vmra.h

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/*
  This file is part of MADNESS.
 
  Copyright (C) 2007,2010 Oak Ridge National Laboratory
 
  This program is free software; you can redistribute it and/or modify
  it under the terms of the GNU General Public License as published by
  the Free Software Foundation; either version 2 of the License, or
  (at your option) any later version.
 
  This program is distributed in the hope that it will be useful,
  but WITHOUT ANY WARRANTY; without even the implied warranty of
  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
  GNU General Public License for more details.
 
  You should have received a copy of the GNU General Public License
  along with this program; if not, write to the Free Software
  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
 
  For more information please contact:
 
  Robert J. Harrison
  Oak Ridge National Laboratory
  One Bethel Valley Road
  P.O. Box 2008, MS-6367
 
  email: harrisonrj@ornl.gov
  tel:   865-241-3937
  fax:   865-572-0680
 
  $Id$
*/
#ifndef MADNESS_MRA_VMRA_H__INCLUDED
#define MADNESS_MRA_VMRA_H__INCLUDED
 
/*!
    \file vmra.h
    \brief Defines operations on vectors of Functions
    \ingroup mra
 
    This file defines a number of operations on vectors of functions.
    Assume v is a vector of NDIM-D functions of a certain type.
 
 
    Operations on array of functions
 
    *) copying: deep copying of vectors of functions to vector of functions
    \code
    vector2 = copy(world, vector1,fence);
    \endcode
 
    *) compress: convert multiwavelet representation to legendre representation
    \code
    compress(world, vector, fence);
    \endcode
 
    *) reconstruct: convert representation to multiwavelets
    \code
    reconstruct(world, vector, fence);
    \endcode
 
    *) make_nonstandard: convert to non-standard form
    \code
    make_nonstandard(world, v, fence);
    \endcode
 
    *) standard: convert to standard form
    \code
    standard(world, v, fence);
    \endcode
 
    *) truncate: truncating vectors of functions to desired precision
    \code
    truncate(world, v, tolerance, fence);
    \endcode
 
 
    *) zero function: create a vector of zero functions of length n
    \code
    v=zero(world, n);
    \endcode
 
    *) transform: transform a representation from one basis to another
    \code
    transform(world, vector, tensor, tolerance, fence )
    \endcode
 
    Setting thresh-hold for precision
 
    *) set_thresh: setting a finite thresh-hold for a vector of functions
    \code
    void set_thresh(World& world, std::vector< Function<T,NDIM> >& v, double thresh, bool fence=true);
    \endcode
 
    Arithmetic Operations on arrays of functions
 
    *) conjugation: conjugate a vector of complex functions
 
    *) add
    *) sub
    *) mul
       - mul_sparse
    *) square
    *) gaxpy
    *) apply
 
    Norms, inner-products, blas-1 like operations on vectors of functions
 
    *) inner
    *) matrix_inner
    *) norm_tree
    *) normalize
    *) norm2
        - norm2s
    *) scale(world, v, alpha);
 
 
 
 
*/
 
#include <madness/mra/mra.h>
#include <madness/mra/derivative.h>
#include <madness/tensor/distributed_matrix.h>
#include <cstdio>
#include <algorithm>
 
namespace madness {
 
 
    /// get tree state of a vector of functions
 
    /// @return TreeState::unknown if the vector is empty or if the functions have different tree states
    template <typename T, std::size_t NDIM>
    TreeState get_tree_state(const std::vector<Function<T,NDIM>>& v) {
        if (v.size()==0) return TreeState::unknown;
        // return unknown if any function is not initialized
        if (std::any_of(v.begin(), v.end(), [](const Function<T,NDIM>& f) {return not f.is_initialized();})) {
            return TreeState::unknown;
        }
        TreeState state=v[0].get_impl()->get_tree_state();
        for (const auto& f : v) {
            if (f.get_impl()->get_tree_state()!=state) state=TreeState::unknown;
        }
        return state;
    }
 
    /// Compress a vector of functions
    template <typename T, std::size_t NDIM>
    void compress(World& world,
                  const std::vector< Function<T,NDIM> >& v,
                  bool fence=true) {
        PROFILE_BLOCK(Vcompress);
        change_tree_state(v, TreeState::compressed, fence);
    }
 
 
    /// reconstruct a vector of functions
 
    /// implies fence
    /// return v for chaining
    template <typename T, std::size_t NDIM>
    const std::vector< Function<T,NDIM> >& reconstruct(const std::vector< Function<T,NDIM> >& v) {
        return change_tree_state(v, TreeState::reconstructed, true);
    }
 
    /// compress a vector of functions
 
    /// implies fence
    /// return v for chaining
    template <typename T, std::size_t NDIM>
    const std::vector< Function<T,NDIM> >& compress(const std::vector< Function<T,NDIM> >& v) {
        return change_tree_state(v, TreeState::compressed, true);
    }
 
    /// Reconstruct a vector of functions
    template <typename T, std::size_t NDIM>
    void reconstruct(World& world,
                     const std::vector< Function<T,NDIM> >& v,
                     bool fence=true) {
        PROFILE_BLOCK(Vreconstruct);
        change_tree_state(v, TreeState::reconstructed, fence);
    }
 
    /// change tree_state of a vector of functions to redundant
    template <typename T, std::size_t NDIM>
    void make_redundant(World& world,
                  const std::vector< Function<T,NDIM> >& v,
                  bool fence=true) {
 
        PROFILE_BLOCK(Vcompress);
        change_tree_state(v, TreeState::redundant, fence);
    }
 
    /// refine the functions according to the autorefine criteria
    template <typename T, std::size_t NDIM>
    void refine(World& world, const std::vector<Function<T,NDIM> >& vf,
            bool fence=true) {
        for (const auto& f : vf) f.refine(false);
        if (fence) world.gop.fence();
    }
 
    /// refine all functions to a common (finest) level
 
    /// if functions are not initialized (impl==NULL) they are ignored
    template <typename T, std::size_t NDIM>
    void refine_to_common_level(World& world, std::vector<Function<T,NDIM> >& vf,
            bool fence=true) {
 
        reconstruct(world,vf);
        Key<NDIM> key0(0, Vector<Translation, NDIM> (0));
        std::vector<FunctionImpl<T,NDIM>*> v_ptr;
 
