FChebInterpolator.hpp 53.1 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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	////////////////////////////////////////////////////////////////////
	// needed for L2P
	struct IJK2JKI {
		enum {size = ORDER * ORDER * ORDER};
		unsigned int ijk[size], jki[size];
		IJK2JKI() {
			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) {
						ijk[counter] = k*ORDER*ORDER + j*ORDER + i;
						jki[counter] = i*ORDER*ORDER + k*ORDER + j;
						counter++;
					}
				}
			}
		}
		void permute(const FReal *const IN, FReal *const OUT) const
		{ for (unsigned int i=0; i<size; ++i) OUT[jki[i]] = IN[ijk[i]]; }
	} perm3;

	struct IJK2KIJ {
		enum {size = ORDER * ORDER * ORDER};
		unsigned int ijk[size], kij[size];
		IJK2KIJ() {
			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) {
						ijk[counter] = k*ORDER*ORDER + j*ORDER + i;
						kij[counter] = j*ORDER*ORDER + i*ORDER + k;
						counter++;
					}
				}
			}
		}
		void permute(const FReal *const IN, FReal *const OUT) const
		{ for (unsigned int i=0; i<size; ++i) OUT[kij[i]] = IN[ijk[i]]; }
	} perm4;

	struct LJK2JKL {
		enum {size = (ORDER-1) * ORDER * ORDER};
		unsigned int ljk[size], jkl[size];
		LJK2JKL() {
			unsigned int counter = 0;
			for (unsigned int l=0; l<ORDER-1; ++l) {
				for (unsigned int j=0; j<ORDER; ++j) {
					for (unsigned int k=0; k<ORDER; ++k) {
						ljk[counter] = k*ORDER*(ORDER-1) + j*(ORDER-1) + l;
						jkl[counter] = l*ORDER*ORDER + k*ORDER + j;
						counter++;
					}
				}
			}
		}
		void permute(const FReal *const IN, FReal *const OUT) const
		{ for (unsigned int i=0; i<size; ++i) OUT[jkl[i]] = IN[ljk[i]]; }
	} perm5;

	struct LJK2KLJ {
		enum {size = (ORDER-1) * ORDER * ORDER};
		unsigned int ljk[size], klj[size];
		LJK2KLJ() {
			unsigned int counter = 0;
			for (unsigned int l=0; l<ORDER-1; ++l) {
				for (unsigned int j=0; j<ORDER; ++j) {
					for (unsigned int k=0; k<ORDER; ++k) {
						ljk[counter] = k*ORDER*(ORDER-1) + j*(ORDER-1) + l;
						klj[counter] = j*(ORDER-1)*ORDER + l*ORDER + k;
						counter++;
					}
				}
			}
		}
		void permute(const FReal *const IN, FReal *const OUT) const
		{ for (unsigned int i=0; i<size; ++i) OUT[klj[i]] = IN[ljk[i]]; }
	} perm6;

	struct MKI2KIM {
		enum {size = (ORDER-1) * ORDER * ORDER};
		unsigned int mki[size], kim[size];
		MKI2KIM() {
			unsigned int counter = 0;
			for (unsigned int m=0; m<ORDER-1; ++m) {
				for (unsigned int k=0; k<ORDER; ++k) {
					for (unsigned int i=0; i<ORDER; ++i) {
						mki[counter] = i*ORDER*(ORDER-1) + k*(ORDER-1) + m;
						kim[counter] = m*ORDER*ORDER + i*ORDER + k;
						counter++;
					}
				}
			}
		}
		void permute(const FReal *const IN, FReal *const OUT) const
		{ for (unsigned int i=0; i<size; ++i) OUT[kim[i]] = IN[mki[i]]; }
	} perm7;

	struct MKL2KLM {
		enum {size = (ORDER-1) * ORDER * (ORDER-1)};
		unsigned int mkl[size], klm[size];
		MKL2KLM() {
			unsigned int counter = 0;
			for (unsigned int m=0; m<ORDER-1; ++m) {
				for (unsigned int k=0; k<ORDER; ++k) {
					for (unsigned int l=0; l<ORDER-1; ++l) {
						mkl[counter] = l*ORDER*(ORDER-1) + k*(ORDER-1) + m;
						klm[counter] = m*(ORDER-1)*ORDER + l*ORDER + k;
						counter++;
					}
				}
			}
		}
		void permute(const FReal *const IN, FReal *const OUT) const
		{ for (unsigned int i=0; i<size; ++i) OUT[klm[i]] = IN[mkl[i]]; }
	} perm8;

