//																							 const ContainerClass *const sourceParticles) const
//{
//	// set all multipole expansions to zero
//	FBlas::setzero(nnodes, multipoleExpansion);
//
//	// allocate stuff
//	const map_glob_loc map(center, width);
//	FPoint localPosition;
//	FReal T_of_x[ORDER][3];
//	FReal S[3], c1;
//	//
//	FReal xpx,ypy,zpz ;
//	c1 = FReal(8.) / nnodes ; // 1 flop
//	// loop over source particles
//	typename ContainerClass::ConstBasicIterator iter(*sourceParticles);
//	while(iter.hasNotFinished()){
//
//		// map global position to [-1,1]
//		map(iter.data().getPosition(), localPosition); // 15 flops
//
//		// evaluate chebyshev polynomials of source particle: T_o(x_i)
//		T_of_x[0][0] = FReal(1.);	T_of_x[1][0] = localPosition.getX();
//		T_of_x[0][1] = FReal(1.);	T_of_x[1][1] = localPosition.getY();
//		T_of_x[0][2] = FReal(1.);	T_of_x[1][2] = localPosition.getZ();
//		xpx = FReal(2.) * localPosition.getX() ; // 1 flop
//		ypy = FReal(2.) * localPosition.getY() ; // 1 flop
//		zpz = FReal(2.) * localPosition.getZ() ; // 1 flop
//
//		for (unsigned int o=2; o<ORDER; ++o) {
//			T_of_x[o][0] = xpx * T_of_x[o-1][0] - T_of_x[o-2][0]; // 2 flops
//			T_of_x[o][1] = ypy * T_of_x[o-1][1] - T_of_x[o-2][1];	// 2 flops
//			T_of_x[o][2] = zpz * T_of_x[o-1][2] - T_of_x[o-2][2]; // 2 flops
//		} // flops: (ORDER-1) * 6
//		
//		// anterpolate
//		const FReal sourceValue = iter.data().getPhysicalValue();
//		for (unsigned int n=0; n<nnodes; ++n) {
//			const unsigned int j[3] = {node_ids[n][0], node_ids[n][1], node_ids[n][2]};
//			S[0] = FReal(0.5) + T_of_x[1][0] * T_of_roots[1][j[0]]; // 2 flops 
//			S[1] = FReal(0.5) + T_of_x[1][1] * T_of_roots[1][j[1]]; // 2 flops
//			S[2] = FReal(0.5) + T_of_x[1][2] * T_of_roots[1][j[2]]; // 2 flops
//			for (unsigned int o=2; o<ORDER; ++o) {
//				S[0] += T_of_x[o][0] * T_of_roots[o][j[0]]; // 2 flops
//				S[1] += T_of_x[o][1] * T_of_roots[o][j[1]]; // 2 flops
//				S[2] += T_of_x[o][2] * T_of_roots[o][j[2]]; // 2 flops
//			} // flops: (ORDER-2) * 6
//
//			// gather contributions
//			multipoleExpansion[n]	+= c1 *	S[0] * S[1] * S[2] *	sourceValue; // 4 flops
//		} // flops: ORDER*ORDER*ORDER * (10 + (ORDER-2) * 6)
//
//		// increment source iterator
//		iter.gotoNext();
//	} // flops: M * (18 + (ORDER-1) * 6 + ORDER*ORDER*ORDER * (10 + (ORDER-2) * 6))
//}






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/**
 * Local to particle operation: application of \f$S_\ell(x,\bar x_m)\f$ (interpolation)
 */
template <int ORDER>
template <class ContainerClass>
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inline void FChebInterpolator<ORDER>::applyL2P(const FPoint& center,
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																							 const FReal width,
																							 const FReal *const localExpansion,
																							 ContainerClass *const localParticles) const
{
	// allocate stuff
	const map_glob_loc map(center, width);
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	FPoint localPosition;
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	FReal T_of_x[ORDER][3];
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	FReal xpx,ypy,zpz ;
	FReal S[3],c1;
	//
	c1 = FReal(8.) / nnodes ;
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	typename ContainerClass::BasicIterator iter(*localParticles);
	while(iter.hasNotFinished()){
			
		// map global position to [-1,1]
		map(iter.data().getPosition(), localPosition);