        // push initialized function pointers into the vector v_ptr
        for (unsigned int i=0; i<vf.size(); ++i) {
            if (vf[i].is_initialized()) v_ptr.push_back(vf[i].get_impl().get());
        }
 
        // sort and remove duplicates to not confuse the refining function
        std::sort(v_ptr.begin(),v_ptr.end());
        typename std::vector<FunctionImpl<T, NDIM>*>::iterator it;
        it = std::unique(v_ptr.begin(), v_ptr.end());
        v_ptr.resize( std::distance(v_ptr.begin(),it) );
 
        std::vector< Tensor<T> > c(v_ptr.size());
        v_ptr[0]->refine_to_common_level(v_ptr, c, key0);
        if (fence) v_ptr[0]->world.gop.fence();
        if (VERIFY_TREE)
            for (unsigned int i=0; i<vf.size(); i++) vf[i].verify_tree();
    }
 
    /// Generates non-standard form of a vector of functions
    template <typename T, std::size_t NDIM>
    void make_nonstandard(World& world,
                          std::vector< Function<T,NDIM> >& v,
                          bool fence= true) {
        PROFILE_BLOCK(Vnonstandard);
        change_tree_state(v, TreeState::nonstandard, fence);
    }
 
 
    /// Generates standard form of a vector of functions
    template <typename T, std::size_t NDIM>
    void standard(World& world,
                  std::vector< Function<T,NDIM> >& v,
                  bool fence=true) {
        PROFILE_BLOCK(Vstandard);
        change_tree_state(v, TreeState::compressed, fence);
    }
 
 
    /// change tree state of the functions
 
    /// might not respect fence
    /// @return v   for chaining
    template <typename T, std::size_t NDIM>
    const std::vector<Function<T,NDIM>>& change_tree_state(const std::vector<Function<T,NDIM>>& v,
                                                     const TreeState finalstate,
                                                     const bool fence=true) {
        // fast return
        if (v.size()==0) return v;
        if (get_tree_state(v)==finalstate) return v;
 
        // find initialized function with world
        Function<T,NDIM> dummy;
        for (const auto& f : v)
            if (f.is_initialized()) {
                dummy=f;
                break;
            }
        if (not dummy.is_initialized()) return v;
        World& world=dummy.world();
 
 
        // if a tree state cannot directly be changed to finalstate, we need to go via intermediate
        auto change_initial_to_intermediate =[](const std::vector<Function<T,NDIM>>& v,
                                                  const TreeState initialstate,
                                                  const TreeState intermediatestate) {
            int must_fence=0;
            for (auto& f : v) {
                if (f.is_initialized() and f.get_impl()->get_tree_state()==initialstate) {
                    f.change_tree_state(intermediatestate,false);
                    must_fence=1;
                }
            }
            return must_fence;
        };
 
        int do_fence=0;
        if (finalstate==compressed) {
            do_fence+=change_initial_to_intermediate(v,redundant,TreeState::reconstructed);
        }
        if (finalstate==nonstandard) {
            do_fence+=change_initial_to_intermediate(v,compressed,TreeState::reconstructed);
            do_fence+=change_initial_to_intermediate(v,redundant,TreeState::reconstructed);
        }
        if (finalstate==nonstandard_with_leaves) {
            do_fence+=change_initial_to_intermediate(v,compressed,TreeState::reconstructed);
            do_fence+=change_initial_to_intermediate(v,nonstandard,TreeState::reconstructed);
            do_fence+=change_initial_to_intermediate(v,redundant,TreeState::reconstructed);
        }
        if (finalstate==redundant) {
            do_fence+=change_initial_to_intermediate(v,compressed,TreeState::reconstructed);
            do_fence+=change_initial_to_intermediate(v,nonstandard,TreeState::reconstructed);
            do_fence+=change_initial_to_intermediate(v,nonstandard_with_leaves,TreeState::reconstructed);
        }
        if (do_fence>0) world.gop.fence();
 
        for (unsigned int i=0; i<v.size(); ++i) v[i].change_tree_state(finalstate,fence);
        if (fence) world.gop.fence();
 
        return v;
    }
 
    /// ensure v has the requested tree state, change the tree state of v if necessary and no fence is given
    template<typename T, std::size_t NDIM>
    bool ensure_tree_state_respecting_fence(const std::vector<Function<T,NDIM>>& v,
        const TreeState state, bool fence) {
        // fast return
        if (get_tree_state(v)==state) return true;;
 
        // if there is a fence we can simply change the tree state, might be a no-op
        if (fence) change_tree_state(v,state,true);
 
        // check success, throw if not
        bool ok=get_tree_state(v)==state;
        if (not ok) {
            print("ensure_tree_state_respecting_fence failed");
            throw std::runtime_error("ensure_tree_state_respecting_fence failed");
        }
        return ok;
    }
 
    /// Truncates a vector of functions
    template <typename T, std::size_t NDIM>
    void truncate(World& world,
                  std::vector< Function<T,NDIM> >& v,
                  double tol=0.0,
                  bool fence=true) {
        PROFILE_BLOCK(Vtruncate);
 
        // truncate in compressed form only for low-dimensional functions
        // compression is very expensive if low-rank tensor approximations are used
        if (NDIM<4) compress(world, v);
 
        for (auto& vv: v) {
            vv.truncate(tol, false);
        }
 
        if (fence) world.gop.fence();
    }
 
    /// Truncates a vector of functions
 
    /// @return the truncated vector for chaining
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> > truncate(std::vector< Function<T,NDIM> > v,
                  double tol=0.0, bool fence=true) {
        if (v.size()>0) truncate(v[0].world(),v,tol,fence);
        return v;
    }
 
    /// reduces the tensor rank of the coefficient tensor (if applicable)
 
    /// @return the vector for chaining
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> > reduce_rank(std::vector< Function<T,NDIM> > v,
                  double thresh=0.0, bool fence=true) {
        if (v.size()==0) return v;
        for (auto& vv : v) vv.reduce_rank(thresh,false);
        if (fence) v[0].world().gop.fence();
        return v;
    }
 
 
    /// Pre-stages the neighbor coefficients that differentiating v with each of grad will need
 