	struct NLM2LMN {
		enum {size = (ORDER-1) * (ORDER-1) * (ORDER-1)};
		unsigned int nlm[size], lmn[size];
		NLM2LMN() {
			unsigned int counter = 0;
			for (unsigned int n=0; n<ORDER-1; ++n) {
				for (unsigned int l=0; l<ORDER-1; ++l) {
					for (unsigned int m=0; m<ORDER-1; ++m) {
						nlm[counter] = m*(ORDER-1)*(ORDER-1) + l*(ORDER-1) + n;
						lmn[counter] = n*(ORDER-1)*(ORDER-1) + m*(ORDER-1) + l;
						counter++;
					}
				}
			}
		}
		void permute(const FReal *const IN, FReal *const OUT) const
		{ for (unsigned int i=0; i<size; ++i) OUT[lmn[i]] = IN[nlm[i]]; }
	} perm9;

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


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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
{
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	FReal f1;
	FReal W2[3][ ORDER-1];
	FReal W4[3][(ORDER-1)*(ORDER-1)];
	FReal W8[   (ORDER-1)*(ORDER-1)*(ORDER-1)];
	{ // sum over interpolation points
		f1 = FReal(0.);
		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.);
		
		for (unsigned int idx=0; idx<nnodes; ++idx) {
			const unsigned int i = node_ids[idx][0];
			const unsigned int j = node_ids[idx][1];
			const unsigned int k = node_ids[idx][2];
			
			f1 += localExpansion[idx]; // 1 flop

			for (unsigned int l=0; l<ORDER-1; ++l) {
				const FReal wx = T[l*ORDER+i] * localExpansion[idx]; // 1 flops
				const FReal wy = T[l*ORDER+j] * localExpansion[idx]; // 1 flops
				const FReal wz = T[l*ORDER+k] * localExpansion[idx]; // 1 flops
				W2[0][l] += wx; // 1 flops
				W2[1][l] += wy; // 1 flops
				W2[2][l] += wz; // 1 flops
				for (unsigned int m=0; m<ORDER-1; ++m) {
					const FReal wxy = wx * T[m*ORDER + j]; // 1 flops
					const FReal wxz = wx * T[m*ORDER + k]; // 1 flops
					const FReal wyz = wy * T[m*ORDER + k]; // 1 flops
					W4[0][m*(ORDER-1)+l] += wxy; // 1 flops
					W4[1][m*(ORDER-1)+l] += wxz; // 1 flops
					W4[2][m*(ORDER-1)+l] += wyz; // 1 flops
					for (unsigned int n=0; n<ORDER-1; ++n) {
						const FReal wxyz = wxy * T[n*ORDER + k]; // 1 flops
						W8[n*(ORDER-1)*(ORDER-1) + m*(ORDER-1) + l]	+= wxyz; // 1 flops
					} // (ORDER-1) * 2 flops
				} // (ORDER-1) * (6 + (ORDER-1)*2) flops
			} // (ORDER-1) * (6 + (ORDER-1) * (6 + (ORDER-1)*2)) flops
		} // ORDER*ORDER*ORDER * (1 + (ORDER-1) * (6 + (ORDER-1) * (6 + (ORDER-1)*2))) flops
		
	}


	// loop over particles
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	const map_glob_loc map(center, width);
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	FPoint localPosition;
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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

		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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		}