		// evaluate chebyshev polynomials of source particle: T_o(x_i)
		T_of_x[0][0] = FReal(1.);	T_of_x[1][0] = localPosition.getX();
		T_of_x[0][1] = FReal(1.);	T_of_x[1][1] = localPosition.getY();
		T_of_x[0][2] = FReal(1.);	T_of_x[1][2] = localPosition.getZ();
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		xpx = FReal(2.) * localPosition.getX() ;
		ypy = FReal(2.) * localPosition.getY() ;
		zpz = FReal(2.) * localPosition.getZ() ;
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		for (unsigned int o=2; o<ORDER; ++o) {
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			T_of_x[o][0] = xpx * T_of_x[o-1][0] - T_of_x[o-2][0];
			T_of_x[o][1] = ypy * T_of_x[o-1][1] - T_of_x[o-2][1];
			T_of_x[o][2] = zpz * T_of_x[o-1][2] - T_of_x[o-2][2];
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		}

		// interpolate and increment target value
		FReal targetValue = iter.data().getPotential();
		for (unsigned int n=0; n<nnodes; ++n) {
			const unsigned int j[3] = {node_ids[n][0], node_ids[n][1], node_ids[n][2]};
			S[0] = T_of_x[1][0] * T_of_roots[1][j[0]];
			S[1] = T_of_x[1][1] * T_of_roots[1][j[1]];
			S[2] = T_of_x[1][2] * T_of_roots[1][j[2]];
			for (unsigned int o=2; o<ORDER; ++o) {
				S[0] += T_of_x[o][0] * T_of_roots[o][j[0]];
				S[1] += T_of_x[o][1] * T_of_roots[o][j[1]];
				S[2] += T_of_x[o][2] * T_of_roots[o][j[2]];
			}
			// gather contributions
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			// S[0] *= FReal(2.); S[0] += FReal(1.);
			// S[1] *= FReal(2.); S[1] += FReal(1.);
			// S[2] *= FReal(2.); S[2] += FReal(1.);
			// targetValue	+= S[0] * S[1] * S[2] * localExpansion[n];
			S[0] += FReal(0.5);
			S[1] += FReal(0.5);
			S[2] += FReal(0.5);
			//
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			targetValue	+= S[0] * S[1] * S[2] * localExpansion[n];
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		}
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		// scale
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		//		targetValue /= nnodes;
		targetValue *= c1;
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		// set potential
		iter.data().setPotential(targetValue);
		// increment target iterator
		iter.gotoNext();
	}
}






/**
 * Local to particle operation: application of \f$\nabla_x S_\ell(x,\bar x_m)\f$ (interpolation)
 */
template <int ORDER>
template <class ContainerClass>
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inline void FChebInterpolator<ORDER>::applyL2PGradient(const FPoint& center,
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																											 const FReal width,
																											 const FReal *const localExpansion,
																											 ContainerClass *const localParticles) const
{
	// setup local to global mapping
	const map_glob_loc map(center, width);
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	FPoint Jacobian;
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	map.computeJacobian(Jacobian);
	const FReal jacobian[3] = {Jacobian.getX(), Jacobian.getY(), Jacobian.getZ()}; 
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	FPoint localPosition;
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	FReal T_of_x[ORDER][3];
	FReal U_of_x[ORDER][3];
	FReal P[3];

	typename ContainerClass::BasicIterator iter(*localParticles);
	while(iter.hasNotFinished()){
			
		// map global position to [-1,1]
		map(iter.data().getPosition(), localPosition);
			
		// evaluate chebyshev polynomials of source particle
		// T_0(x_i) and T_1(x_i)
		T_of_x[0][0] = FReal(1.);	T_of_x[1][0] = localPosition.getX();
		T_of_x[0][1] = FReal(1.);	T_of_x[1][1] = localPosition.getY();
		T_of_x[0][2] = FReal(1.);	T_of_x[1][2] = localPosition.getZ();
		// U_0(x_i) and U_1(x_i)
		U_of_x[0][0] = FReal(1.);	U_of_x[1][0] = localPosition.getX() * FReal(2.);
		U_of_x[0][1] = FReal(1.);	U_of_x[1][1] = localPosition.getY() * FReal(2.);
		U_of_x[0][2] = FReal(1.);	U_of_x[1][2] = localPosition.getZ() * FReal(2.);
		for (unsigned int o=2; o<ORDER; ++o) {
			// T_o(x_i)
			T_of_x[o][0] = FReal(2.)*localPosition.getX()*T_of_x[o-1][0] - T_of_x[o-2][0];
			T_of_x[o][1] = FReal(2.)*localPosition.getY()*T_of_x[o-1][1] - T_of_x[o-2][1];
			T_of_x[o][2] = FReal(2.)*localPosition.getZ()*T_of_x[o-1][2] - T_of_x[o-2][2];
			// U_o(x_i)
			U_of_x[o][0] = FReal(2.)*localPosition.getX()*U_of_x[o-1][0] - U_of_x[o-2][0];
			U_of_x[o][1] = FReal(2.)*localPosition.getY()*U_of_x[o-1][1] - U_of_x[o-2][1];
			U_of_x[o][2] = FReal(2.)*localPosition.getZ()*U_of_x[o-1][2] - U_of_x[o-2][2];
		}