    /// Differentiating then serves those neighbors locally instead of fetching them one at a time.
    /// One halo per function holds every operator's pushes, so it pays when several functions are
    /// differentiated together. `clear_halo` frees them afterwards.
    template <typename T, std::size_t NDIM>
    void stage_halo(World& world,
                    const std::vector< std::shared_ptr< Derivative<T,NDIM> > >& grad,
                    const std::vector< Function<T,NDIM> >& v,
                    bool fence=true)
    {
        for (const auto& f : v) MADNESS_CHECK(f.is_reconstructed());
        for (const auto& D : grad)
            for (const auto& f : v) D->stage_halo(f.get_impl().get(), false);
        if (fence) world.gop.fence();
    }
 
    /// Discards the neighbor halos staged on v
 
    /// Requires a quiescent window: it frees tables the derivative may still be reading.
    template <typename T, std::size_t NDIM>
    void clear_halo(const std::vector< Function<T,NDIM> >& v)
    {
        for (const auto& f : v) f.get_impl()->halo_clear();
    }
 
    /// Applies a derivative operator to a vector of functions
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    apply(World& world,
          const Derivative<T,NDIM>& D,
          const std::vector< Function<T,NDIM> >& v,
          bool fence=true)
    {
        reconstruct(world, v);
        std::vector< Function<T,NDIM> > df(v.size());
        for (unsigned int i=0; i<v.size(); ++i) {
            df[i] = D(v[i],false);
        }
        if (fence) world.gop.fence();
        return df;
    }
 
    /// Generates a vector of zero functions with a given tree state
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    zero_functions_tree_state(World& world, int n, const TreeState state, bool fence=true) {
        std::vector< Function<T,NDIM> > r(n);
        for (int i=0; i<n; ++i) {
            if (state==compressed)
                r[i] = Function<T,NDIM>(FunctionFactory<T,NDIM>(world).fence(false).compressed(true).initial_level(1));
            else if (state==reconstructed)
                r[i] = Function<T,NDIM>(FunctionFactory<T,NDIM>(world).fence(false));
            else {
                print("zero_functions_tree_state: unknown tree state");
                throw std::runtime_error("zero_functions_tree_state: unknown tree state");
            }
        }
 
        if (n && fence) world.gop.fence();
        return r;
 
    }
 
    /// Generates a vector of zero functions (reconstructed)
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    zero_functions(World& world, int n, bool fence=true) {
        return zero_functions_tree_state<T,NDIM>(world,n,reconstructed,fence);
    }
 
    /// Generates a vector of zero functions (compressed)
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    zero_functions_compressed(World& world, int n, bool fence=true) {
        return zero_functions_tree_state<T,NDIM>(world,n,compressed,fence);
    }
 
    /// Generates a vector of zero functions, either compressed or reconstructed, depending on tensor type
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    zero_functions_auto_tree_state(World& world, int n, bool fence=true) {
        TreeState state=FunctionDefaults<NDIM>::get_tensor_type()==TT_FULL ? compressed : reconstructed;
        return zero_functions_tree_state<T,NDIM>(world,n,state,fence);
    }
 
 
 
    /// orthonormalize the vectors
    template<typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM>> orthonormalize(const std::vector<Function<T,NDIM> >& vf_in) {
        if (vf_in.size()==0) return std::vector<Function<T,NDIM>>();
        World& world=vf_in.front().world();
        auto vf=copy(world,vf_in);
        normalize(world,vf);
        if (vf.size()==1) return copy(world,vf_in);
        double maxq;
        double trantol=0.0;
        auto Q2=[](const Tensor<T>& s) {
            Tensor<T> Q = -0.5*s;
            for (int i=0; i<s.dim(0); ++i) Q(i,i) += 1.5;
            return Q;
        };
 
        do {
            Tensor<T> Q = Q2(matrix_inner(world, vf, vf));
            maxq=0.0;
            for (int i=0; i<Q.dim(0); ++i)
                for (int j=0; j<i; ++j)
                    maxq = std::max(maxq,std::abs(Q(i,j)));
 
            vf = transform(world, vf, Q, trantol, true);
            truncate(world, vf);
 
        } while (maxq>0.01);
        normalize(world,vf);
        return vf;
    }
 
 
    /// symmetric orthonormalization (see e.g. Szabo/Ostlund)
 
    /// @param[in] the vector to orthonormalize
    /// @param[in] overlap matrix
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > orthonormalize_symmetric(
            const std::vector<Function<T,NDIM> >& v,
            const Tensor<T>& ovlp,
            double lindep = 1e-12) {
        if(v.empty()) return v;
 
        World& world = v.front().world();
        const size_t n = v.size();
 
        Tensor<T> U;
        Tensor< typename Tensor<T>::scalar_type > s;
        syev(ovlp, U, s);
        lindep *= s(s.size() - 1);  // eigenvalues are in ascending order
 
        // transform s to s^{-1/2} in-place
        int rank = 0, nlindep = 0;
        for(size_t i = 0; i < n; ++i) {
            const auto s_i = s(i);
            s(i) = 1.0 / sqrt(s_i);
            (s_i > lindep) ? rank++ : nlindep++;
        }
        MADNESS_ASSERT(size_t(nlindep + rank) == n);
 
        // warn of linearly dependent vectors and values
        if (nlindep > 0) {
            if (world.rank() == 0)
                print("WARNING: linear dependencies detected in ", nlindep,
                      " functions, rank = ", rank);
        }
 
        // save Ut before U gets modified with s^{-1/2}
        const Tensor<T> Ut = conj_transpose(U);
 
        for(size_t i = 0; i < n; ++i){
            for(size_t j = 0; j < n; ++j){
                U(i, j) = U(i, j) * s(j);
            }
        }
 
        Tensor<T> X = inner(U, Ut, 1, 0);
 
        return transform(world, v, X);
    }
 
    /// convenience routine for symmetric orthonormalization (see e.g. Szabo/Ostlund)
    /// overlap matrix is calculated
    /// @param[in] the vector to orthonormalize
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > orthonormalize_symmetric(const std::vector<Function<T,NDIM> >& v,
            double lindep = 1e-12){
        if(v.empty()) return v;
 
        Tensor<T> ovlp = matrix_inner(v.front().world(), v, v, /* sym= */ true);
 
        return orthonormalize_symmetric(v, ovlp, lindep);
    }
 
    /// canonical orthonormalization (see e.g. Szabo/Ostlund)
    /// @param[in] the vector to orthonormalize
    /// @param[in] overlap matrix
    /// @param[in]  lindep  linear dependency threshold relative to largest eigenvalue
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > orthonormalize_canonical(
            const std::vector<Function<T,NDIM> >& v,
            const Tensor<T>& ovlp,
            double lindep = 1e-12) {
        if(v.empty()) return v;
 