		// interpolate and increment target value
		FReal targetValue = iter.data().getPotential();
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		{
			FReal f2, f4, f8;
			{
				f2 = f4 = f8 = FReal(0.);
				for (unsigned int l=1; l<ORDER; ++l) {
					f2 +=
						T_of_x[0][l] * W2[0][l-1] +
						T_of_x[1][l] * W2[1][l-1] +
						T_of_x[2][l] * W2[2][l-1]; // 6 flops
					for (unsigned int m=1; m<ORDER; ++m) {
						f4 +=
							T_of_x[0][l] * T_of_x[1][m] * W4[0][(m-1)*(ORDER-1)+(l-1)] +
							T_of_x[0][l] * T_of_x[2][m] * W4[1][(m-1)*(ORDER-1)+(l-1)] +
							T_of_x[1][l] * T_of_x[2][m] * W4[2][(m-1)*(ORDER-1)+(l-1)]; // 9 flops
						for (unsigned int n=1; n<ORDER; ++n) {
							f8 +=
								T_of_x[0][l] * T_of_x[1][m] * T_of_x[2][n] *
								W8[(n-1)*(ORDER-1)*(ORDER-1) + (m-1)*(ORDER-1) + (l-1)];
						} // ORDER * 4 flops
					} // ORDER * (9 + ORDER * 4) flops
				} // ORDER * (ORDER * (9 + ORDER * 4)) flops
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			}
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			targetValue = (f1 + FReal(2.)*f2 + FReal(4.)*f4 + FReal(8.)*f8) / nnodes; // 7 flops
		} // 7 + ORDER * (ORDER * (9 + ORDER * 4)) flops
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		// set potential
		iter.data().setPotential(targetValue);
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		// increment target iterator
		iter.gotoNext();
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	} // N * (7 + ORDER * (ORDER * (9 + ORDER * 4))) flops
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}


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//	FReal F2[3][ORDER-1];
//	FBlas::gemtv(ORDER, ORDER-1, FReal(1.), const_cast<FReal*>(T), const_cast<FReal*>(localExpansion), F2[0]);
//	for (unsigned int i=1; i<ORDER*ORDER; ++i)
//		FBlas::gemtva(ORDER, ORDER-1, FReal(1.), const_cast<FReal*>(T),
//									const_cast<FReal*>(localExpansion) + ORDER*i, F2[0]);
//	for (unsigned int i=0; i<ORDER-1; ++i)
//		std::cout << W2[0][i] << "\t" << F2[0][i] << std::endl;
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//	FReal F2[(ORDER-1) * ORDER*ORDER];
//	FBlas::gemtm(ORDER, ORDER-1, ORDER*ORDER, FReal(1.), const_cast<FReal*>(T), ORDER,
//							 const_cast<FReal*>(localExpansion), ORDER, F2, ORDER-1);
//	FReal F[ORDER-1]; FBlas::setzero(ORDER-1, F);
//	for (unsigned int i=0; i<ORDER-1; ++i)
//		for (unsigned int j=0; j<ORDER*ORDER; ++j) F[i] += F2[j*(ORDER-1) + i];
//	for (unsigned int i=0; i<ORDER-1; ++i)
//		std::cout << W2[0][i] << "\t" << F[i] << std::endl;
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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>
//inline void FChebInterpolator<ORDER>::applyL2P(const FPoint& center,
//																							 const FReal width,
//																							 const FReal *const localExpansion,
//																							 ContainerClass *const localParticles) const
//{
//	// allocate stuff
//	const map_glob_loc map(center, width);
//	FPoint localPosition;
//	FReal T_of_x[ORDER][3];
//	FReal xpx,ypy,zpz ;
//	FReal S[3],c1;
//	//
//	c1 = FReal(8.) / nnodes ;
//	typename ContainerClass::BasicIterator iter(*localParticles);
//	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 flop
//			T_of_x[o][1] = ypy * T_of_x[o-1][1] - T_of_x[o-2][1]; // 2 flop
//			T_of_x[o][2] = zpz * T_of_x[o-1][2] - T_of_x[o-2][2]; // 2 flop
//		} // (ORDER-2) * 6 flops
//
//		// 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]]; // 1 flops
//			S[1] = T_of_x[1][1] * T_of_roots[1][j[1]]; // 1 flops
//			S[2] = T_of_x[1][2] * T_of_roots[1][j[2]]; // 1 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
//			} // (ORDER-2) * 6 flops 
//			// gather contributions
//			S[0] += FReal(0.5); // 1 flops
//			S[1] += FReal(0.5); // 1 flops
//			S[2] += FReal(0.5); // 1 flops
//			targetValue	+= S[0] * S[1] * S[2] * localExpansion[n]; // 4 flops
//		} // ORDER*ORDER*ORDER * (10 + (ORDER-2) * 6) flops
//		// scale
//		targetValue *= c1; // 1 flops
//
//		// set potential
//		iter.data().setPotential(targetValue);
//
//		// increment target iterator
//		iter.gotoNext();
//	} // N * ORDER*ORDER*ORDER * (10 + (ORDER-2) * 6) flops
//}
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///**
// * Local to particle operation: application of \f$\nabla_x S_\ell(x,\bar x_m)\f$ (interpolation)
// */
//template <int ORDER>
//template <class ContainerClass>
//inline void FChebInterpolator<ORDER>::applyL2PGradient(const FPoint& center,
//																											 const FReal width,
//																											 const FReal *const localExpansion,
//																											 ContainerClass *const localParticles) const
//{
//	// setup local to global mapping
//	const map_glob_loc map(center, width);
//	FPoint Jacobian;
//	map.computeJacobian(Jacobian);
//	const FReal jacobian[3] = {Jacobian.getX(), Jacobian.getY(), Jacobian.getZ()}; 
//	FPoint localPosition;
//	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.);
//			forces[0]	+= P[0] * P[1] * P[2] * localExpansion[n];
//
//			// 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.);
//			forces[1]	+= P[0] * P[1] * P[2] * localExpansion[n];
//
//			// 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.);
//			forces[2]	+= P[0] * P[1] * P[2] * localExpansion[n];
//		}
//
//		// 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
{
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	FReal f1;
	FReal W2[3][ ORDER-1];
	FReal W4[3][(ORDER-1)*(ORDER-1)];
	FReal W8[   (ORDER-1)*(ORDER-1)*(ORDER-1)];