		// scale, because dT_o/dx = oU_{o-1}
		for (unsigned int o=2; o<ORDER; ++o) {
			U_of_x[o-1][0] *= FReal(o);
			U_of_x[o-1][1] *= FReal(o);
			U_of_x[o-1][2] *= FReal(o);
		}

		// apply P and increment forces
		FReal forces[3] = {FReal(0.), FReal(0.), FReal(0.)};
		for (unsigned int n=0; n<nnodes; ++n) {
			
			// tensor indices of chebyshev nodes
			const unsigned int j[3] = {node_ids[n][0], node_ids[n][1], node_ids[n][2]};

			// f0 component //////////////////////////////////////
			P[0] = U_of_x[0][0] * T_of_roots[1][j[0]];
			P[1] = T_of_x[1][1] * T_of_roots[1][j[1]];
			P[2] = T_of_x[1][2] * T_of_roots[1][j[2]];
			for (unsigned int o=2; o<ORDER; ++o) {
				P[0] += U_of_x[o-1][0] * T_of_roots[o][j[0]];
				P[1] += T_of_x[o  ][1] * T_of_roots[o][j[1]];
				P[2] += T_of_x[o  ][2] * T_of_roots[o][j[2]];
			}
			P[0] *= FReal(2.);
			P[1] *= FReal(2.); P[1] += FReal(1.);
			P[2] *= FReal(2.); P[2] += FReal(1.);
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			forces[0]	+= P[0] * P[1] * P[2] * localExpansion[n];
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			// f1 component //////////////////////////////////////
			P[0] = T_of_x[1][0] * T_of_roots[1][j[0]];
			P[1] = U_of_x[0][1] * T_of_roots[1][j[1]];
			P[2] = T_of_x[1][2] * T_of_roots[1][j[2]];
			for (unsigned int o=2; o<ORDER; ++o) {
				P[0] += T_of_x[o  ][0] * T_of_roots[o][j[0]];
				P[1] += U_of_x[o-1][1] * T_of_roots[o][j[1]];
				P[2] += T_of_x[o  ][2] * T_of_roots[o][j[2]];
			}
			P[0] *= FReal(2.); P[0] += FReal(1.);
			P[1] *= FReal(2.); 
			P[2] *= FReal(2.); P[2] += FReal(1.);
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			forces[1]	+= P[0] * P[1] * P[2] * localExpansion[n];
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			// f2 component //////////////////////////////////////
			P[0] = T_of_x[1][0] * T_of_roots[1][j[0]];
			P[1] = T_of_x[1][1] * T_of_roots[1][j[1]];
			P[2] = U_of_x[0][2] * T_of_roots[1][j[2]];
			for (unsigned int o=2; o<ORDER; ++o) {
				P[0] += T_of_x[o  ][0] * T_of_roots[o][j[0]];
				P[1] += T_of_x[o  ][1] * T_of_roots[o][j[1]];
				P[2] += U_of_x[o-1][2] * T_of_roots[o][j[2]];
			}
			P[0] *= FReal(2.); P[0] += FReal(1.);
			P[1] *= FReal(2.); P[1] += FReal(1.);
			P[2] *= FReal(2.);
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			forces[2]	+= P[0] * P[1] * P[2] * localExpansion[n];
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		}

		// scale forces
		forces[0] *= jacobian[0] / nnodes;
		forces[1] *= jacobian[1] / nnodes;
		forces[2] *= jacobian[2] / nnodes;

		// set computed forces
		iter.data().incForces(forces[0] * iter.data().getPhysicalValue(),
													forces[1] * iter.data().getPhysicalValue(),
													forces[2] * iter.data().getPhysicalValue());