        World& world = v.front().world();
        const size_t n = v.size();
 
        Tensor<T> U;
        Tensor< typename Tensor<T>::scalar_type > s;
        syev(ovlp, U, s);
        lindep *= s(s.size() - 1);  // eigenvalues are in ascending order
 
        // transform s to s^{-1/2} in-place
        size_t rank = 0, nlindep = 0;
        for(size_t i = 0; i < n; ++i) {
            const auto s_i = s(i);
            if (s_i > lindep) {
                s(i) = 1.0 / sqrt(s_i);
                rank++;
            } else {
                nlindep++;
            }
        }
        MADNESS_ASSERT(size_t(nlindep + rank) == n);
 
        // remove linearly dependent vectors and values
        if (nlindep > 0) {
            if (world.rank() == 0)
                print("Linear dependencies detected: removed ", nlindep,
                      " functions, rank = ", rank);
            U = U(_, Slice(nlindep, -1));
            s = s(Slice(nlindep, -1));
        }
 
        // modify U in-place, U is now transformation matrix (U * s^{-1/2})
        for(size_t i = 0; i < n; ++i){
            for(size_t j = 0; j < rank; ++j){
                U(i, j) = U(i, j) * s(j);
            }
        }
 
        return transform(world, v, U);
    }
 
    /// convenience routine for canonical routine for symmetric orthonormalization (see e.g. Szabo/Ostlund)
    /// overlap matrix is calculated
    /// @param[in] the vector to orthonormalize
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > orthonormalize_canonical(const std::vector<Function<T,NDIM> >& v,
            double lindep = 1e-12){
        if(v.empty()) return v;
 
        Tensor<T> ovlp = matrix_inner(v.front().world(), v, v, /* sym= */ true);
 
        return orthonormalize_canonical(v, ovlp, lindep);
    }
 
    /// cholesky orthonormalization without pivoting
    /// @param[in] the vector to orthonormalize
    /// @param[in] overlap matrix, destroyed on return!
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > orthonormalize_cd(
            const std::vector<Function<T,NDIM> >& v,
            Tensor<T>& ovlp) {
 
        if (v.empty()) return v;
 
        cholesky(ovlp); // destroys ovlp and gives back Upper ∆ Matrix from CD
 
        Tensor<T> L = transpose(ovlp);
        Tensor<T> Linv = inverse(L);
        Tensor<T> U = transpose(Linv);
 
        World& world=v.front().world();
        return transform(world, v, U);
 
    }
 
    /// convenience routine for cholesky orthonormalization without pivoting
    /// @param[in] the vector to orthonormalize
    /// @param[in] overlap matrix
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > orthonormalize_cd(const std::vector<Function<T,NDIM> >& v){
        if(v.empty()) return v;
 
        World& world=v.front().world();
        Tensor<T> ovlp = matrix_inner(world, v, v, /* sym= */ true);
 
        return orthonormalize_cd(v,ovlp);
    }
 
    /// @param[in] the vector to orthonormalize
    /// @param[in] overlap matrix, will be destroyed on return!
    /// @param[in] tolerance for numerical rank reduction
    /// @param[out] pivoting vector, no allocation on input needed
    /// @param[out] rank
    /// @return orthonormalized vector (may or may not be truncated)
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > orthonormalize_rrcd(
            const std::vector<Function<T,NDIM> >& v,
            Tensor<T>& ovlp,
            const double tol,
            Tensor<integer>& piv,
            int& rank) {
 
        if (v.empty()) {
            return v;
        }
 
        rr_cholesky(ovlp,tol,piv,rank); // destroys ovlp and gives back Upper ∆ Matrix from CCD
 
        // rearrange and truncate the functions according to the pivoting of the rr_cholesky
        std::vector<Function<T,NDIM> > pv(rank);
        for(integer i=0;i<rank;++i){
            pv[i]=v[piv[i]];
        }
        ovlp=ovlp(Slice(0,rank-1),Slice(0,rank-1));
 
        Tensor<T> L = transpose(ovlp);
        Tensor<T> Linv = inverse(L);
        Tensor<T> U = transpose(Linv);
 
        World& world=v.front().world();
        return transform(world, pv, U);
    }
 
    /// convenience routine for orthonormalize_cholesky: orthonormalize_cholesky without information on pivoting and rank
    /// @param[in] the vector to orthonormalize
    /// @param[in] overlap matrix
    /// @param[in] tolerance for numerical rank reduction
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > orthonormalize_rrcd(const std::vector<Function<T,NDIM> >& v, Tensor<T> ovlp , const double tol) {
        Tensor<integer> piv;
        int rank;
        return orthonormalize_rrcd(v,ovlp,tol,piv,rank);
    }
 
    /// convenience routine for orthonormalize_cholesky: computes the overlap matrix and then calls orthonormalize_cholesky
    /// @param[in] the vector to orthonormalize
    /// @param[in] tolerance for numerical rank reduction
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > orthonormalize_rrcd(const std::vector<Function<T,NDIM> >& v, const double tol) {
        if (v.empty()) {
            return v;
        }
        // compute overlap
        World& world=v.front().world();
        Tensor<T> ovlp = matrix_inner(world, v, v, /* sym= */ true);
        return orthonormalize_rrcd(v,ovlp,tol);
    }
 
    /// combine two vectors
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > append(const std::vector<Function<T,NDIM> > & lhs, const std::vector<Function<T,NDIM> > & rhs){
        std::vector<Function<T,NDIM> >  v=lhs;
        for (std::size_t i = 0; i < rhs.size(); ++i) v.push_back(rhs[i]);
        return v;
    }
 