	//{ // sum over interpolation points
	//	f1 = FReal(0.);
	//	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.);
	//	
	//	for (unsigned int idx=0; idx<nnodes; ++idx) {
	//		const unsigned int i = node_ids[idx][0];
	//		const unsigned int j = node_ids[idx][1];
	//		const unsigned int k = node_ids[idx][2];
	//		
	//		f1 += localExpansion[idx]; // 1 flop
	//		
	//		for (unsigned int l=0; l<ORDER-1; ++l) {
	//			const FReal wx = T[l*ORDER+i] * localExpansion[idx]; // 1 flops
	//			const FReal wy = T[l*ORDER+j] * localExpansion[idx]; // 1 flops
	//			const FReal wz = T[l*ORDER+k] * localExpansion[idx]; // 1 flops
	//			W2[0][l] += wx; // 1 flops
	//			W2[1][l] += wy; // 1 flops
	//			W2[2][l] += wz; // 1 flops
	//			for (unsigned int m=0; m<ORDER-1; ++m) {
	//				const FReal wxy = wx * T[m*ORDER + j]; // 1 flops
	//				const FReal wxz = wx * T[m*ORDER + k]; // 1 flops
	//				const FReal wyz = wy * T[m*ORDER + k]; // 1 flops
	//				W4[0][m*(ORDER-1)+l] += wxy; // 1 flops
	//				W4[1][m*(ORDER-1)+l] += wxz; // 1 flops
	//				W4[2][m*(ORDER-1)+l] += wyz; // 1 flops
	//				for (unsigned int n=0; n<ORDER-1; ++n) {
	//					const FReal wxyz = wxy * T[n*ORDER + k]; // 1 flops
	//					W8[n*(ORDER-1)*(ORDER-1) + m*(ORDER-1) + l]	+= wxyz; // 1 flops
	//				} // (ORDER-1) * 2 flops
	//			} // (ORDER-1) * (6 + (ORDER-1)*2) flops
	//		} // (ORDER-1) * (6 + (ORDER-1) * (6 + (ORDER-1)*2)) flops
	//
	//	} // ORDER*ORDER*ORDER * (1 + (ORDER-1) * (6 + (ORDER-1) * (6 + (ORDER-1)*2))) flops
	//	
	//}