		// increment iterator
		iter.gotoNext();
	}
}


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/**
 * Local to particle operation: application of \f$S_\ell(x,\bar x_m)\f$ and
 * \f$\nabla_x S_\ell(x,\bar x_m)\f$ (interpolation)
 */
template <int ORDER>
template <class ContainerClass>
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inline void FChebInterpolator<ORDER>::applyL2PTotal(const FPoint& center,
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																										const FReal width,
																										const FReal *const localExpansion,
																										ContainerClass *const localParticles) const
{
	// setup local to global mapping
	const map_glob_loc map(center, width);
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	FPoint Jacobian;
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	map.computeJacobian(Jacobian); // 6 flops
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	const FReal jacobian[3] = {Jacobian.getX(), Jacobian.getY(), Jacobian.getZ()}; 
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	FPoint localPosition;
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	FReal T_of_x[ORDER][3];
	FReal U_of_x[ORDER][3];
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	FReal P[6];
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	//
	FReal xpx,ypy,zpz ;
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	FReal c1 = FReal(8.0) / nnodes; // 1 flop
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	//
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	typename ContainerClass::BasicIterator iter(*localParticles);
	while(iter.hasNotFinished()){
			
		// map global position to [-1,1]
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		map(iter.data().getPosition(), localPosition); // 15 flops
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		// evaluate chebyshev polynomials of source particle
		// T_0(x_i) and T_1(x_i)
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		xpx = FReal(2.) * localPosition.getX(); // 1 flop
		ypy = FReal(2.) * localPosition.getY(); // 1 flop
		zpz = FReal(2.) * localPosition.getZ(); // 1 flop
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		//
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		T_of_x[0][0] = FReal(1.);	T_of_x[1][0] = localPosition.getX();
		T_of_x[0][1] = FReal(1.);	T_of_x[1][1] = localPosition.getY();
		T_of_x[0][2] = FReal(1.);	T_of_x[1][2] = localPosition.getZ();
		// U_0(x_i) and U_1(x_i)
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		// U_of_x[0][0] = FReal(1.);	U_of_x[1][0] = localPosition.getX() * FReal(2.);
		// U_of_x[0][1] = FReal(1.);	U_of_x[1][1] = localPosition.getY() * FReal(2.);
		// U_of_x[0][2] = FReal(1.);	U_of_x[1][2] = localPosition.getZ() * FReal(2.);
		U_of_x[0][0] = FReal(1.);	U_of_x[1][0] = xpx;
		U_of_x[0][1] = FReal(1.);	U_of_x[1][1] = ypy;
		U_of_x[0][2] = FReal(1.);	U_of_x[1][2] = zpz;
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		for (unsigned int o=2; o<ORDER; ++o) {
			// T_o(x_i)
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			// T_of_x[o][0] = FReal(2.)*localPosition.getX()*T_of_x[o-1][0] - T_of_x[o-2][0];
			// T_of_x[o][1] = FReal(2.)*localPosition.getY()*T_of_x[o-1][1] - T_of_x[o-2][1];
			// T_of_x[o][2] = FReal(2.)*localPosition.getZ()*T_of_x[o-1][2] - T_of_x[o-2][2];
			// // U_o(x_i)
			// U_of_x[o][0] = FReal(2.)*localPosition.getX()*U_of_x[o-1][0] - U_of_x[o-2][0];
			// U_of_x[o][1] = FReal(2.)*localPosition.getY()*U_of_x[o-1][1] - U_of_x[o-2][1];
			// U_of_x[o][2] = FReal(2.)*localPosition.getZ()*U_of_x[o-1][2] - U_of_x[o-2][2];
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			T_of_x[o][0] = xpx * T_of_x[o-1][0] - T_of_x[o-2][0]; // 2 flops 
			T_of_x[o][1] = ypy * T_of_x[o-1][1] - T_of_x[o-2][1]; // 2 flops
			T_of_x[o][2] = zpz * T_of_x[o-1][2] - T_of_x[o-2][2]; // 2 flops
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			// U_o(x_i)
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			U_of_x[o][0] = xpx * U_of_x[o-1][0] - U_of_x[o-2][0]; // 2 flops
			U_of_x[o][1] = ypy * U_of_x[o-1][1] - U_of_x[o-2][1]; // 2 flops
			U_of_x[o][2] = zpz * U_of_x[o-1][2] - U_of_x[o-2][2]; // 2 flops
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		}