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > flatten(const std::vector< std::vector<Function<T,NDIM> > >& vv){
        std::vector<Function<T,NDIM> >result;
        for(const auto& x:vv) result=append(result,x);
        return result;
    }
 
    template<typename T, std::size_t NDIM>
    std::vector<std::shared_ptr<FunctionImpl<T,NDIM>>> get_impl(const std::vector<Function<T,NDIM>>& v) {
        std::vector<std::shared_ptr<FunctionImpl<T,NDIM>>> result;
        for (auto& f : v) result.push_back(f.get_impl());
        return result;
    }
 
    template<typename T, std::size_t NDIM>
    void set_impl(std::vector<Function<T,NDIM>>& v, const std::vector<std::shared_ptr<FunctionImpl<T,NDIM>>> vimpl) {
        MADNESS_CHECK(vimpl.size()==v.size());
        for (std::size_t i=0; i<vimpl.size(); ++i) v[i].set_impl(vimpl[i]);
    }
 
    template<typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM>> impl2function(const std::vector<std::shared_ptr<FunctionImpl<T,NDIM>>> vimpl) {
        std::vector<Function<T,NDIM>> v(vimpl.size());
        for (std::size_t i=0; i<vimpl.size(); ++i) v[i].set_impl(vimpl[i]);
        return v;
    }
 
 
    /// Transforms a vector of functions according to new[i] = sum[j] old[j]*c[j,i]
 
    /// Uses sparsity in the transformation matrix --- set small elements to
    /// zero to take advantage of this.
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R),NDIM> >
    transform(World& world,
              const std::vector< Function<T,NDIM> >& v,
              const Tensor<R>& c,
              bool fence=true) {
 
        PROFILE_BLOCK(Vtransformsp);
        typedef TENSOR_RESULT_TYPE(T,R) resultT;
        int n = v.size();  // n is the old dimension
        int m = c.dim(1);  // m is the new dimension
        MADNESS_CHECK(n==c.dim(0));
 
        std::vector< Function<resultT,NDIM> > vc = zero_functions_compressed<resultT,NDIM>(world, m);
        compress(world, v);
 
        for (int i=0; i<m; ++i) {
            for (int j=0; j<n; ++j) {
                if (c(j,i) != R(0.0)) vc[i].gaxpy(resultT(1.0),v[j],resultT(c(j,i)),false);
            }
        }
 
        if (fence) world.gop.fence();
        return vc;
    }
 
    /// Transforms a vector of functions according to new[i] = sum[j] old[j]*c[j,i]
 
    /// all trees are in reconstructed state, final trees have to be summed down if no fence is present
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R),NDIM> >
    transform_reconstructed(World& world,
              const std::vector< Function<T,NDIM> >& v,
              const Tensor<R>& c,
              bool fence=true) {
 
        PROFILE_BLOCK(Vtransformsp);
        typedef TENSOR_RESULT_TYPE(T,R) resultT;
        int n = v.size();  // n is the old dimension
        int m = c.dim(1);  // m is the new dimension
        MADNESS_CHECK(n==c.dim(0));
 
        // if we fence set the right tree state here, otherwise it has to be correct from the start.
        if (fence) change_tree_state(v,reconstructed);
        for (const auto& vv : v) MADNESS_CHECK_THROW(
            vv.get_impl()->get_tree_state()==reconstructed,"trees have to be reconstructed in transform_reconstructed");
 
        std::vector< Function<resultT,NDIM> > result = zero_functions<resultT,NDIM>(world, m);
 
        for (int i=0; i<m; ++i) {
            result[i].get_impl()->set_tree_state(redundant_after_merge);
            for (int j=0; j<n; ++j) {
                if (c(j,i) != R(0.0)) v[j].get_impl()->accumulate_trees(*(result[i].get_impl()),resultT(c(j,i)),true);
            }
        }
 
        // if we fence we can as well finish the job here. Otherwise no harm done, as the tree state is well-defined.
        if (fence) {
            world.gop.fence();
            // for (auto& r : vc) r.sum_down(false);
            for (auto& r : result) r.get_impl()->finalize_sum();
            world.gop.fence();
        }
        return result;
    }
 
    /// this version of transform uses Function::vtransform and screens
    /// using both elements of `c` and `v`
    template <typename L, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(L,R),NDIM> >
    transform(World& world,  const std::vector< Function<L,NDIM> >& v,
            const Tensor<R>& c, double tol, bool fence=true) {
        PROFILE_BLOCK(Vtransform);
        MADNESS_ASSERT(v.size() == (unsigned int)(c.dim(0)));
 
        std::vector< Function<TENSOR_RESULT_TYPE(L,R),NDIM> > vresult
            = zero_functions_compressed<TENSOR_RESULT_TYPE(L,R),NDIM>(world, c.dim(1));
 
        compress(world, v, true);
        vresult[0].vtransform(v, c, vresult, tol, fence);
        return vresult;
    }
 
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R),NDIM> >
    transform(World& world,
              const std::vector< Function<T,NDIM> >& v,
              const DistributedMatrix<R>& c,
              bool fence=true) {
        PROFILE_FUNC;
 
        typedef TENSOR_RESULT_TYPE(T,R) resultT;
        long n = v.size();    // n is the old dimension
        long m = c.rowdim();  // m is the new dimension
        MADNESS_ASSERT(n==c.coldim());
 
        // new(i) = sum(j) old(j) c(j,i)
 
        Tensor<T> tmp(n,m);
        c.copy_to_replicated(tmp); // for debugging
        tmp = transpose(tmp);
 
        std::vector< Function<resultT,NDIM> > vc = zero_functions_compressed<resultT,NDIM>(world, m);
        compress(world, v);
 
        for (int i=0; i<m; ++i) {
            for (int j=0; j<n; ++j) {
                if (tmp(j,i) != R(0.0)) vc[i].gaxpy(1.0,v[j],tmp(j,i),false);
            }
        }
 
        if (fence) world.gop.fence();
        return vc;
    }
 
 
    /// Scales inplace a vector of functions by distinct values
    template <typename T, typename Q, std::size_t NDIM>
    void scale(World& world,
               std::vector< Function<T,NDIM> >& v,
               const std::vector<Q>& factors,
               bool fence=true) {
        PROFILE_BLOCK(Vscale);
        for (unsigned int i=0; i<v.size(); ++i) v[i].scale(factors[i],false);
        if (fence) world.gop.fence();
    }
 
    /// Scales inplace a vector of functions by the same
    template <typename T, typename Q, std::size_t NDIM>
    void scale(World& world,
               std::vector< Function<T,NDIM> >& v,
           const Q factor,
               bool fence=true) {
        PROFILE_BLOCK(Vscale);
        for (unsigned int i=0; i<v.size(); ++i) v[i].scale(factor,false);
        if (fence) world.gop.fence();
    }
 