	{
		// for W2
		FReal lE[nnodes];
		FReal F2[(ORDER-1) * ORDER*ORDER];
		// for W4
		FReal F4[ORDER * ORDER*(ORDER-1)];
		FReal G4[(ORDER-1) * ORDER*(ORDER-1)];
		// for W8
		FReal G8[ORDER * (ORDER-1)*(ORDER-1)];

		// sum local expansions
		f1 = FReal(0.);
		for (unsigned int idx=0; idx<nnodes; ++idx)	f1 += localExpansion[idx]; // 1 flop

		//////////////////////////////////////////////////////////////////
		// IMPORTANT: NOT CHANGE ORDER OF COMPUTATIONS!!! ////////////////
		//////////////////////////////////////////////////////////////////

		// W2[0] /////////////////
		FBlas::gemtm(ORDER, ORDER-1, ORDER*ORDER, FReal(1.), const_cast<FReal*>(T), ORDER,
								 const_cast<FReal*>(localExpansion), ORDER, F2, ORDER-1);
		for (unsigned int l=0; l<ORDER-1; ++l) { W2[0][l] = F2[l];
			for (unsigned int j=1; j<ORDER*ORDER; ++j) W2[0][l] += F2[j*(ORDER-1) + l];	}
		// W4[0] /////////////////
		perm5.permute(F2, F4);
		FBlas::gemtm(ORDER, ORDER-1, ORDER*(ORDER-1), FReal(1.), const_cast<FReal*>(T), ORDER, F4, ORDER, G4, ORDER-1);
		for (unsigned int l=0; l<ORDER-1; ++l)
			for (unsigned int m=0; m<ORDER-1; ++m) { W4[0][m*(ORDER-1)+l] = G4[l*ORDER*(ORDER-1) + m];
				for (unsigned int k=1; k<ORDER; ++k) W4[0][m*(ORDER-1)+l] += G4[l*ORDER*(ORDER-1) + k*(ORDER-1) + m];	}
		// W8 ////////////////////
		perm8.permute(G4, G8);
		FReal F8[(ORDER-1)*(ORDER-1)*(ORDER-1)];
		FBlas::gemtm(ORDER, ORDER-1, (ORDER-1)*(ORDER-1), FReal(1.), const_cast<FReal*>(T), ORDER, G8, ORDER, F8, ORDER-1);
		perm9.permute(F8, W8);
		// W4[1] /////////////////
		perm6.permute(F2, F4);
		FBlas::gemtm(ORDER, ORDER-1, (ORDER-1)*ORDER, FReal(1.), const_cast<FReal*>(T), ORDER, F4, ORDER, G4, ORDER-1);
		for (unsigned int l=0; l<ORDER-1; ++l)
			for (unsigned int n=0; n<ORDER-1; ++n) { W4[1][n*(ORDER-1)+l] = G4[l*(ORDER-1) + n];
				for (unsigned int j=1; j<ORDER; ++j) W4[1][n*(ORDER-1)+l] += G4[j*(ORDER-1)*(ORDER-1) + l*(ORDER-1) + n];	}
		// W2[1] /////////////////
		perm3.permute(localExpansion, lE);
		FBlas::gemtm(ORDER, ORDER-1, ORDER*ORDER, FReal(1.), const_cast<FReal*>(T), ORDER, lE, ORDER, F2, ORDER-1);
		for (unsigned int i=0; i<ORDER-1; ++i) { W2[1][i] = F2[i];
			for (unsigned int j=1; j<ORDER*ORDER; ++j) W2[1][i] += F2[j*(ORDER-1) + i]; }
		// W4[2] /////////////////
		perm7.permute(F2, F4);
		FBlas::gemtm(ORDER, ORDER-1, (ORDER-1)*ORDER, FReal(1.), const_cast<FReal*>(T), ORDER, F4, ORDER, G4, ORDER-1);
		for (unsigned int m=0; m<ORDER-1; ++m)
			for (unsigned int n=0; n<ORDER-1; ++n) { W4[2][n*(ORDER-1)+m] = G4[m*ORDER*(ORDER-1) + n];
				for (unsigned int i=1; i<ORDER; ++i) W4[2][n*(ORDER-1)+m] += G4[m*ORDER*(ORDER-1) + i*(ORDER-1) + n];	}
		// W2[2] /////////////////
		perm4.permute(localExpansion, lE);
		FBlas::gemtm(ORDER, ORDER-1, ORDER*ORDER, FReal(1.), const_cast<FReal*>(T), ORDER, lE, ORDER, F2, ORDER-1);
		for (unsigned int i=0; i<ORDER-1; ++i) { W2[2][i] = F2[i];
			for (unsigned int j=1; j<ORDER*ORDER; ++j) W2[2][i] += F2[j*(ORDER-1) + i]; }
	}