		// scale, because dT_o/dx = oU_{o-1}
		for (unsigned int o=2; o<ORDER; ++o) {
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			U_of_x[o-1][0] *= FReal(o); // 1 flops
			U_of_x[o-1][1] *= FReal(o); // 1 flops
			U_of_x[o-1][2] *= FReal(o); // 1 flops
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		}

		// apply P and increment forces
		FReal potential = FReal(0.);
		FReal forces[3] = {FReal(0.), FReal(0.), FReal(0.)};
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		//
		// Optimization:
		//   Here we compute 1/2 S and 1/2 P  rather S and F like in the paper
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		for (unsigned int n=0; n<nnodes; ++n) {
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		  // tensor indices of chebyshev nodes
		  const unsigned int j[3] = {node_ids[n][0], node_ids[n][1], node_ids[n][2]};
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		  //
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		  P[0] = FReal(0.5) + T_of_x[1][0] * T_of_roots[1][j[0]]; // 2 flops 
		  P[1] = FReal(0.5) + T_of_x[1][1] * T_of_roots[1][j[1]]; // 2 flops
		  P[2] = FReal(0.5) + T_of_x[1][2] * T_of_roots[1][j[2]]; // 2 flops
		  P[3] = U_of_x[0][0] * T_of_roots[1][j[0]]; // 1 flop
		  P[4] = U_of_x[0][1] * T_of_roots[1][j[1]]; // 1 flop
		  P[5] = U_of_x[0][2] * T_of_roots[1][j[2]]; // 1 flop
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		  for (unsigned int o=2; o<ORDER; ++o) {
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		    P[0] += T_of_x[o  ][0] * T_of_roots[o][j[0]]; // 2 flop
		    P[1] += T_of_x[o  ][1] * T_of_roots[o][j[1]]; // 2 flop
		    P[2] += T_of_x[o  ][2] * T_of_roots[o][j[2]]; // 2 flop
		    P[3] += U_of_x[o-1][0] * T_of_roots[o][j[0]]; // 2 flop
		    P[4] += U_of_x[o-1][1] * T_of_roots[o][j[1]]; // 2 flop
		    P[5] += U_of_x[o-1][2] * T_of_roots[o][j[2]]; // 2 flop
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		  }
		  //
889 890 891 892
		  potential	+= P[0] * P[1] * P[2] * localExpansion[n]; // 4 flops
		  forces[0]	+= P[3] * P[1] * P[2] * localExpansion[n]; // 4 flops
		  forces[1]	+= P[0] * P[4] * P[2] * localExpansion[n]; // 4 flops
		  forces[2]	+= P[0] * P[1] * P[5] * localExpansion[n]; // 4 flops
893
		}
894
		//
895 896 897 898
		potential *= c1 ; // 1 flop
		forces[0] *= jacobian[0] *c1; // 2 flops 
		forces[1] *= jacobian[1] *c1; // 2 flops
		forces[2] *= jacobian[2] *c1; // 2 flops
899
		// set computed potential
900 901
		iter.data().incPotential(potential); // 1 flop
		
902 903
		// set computed forces
		iter.data().incForces(forces[0] * iter.data().getPhysicalValue(),
904 905
													forces[1] * iter.data().getPhysicalValue(),
													forces[2] * iter.data().getPhysicalValue()); // 6 flops
906 907 908 909 910 911 912

		// increment iterator
		iter.gotoNext();
	}
}


913
#endif
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		////struct IMN2MNI {
		////	enum {size = ORDER * (ORDER-1) * (ORDER-1)};
		////	unsigned int imn[size], mni[size];
		////	IMN2MNI() {
		////		unsigned int counter = 0;
		////		for (unsigned int i=0; i<ORDER; ++i) {
		////			for (unsigned int m=0; m<ORDER-1; ++m) {
		////				for (unsigned int n=0; n<ORDER-1; ++n) {
		////					imn[counter] = n*(ORDER-1)*ORDER + m*ORDER + i;
		////					mni[counter] = i*(ORDER-1)*(ORDER-1) + n*(ORDER-1) + m;
		////					counter++;
		////				}
		////			}
		////		}
		////	}
		////} perm0;
		//
		////for (unsigned int i=0; i<ORDER; ++i) {
		////	for (unsigned int m=0; m<ORDER-1; ++m) {
		////		for (unsigned int n=0; n<ORDER-1; ++n) {
		////			const unsigned int a = n*(ORDER-1)*ORDER + m*ORDER + i;
		////			const unsigned int b = i*(ORDER-1)*(ORDER-1) + n*(ORDER-1) + m;
		////			E[b] = D[a];
		////		}
		////	}
		////}