    /// Computes the 2-norms of a vector of functions
    template <typename T, std::size_t NDIM>
    std::vector<double> norm2s(World& world,
                              const std::vector< Function<T,NDIM> >& v) {
        PROFILE_BLOCK(Vnorm2);
        std::vector<double> norms(v.size());
        if (not (get_tree_state(v)==compressed or get_tree_state(v)==reconstructed)) reconstruct(world,v);
        for (unsigned int i=0; i<v.size(); ++i) norms[i] = v[i].norm2sq_local();
        world.gop.sum(&norms[0], norms.size());
        for (unsigned int i=0; i<v.size(); ++i) norms[i] = sqrt(norms[i]);
        world.gop.fence();
        return norms;
    }
    /// Computes the 2-norms of a vector of functions
    template <typename T, std::size_t NDIM>
    Tensor<double> norm2s_T(World& world, const std::vector<Function<T, NDIM>>& v) {
        PROFILE_BLOCK(Vnorm2);
        Tensor<double> norms(v.size());
        if (not (get_tree_state(v)==compressed or get_tree_state(v)==reconstructed)) reconstruct(world,v);
        for (unsigned int i = 0; i < v.size(); ++i) norms[i] = v[i].norm2sq_local();
        world.gop.sum(&norms[0], norms.size());
        for (unsigned int i = 0; i < v.size(); ++i) norms[i] = sqrt(norms[i]);
        world.gop.fence();
        return norms;
    }
 
    /// Computes the 2-norm of a vector of functions
    template <typename T, std::size_t NDIM>
    double norm2(World& world,const std::vector< Function<T,NDIM> >& v) {
        PROFILE_BLOCK(Vnorm2);
        if (v.size()==0) return 0.0;
        if (not (get_tree_state(v)==compressed or get_tree_state(v)==reconstructed)) reconstruct(world,v);
        std::vector<double> norms(v.size());
        for (unsigned int i=0; i<v.size(); ++i) norms[i] = v[i].norm2sq_local();
        world.gop.sum(&norms[0], norms.size());
        for (unsigned int i=1; i<v.size(); ++i) norms[0] += norms[i];
        world.gop.fence();
        return sqrt(norms[0]);
    }
 
    inline double conj(double x) {
        return x;
    }
 
    inline double conj(float x) {
        return x;
    }
 
// !!! FIXME: this task is broken because FunctionImpl::inner_local forces a
// future on return from WorldTaskQueue::reduce, which will causes a deadlock if
// run inside a task. This behavior must be changed before this task can be used
// again.
//
//    template <typename T, typename R, std::size_t NDIM>
//    struct MatrixInnerTask : public TaskInterface {
//        Tensor<TENSOR_RESULT_TYPE(T,R)> result; // Must be a copy
//        const Function<T,NDIM>& f;
//        const std::vector< Function<R,NDIM> >& g;
//        long jtop;
//
//        MatrixInnerTask(const Tensor<TENSOR_RESULT_TYPE(T,R)>& result,
//                        const Function<T,NDIM>& f,
//                        const std::vector< Function<R,NDIM> >& g,
//                        long jtop)
//                : result(result), f(f), g(g), jtop(jtop) {}
//
//        void run(World& world) {
//            for (long j=0; j<jtop; ++j) {
//                result(j) = f.inner_local(g[j]);
//            }
//        }
//
//    private:
//        /// Get the task id
//
//        /// \param id The id to set for this task
//        virtual void get_id(std::pair<void*,unsigned short>& id) const {
//            PoolTaskInterface::make_id(id, *this);
//        }
//    }; // struct MatrixInnerTask
 
 
 
    template <typename T, std::size_t NDIM>
    DistributedMatrix<T> matrix_inner(const DistributedMatrixDistribution& d,
                                      const std::vector< Function<T,NDIM> >& f,
                                      const std::vector< Function<T,NDIM> >& g,
                                      bool sym=false)
    {
        PROFILE_FUNC;
        DistributedMatrix<T> A(d);
        const int64_t n = A.coldim();
        const int64_t m = A.rowdim();
        MADNESS_ASSERT(int64_t(f.size()) == n && int64_t(g.size()) == m);
 
        // Assume we can always create an ichunk*jchunk matrix locally
        const int ichunk = 1000;
        const int jchunk = 1000; // 1000*1000*8 = 8 MBytes
        for (int64_t ilo=0; ilo<n; ilo+=ichunk) {
            int64_t ihi = std::min(ilo + ichunk, n);
            std::vector< Function<T,NDIM> > ivec(f.begin()+ilo, f.begin()+ihi);
            for (int64_t jlo=0; jlo<m; jlo+=jchunk) {
                int64_t jhi = std::min(jlo + jchunk, m);
                std::vector< Function<T,NDIM> > jvec(g.begin()+jlo, g.begin()+jhi);
 
                Tensor<T> P = matrix_inner(A.get_world(), ivec, jvec);
                A.copy_from_replicated_patch(ilo, ihi - 1, jlo, jhi - 1, P);
            }
        }
        return A;
    }
 
    /// Computes the matrix inner product of two function vectors - q(i,j) = inner(f[i],g[j])
 
    /// For complex types symmetric is interpreted as Hermitian.
 
    /// The current parallel loop is non-optimal but functional.
    template <typename T, typename R, std::size_t NDIM>
    Tensor< TENSOR_RESULT_TYPE(T,R) > matrix_inner(World& world,
                                                   const std::vector< Function<T,NDIM> >& f,
                                                   const std::vector< Function<R,NDIM> >& g,
                                                   bool sym=false)
    {
        world.gop.fence();
        auto tensor_type = [](const std::vector<Function<T,NDIM>>& v) {
            return v.front().get_impl()->get_tensor_type();
        };
        TreeState operating_state=tensor_type(f)==TT_FULL ? compressed : redundant;
        ensure_tree_state_respecting_fence(f,operating_state,true);
        ensure_tree_state_respecting_fence(g,operating_state,true);
 
        std::vector<const FunctionImpl<T,NDIM>*> left(f.size());
        std::vector<const FunctionImpl<R,NDIM>*> right(g.size());
        for (unsigned int i=0; i<f.size(); i++) left[i] = f[i].get_impl().get();
        for (unsigned int i=0; i<g.size(); i++) right[i]= g[i].get_impl().get();
 