	
	// loop over particles
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	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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	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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		FReal U_of_x[3][ORDER];
		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
			U_of_x[0][0] = FReal(1.);	U_of_x[0][1] = x2;
			U_of_x[1][0] = FReal(1.);	U_of_x[1][1] = y2;
			U_of_x[2][0] = FReal(1.);	U_of_x[2][1] = z2;
			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
				U_of_x[0][j] = x2 * U_of_x[0][j-1] - U_of_x[0][j-2]; // 2 flops
				U_of_x[1][j] = y2 * U_of_x[1][j-1] - U_of_x[1][j-2]; // 2 flops
				U_of_x[2][j] = z2 * U_of_x[2][j-1] - U_of_x[2][j-2]; // 2 flops
			}
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			// scale, because dT_j/dx = jU_{j-1}
			for (unsigned int j=2; j<ORDER; ++j) {
				U_of_x[0][j-1] *= FReal(j); // 1 flops
				U_of_x[1][j-1] *= FReal(j); // 1 flops
				U_of_x[2][j-1] *= FReal(j); // 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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		{
			FReal f2[4], f4[4], f8[4];
			for (unsigned int i=0; i<4; ++i) f2[i] = f4[i] = f8[i] = FReal(0.);
			{
				for (unsigned int l=1; l<ORDER; ++l) {
					const FReal w2[3] = {W2[0][l-1], W2[1][l-1], W2[2][l-1]};
					f2[0] += T_of_x[0][l  ] * w2[0] + T_of_x[1][l] * w2[1] + T_of_x[2][l] * w2[2]; // 6 flops
					f2[1] += U_of_x[0][l-1] * w2[0]; // 2 flops
					f2[2] += U_of_x[1][l-1] * w2[1]; // 2 flops
					f2[3] += U_of_x[2][l-1] * w2[2]; // 2 flops
					for (unsigned int m=1; m<ORDER; ++m) {
						const unsigned int w4idx = (m-1)*(ORDER-1)+(l-1);
						const FReal w4[3] = {W4[0][w4idx], W4[1][w4idx], W4[2][w4idx]};
						f4[0] +=
							T_of_x[0][l] * T_of_x[1][m] * w4[0] +
							T_of_x[0][l] * T_of_x[2][m] * w4[1] +
							T_of_x[1][l] * T_of_x[2][m] * w4[2]; // 9 flops
						f4[1] += U_of_x[0][l-1] * T_of_x[1][m]   * w4[0] + U_of_x[0][l-1] * T_of_x[2][m]   * w4[1]; // 6 flops
						f4[2] += T_of_x[0][l]   * U_of_x[1][m-1] * w4[0] + U_of_x[1][l-1] * T_of_x[2][m]   * w4[2]; // 6 flops
						f4[3] += T_of_x[0][l]   * U_of_x[2][m-1] * w4[1] + T_of_x[1][l]   * U_of_x[2][m-1] * w4[2]; // 6 flops
						for (unsigned int n=1; n<ORDER; ++n) {
							const FReal w8 = W8[(n-1)*(ORDER-1)*(ORDER-1) + (m-1)*(ORDER-1) + (l-1)];
							f8[0] += T_of_x[0][l]   * T_of_x[1][m]   * T_of_x[2][n]   * w8;
							f8[1] += U_of_x[0][l-1] * T_of_x[1][m]   * T_of_x[2][n]   * w8;
							f8[2] += T_of_x[0][l]   * U_of_x[1][m-1] * T_of_x[2][n]   * w8;
							f8[3] += T_of_x[0][l]   * T_of_x[1][m]   * U_of_x[2][n-1] * w8;
						} // ORDER * 4 flops
					} // ORDER * (9 + ORDER * 4) flops
				} // ORDER * (ORDER * (9 + ORDER * 4)) flops
			}
			potential = (f1 + FReal(2.)*f2[0] + FReal(4.)*f4[0] + FReal(8.)*f8[0]) / nnodes; // 7 flops
			forces[0] = (     FReal(2.)*f2[1] + FReal(4.)*f4[1] + FReal(8.)*f8[1]) * jacobian[0] / nnodes; // 6 flops
			forces[1] = (     FReal(2.)*f2[2] + FReal(4.)*f4[2] + FReal(8.)*f8[2]) * jacobian[1] / nnodes; // 6 flops
			forces[2] = (     FReal(2.)*f2[3] + FReal(4.)*f4[3] + FReal(8.)*f8[3]) * jacobian[2] / nnodes; // 6 flops
		} // 7 + ORDER * (ORDER * (9 + ORDER * 4)) flops