		////struct JNI2NIJ {
		////	enum {size = ORDER * ORDER * (ORDER-1)};
		////	unsigned int jni[size], nij[size];
		////	JNI2NIJ() {
		////		unsigned int counter = 0;
		////		for (unsigned int i=0; i<ORDER; ++i) {
		////			for (unsigned int j=0; j<ORDER; ++j) {
		////				for (unsigned int n=0; n<ORDER-1; ++n) {
		////					jni[counter] = i*(ORDER-1)*ORDER + n*ORDER + j;
		////					nij[counter] = j*ORDER*(ORDER-1) + i*(ORDER-1) + n;
		////					counter++;
		////				}
		////			}
		////		}
		////	}
		////} perm1;
		//
		////for (unsigned int i=0; i<ORDER; ++i) {
		////	for (unsigned int j=0; j<ORDER; ++j) {
		////		for (unsigned int n=0; n<ORDER-1; ++n) {
		////			const unsigned int a = i*(ORDER-1)*ORDER + n*ORDER + j;
		////			const unsigned int b = j*ORDER*(ORDER-1) + i*(ORDER-1) + n;
		////			G[b] = F[a];
		////		}
		////	}
		////}

		////struct KIJ2IJK {
		////	enum {size = ORDER * ORDER * ORDER};
		////	unsigned int kij[size], ijk[size];
		////	KIJ2IJK() {
		////		unsigned int counter = 0;
		////		for (unsigned int i=0; i<ORDER; ++i) {
		////			for (unsigned int j=0; j<ORDER; ++j) {
		////				for (unsigned int k=0; k<ORDER; ++k) {
		////					kij[counter] = j*ORDER*ORDER + i*ORDER + k;
		////					ijk[counter] = k*ORDER*ORDER + j*ORDER + i;
		////					counter++;
		////				}
		////			}
		////		}
		////	}
		////} perm2;
		//
		////for (unsigned int i=0; i<ORDER; ++i) {
		////	for (unsigned int j=0; j<ORDER; ++j) {
		////		for (unsigned int k=0; k<ORDER; ++k) {
		////			const unsigned int a = j*ORDER*ORDER + i*ORDER + k;
		////			const unsigned int b = k*ORDER*ORDER + j*ORDER + i;
		////			F8[b] = H[a];
		////		}
		////	}
		////}

		//FReal T_of_y[ORDER * (ORDER-1)];
		//for (unsigned int o=1; o<ORDER; ++o)
		//	for (unsigned int j=0; j<ORDER; ++j)
		//		T_of_y[(o-1)*ORDER + j] = FReal(FChebRoots<ORDER>::T(o, FReal(FChebRoots<ORDER>::roots[j])));

	//struct SumP2M {
	//	unsigned int f2[3][nnodes], f4[3][nnodes];
	//	SumP2M() {
	//		for (unsigned int i=0; i<ORDER; ++i) {
	//			for (unsigned int j=0; j<ORDER; ++j) {
	//				for (unsigned int k=0; k<ORDER; ++k) {
	//					const unsigned int idx = k*ORDER*ORDER + j*ORDER + i;
	//					f2[0][idx] = i;
	//					f2[1][idx] = j;
	//					f2[2][idx] = k;
	//					f4[0][idx] = j*ORDER+i;
	//					f4[1][idx] = k*ORDER+i;
	//					f4[2][idx] = k*ORDER+j;
	//				}
	//			}
	//		}
	//	}
	//} idx0;
	//
	//for (unsigned int i=0; i<nnodes; ++i)
	//	multipoleExpansion[i] = (W1 + 
	//													 FReal(2.) * (F2[0][idx0.f2[0][i]] + F2[1][idx0.f2[1][i]] + F2[2][idx0.f2[2][i]]) +
	//													 FReal(4.) * (F4[0][idx0.f4[0][i]] + F4[1][idx0.f4[1][i]] + F4[2][idx0.f4[2][i]]) +
	//													 FReal(8.) *  F8[i]) / nnodes;