        Tensor< TENSOR_RESULT_TYPE(T,R) > r= FunctionImpl<T,NDIM>::inner_local(left, right, sym);
 
        world.gop.fence();
        world.gop.sum(r.ptr(),f.size()*g.size());
 
        return r;
    }
 
    /// Computes the matrix inner product of two function vectors - q(i,j) = inner(f[i],g[j])
 
    /// For complex types symmetric is interpreted as Hermitian.
    ///
    /// The current parallel loop is non-optimal but functional.
    template <typename T, typename R, std::size_t NDIM>
    Tensor< TENSOR_RESULT_TYPE(T,R) > matrix_inner_old(World& world,
            const std::vector< Function<T,NDIM> >& f,
            const std::vector< Function<R,NDIM> >& g,
            bool sym=false) {
        PROFILE_BLOCK(Vmatrix_inner);
        long n=f.size(), m=g.size();
        Tensor< TENSOR_RESULT_TYPE(T,R) > r(n,m);
        if (sym) MADNESS_ASSERT(n==m);
 
        world.gop.fence();
        compress(world, f);
        if ((void*)(&f) != (void*)(&g)) compress(world, g);
 
        for (long i=0; i<n; ++i) {
            long jtop = m;
            if (sym) jtop = i+1;
            for (long j=0; j<jtop; ++j) {
                r(i,j) = f[i].inner_local(g[j]);
                if (sym) r(j,i) = conj(r(i,j));
            }
         }
 
//        for (long i=n-1; i>=0; --i) {
//            long jtop = m;
//            if (sym) jtop = i+1;
//            world.taskq.add(new MatrixInnerTask<T,R,NDIM>(r(i,_), f[i], g, jtop));
//        }
        world.gop.fence();
        world.gop.sum(r.ptr(),n*m);
 
//        if (sym) {
//            for (int i=0; i<n; ++i) {
//                for (int j=0; j<i; ++j) {
//                    r(j,i) = conj(r(i,j));
//                }
//            }
//        }
        return r;
    }
 
    /// Computes the element-wise inner product of two function vectors - q(i) = inner(f[i],g[i])
 
    /// works in reconstructed or compressed state, state is chosen based on TensorType
    template <typename T, typename R, std::size_t NDIM>
    Tensor< TENSOR_RESULT_TYPE(T,R) > inner(World& world,
                                            const std::vector< Function<T,NDIM> >& f,
                                            const std::vector< Function<R,NDIM> >& g) {
        PROFILE_BLOCK(Vinnervv);
        long n=f.size(), m=g.size();
        MADNESS_CHECK(n==m);
        Tensor< TENSOR_RESULT_TYPE(T,R) > r(n);
        if (n==0) return r;
 
        auto tensor_type = [](const std::vector<Function<T,NDIM>>& v) {
            return v.front().get_impl()->get_tensor_type();
        };
        TreeState operating_state=tensor_type(f)==TT_FULL ? compressed : redundant;
        ensure_tree_state_respecting_fence(f,operating_state,true);
        ensure_tree_state_respecting_fence(g,operating_state,true);
 
        for (long i=0; i<n; ++i) r(i) = f[i].inner_local(g[i]);
 
        world.taskq.fence();
        world.gop.sum(r.ptr(),n);
        world.gop.fence();
        return r;
    }
 
 
    /// Computes the inner product of a function with a function vector - q(i) = inner(f,g[i])
 
    /// works in reconstructed or compressed state, state is chosen based on TensorType
    template <typename T, typename R, std::size_t NDIM>
    Tensor< TENSOR_RESULT_TYPE(T,R) > inner(World& world,
                                            const Function<T,NDIM>& f,
                                            const std::vector< Function<R,NDIM> >& g) {
        PROFILE_BLOCK(Vinner);
        long n=g.size();
        Tensor< TENSOR_RESULT_TYPE(T,R) > r(n);
 
        auto tensor_type = [](const std::vector<Function<T,NDIM>>& v) {
            return v.front().get_impl()->get_tensor_type();
        };
        TreeState operating_state=tensor_type(g)==TT_FULL ? compressed : redundant;
        f.change_tree_state(operating_state,false);
        ensure_tree_state_respecting_fence(g,operating_state,true);
        world.gop.fence();
 
        for (long i=0; i<n; ++i) {
            r(i) = f.inner_local(g[i]);
        }
 
        world.taskq.fence();
        world.gop.sum(r.ptr(),n);
        world.gop.fence();
        return r;
    }
 
    /// inner function with right signature for the nonlinear solver
    /// this is needed for the KAIN solvers and other functions
    template <typename T, typename R, std::size_t NDIM>
    TENSOR_RESULT_TYPE(T,R) inner( const std::vector< Function<T,NDIM> >& f,
                                                const std::vector< Function<R,NDIM> >& g){
      MADNESS_ASSERT(f.size()==g.size());
      if(f.empty()) return 0.0;
      else return inner(f[0].world(),f,g).sum();
    }
 
 
    /// Multiplies a function against a vector of functions --- q[i] = a * v[i]
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    mul(World& world,
        const Function<T,NDIM>& a,
        const std::vector< Function<R,NDIM> >& v,
        bool fence=true) {
        PROFILE_BLOCK(Vmul);
        make_redundant(world, v, false);
        a.make_redundant(false);
        world.gop.fence();
        return vmulXX(a, v, 0.0, fence);
    }
 