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		// set computed potential
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		iter.data().incPotential(potential); // 1 flop
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		// set computed forces
		iter.data().incForces(forces[0] * iter.data().getPhysicalValue(),
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													forces[1] * iter.data().getPhysicalValue(),
													forces[2] * iter.data().getPhysicalValue()); // 6 flops
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		// increment target iterator
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		iter.gotoNext();
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	} // N * (7 + ORDER * (ORDER * (9 + ORDER * 4))) flops
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}


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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>
//inline void FChebInterpolator<ORDER>::applyL2PTotal(const FPoint& center,
//																										const FReal width,
//																										const FReal *const localExpansion,
//																										ContainerClass *const localParticles) const
//{
//	// setup local to global mapping
//	const map_glob_loc map(center, width);
//	FPoint Jacobian;
//	map.computeJacobian(Jacobian); // 6 flops
//	const FReal jacobian[3] = {Jacobian.getX(), Jacobian.getY(), Jacobian.getZ()}; 
//	FPoint localPosition;
//	FReal T_of_x[ORDER][3];
//	FReal U_of_x[ORDER][3];
//	FReal P[6];
//	//
//	FReal xpx,ypy,zpz ;
//	FReal c1 = FReal(8.0) / nnodes; // 1 flop
//	//
//	typename ContainerClass::BasicIterator iter(*localParticles);
//	while(iter.hasNotFinished()){
//			
//		// map global position to [-1,1]
//		map(iter.data().getPosition(), localPosition); // 15 flops
//			
//		// evaluate chebyshev polynomials of source particle
//		// T_0(x_i) and T_1(x_i)
//		xpx = FReal(2.) * localPosition.getX(); // 1 flop
//		ypy = FReal(2.) * localPosition.getY(); // 1 flop
//		zpz = FReal(2.) * localPosition.getZ(); // 1 flop
//		//
//		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_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;
//		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
//			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
//		}
//
//		// scale, because dT_o/dx = oU_{o-1}
//		for (unsigned int o=2; o<ORDER; ++o) {
//			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
//		}
//
//		// apply P and increment forces
//		FReal potential = FReal(0.);
//		FReal forces[3] = {FReal(0.), FReal(0.), FReal(0.)};
//		//
//		// Optimization:
//		//   Here we compute 1/2 S and 1/2 P  rather S and F like in the paper
//		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]};
//		  //
//		  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
//		  for (unsigned int o=2; o<ORDER; ++o) {
//		    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
//		  }
//		  //
//		  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
//		}
//		//
//		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
//		// set computed potential
//		iter.data().incPotential(potential); // 1 flop
//		
//		// set computed forces
//		iter.data().incForces(forces[0] * iter.data().getPhysicalValue(),
//													forces[1] * iter.data().getPhysicalValue(),
//													forces[2] * iter.data().getPhysicalValue()); // 6 flops
//
//		// increment iterator
//		iter.gotoNext();
//	}
//}


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#endif
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