    /// Multiplies a function against a vector of functions using sparsity of a and v[i] --- q[i] = a * v[i]
    ///
    /// Box pairs whose estimated contribution falls below the tolerance are skipped instead
    /// of being multiplied. Both inputs are made redundant; the screening reads their
    /// norm_tree and dnorm_tree.
    ///
    /// Leaves both inputs in redundant form. Function is a shallow handle, so this is visible
    /// to the caller: logically const, not bitwise const. Converting back is not free, so a
    /// caller that reuses the operands afterwards must do it itself.
    ///
    /// @param[in] tol  target absolute accuracy of the product; the safety margin is applied
    ///                 internally (FunctionImpl::MUL_SCREENING_SAFETY), so pass the accuracy
    ///                 wanted, not a pre-scaled value. tol=0 multiplies exactly. The criterion
    ///                 estimates the neglected cross terms rather than bounding them: the error
    ///                 tracks tol up to a measured O(1-20) constant and decays as ~tol^0.75
    ///                 rather than ~tol (see test_mul_sparse.cc). The meaning differs from the
    ///                 earlier norm_tree-based screen, so a previously tuned value needs
    ///                 re-checking.
    /// @param[in] do_make_redundant  if false, both inputs must already be redundant
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    mul_sparse(World& world,
               const Function<T,NDIM>& a,
               const std::vector< Function<R,NDIM> >& v,
               double tol,
               bool fence=true,
               bool do_make_redundant=true) {
        PROFILE_BLOCK(Vmulsp);
        if (do_make_redundant) {
            try {
                ensure_tree_state_respecting_fence(std::vector<Function<T,NDIM>>({a}), TreeState::redundant, fence);
                ensure_tree_state_respecting_fence(v, TreeState::redundant, fence);
            } catch (...) {
                print("could not respect fence in mul_sparse");
                a.make_redundant(false);
                make_redundant(world, v, false);
                world.gop.fence();
            }
        } else if (!v.empty()) {
            MADNESS_CHECK_THROW(a.get_impl()->get_tree_state() == TreeState::redundant,
                                "mul_sparse: left input must be redundant when do_make_redundant=false");
            MADNESS_CHECK_THROW(get_tree_state(v) == TreeState::redundant,
                                "mul_sparse: right inputs must be redundant when do_make_redundant=false");
        }
        return vmulXX(a, v, tol, fence);
    }
 
    /// Multiplies two vectors of functions using sparsity of a[i] and b[i] --- q[i] = a[i] * b[i]
    ///
    /// Box pairs whose estimated contribution falls below the tolerance are skipped instead
    /// of being multiplied. Both inputs are made redundant; the screening reads their
    /// norm_tree and dnorm_tree.
    ///
    /// Leaves both inputs in redundant form. Function is a shallow handle, so this is visible
    /// to the caller: logically const, not bitwise const. Converting back is not free, so a
    /// caller that reuses the operands afterwards must do it itself.
    ///
    /// @param[in] tol  target absolute accuracy of the product; the safety margin is applied
    ///                 internally (FunctionImpl::MUL_SCREENING_SAFETY), so pass the accuracy
    ///                 wanted, not a pre-scaled value. tol=0 multiplies exactly. The criterion
    ///                 estimates the neglected cross terms rather than bounding them: the error
    ///                 tracks tol up to a measured O(1-20) constant and decays as ~tol^0.75
    ///                 rather than ~tol (see test_mul_sparse.cc). The meaning differs from the
    ///                 earlier norm_tree-based screen, so a previously tuned value needs
    ///                 re-checking.
    /// @param[in] do_make_redundant  if false, both inputs must already be redundant
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    mul_sparse(World& world,
        const std::vector< Function<T,NDIM> >& a,
        const std::vector< Function<R,NDIM> >& b,
        double tol,
        bool fence=true,
        bool do_make_redundant=true) {
        PROFILE_BLOCK(Vmulvv);
        if (do_make_redundant) {
            try {
                ensure_tree_state_respecting_fence(a, TreeState::redundant, fence);
                ensure_tree_state_respecting_fence(b, TreeState::redundant, fence);
            } catch (...) {
                print("could not respect fence in mul_sparse");
                make_redundant(world, a, false);
                make_redundant(world, b, false);
                world.gop.fence();
            }
        }
        std::vector< Function<TENSOR_RESULT_TYPE(T,R),NDIM> > q(a.size());
        for (unsigned int i=0; i<a.size(); ++i) {
            q[i] = mul_sparse(a[i], b[i], tol, false, false);
        }
        if (fence) world.gop.fence();
        return q;
    }
 
 
    /// Outer product of a vector of functions with a vector of functions using sparsity
 
    /// \tparam T       type parameter for first factor
    /// \tparam R       type parameter for second factor
    /// \tparam NDIM    dimension of first and second factors
    /// \param world    the world
    /// \param f        first vector of functions
    /// \param g        second vector of functions
    /// \param tol      target absolute accuracy of each product; see mul_sparse for the
    ///                 semantics, including the internal safety margin and tol=0
    /// \param fence    force fence (will always fence if necessary)
    /// \param symm     if true, only compute f(i) * g(j) for j<=i
    /// \return         fg(i,j) = f(i) * g(j), as a vector of vectors
    template <typename T, typename R, std::size_t NDIM>
    std::vector<std::vector<Function<TENSOR_RESULT_TYPE(T, R), NDIM> > >
    matrix_mul_sparse(World &world,
                      const std::vector<Function<R, NDIM> > &f,
                      const std::vector<Function<R, NDIM> > &g,
                      double tol,
                      bool fence = true,
                      bool symm = false) {
        PROFILE_BLOCK(Vmulsp);
        bool same=(&f == &g);
        make_redundant(world, f, false);
        if (not same) make_redundant(world, g, false);
        world.gop.fence();
 
        std::vector<std::vector<Function<R,NDIM> > >result(f.size());
        std::vector<Function<R,NDIM>> g_i;
        for (int64_t i=f.size()-1; i>=0; --i) {
          if (!symm)
            result[i]= vmulXX(f[i], g, tol, false);
          else {
            if (g_i.empty()) g_i = g;
            g_i.resize(i+1);  // this shrinks g_i down to single function for i=0
            result[i]= vmulXX(f[i], g_i, tol, false);
          }
        }
        if (fence) world.gop.fence();
        return result;
    }
 
    /// Makes the norm tree for all functions in a vector
    template <typename T, std::size_t NDIM>
    void norm_tree(World& world,
                   const std::vector< Function<T,NDIM> >& v,
                   bool fence=true)
    {
        PROFILE_BLOCK(Vnorm_tree);
        for (unsigned int i=0; i<v.size(); ++i) {
            v[i].norm_tree(false);
        }
        if (fence) world.gop.fence();
    }
 
    /// Multiplies two vectors of functions q[i] = a[i] * b[i]; see mul_sparse to screen
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    mul(World& world,
        const std::vector< Function<T,NDIM> >& a,
        const std::vector< Function<R,NDIM> >& b,
        bool fence=true,
        bool do_make_redundant=true) {
        PROFILE_BLOCK(Vmulvv);
        if (do_make_redundant) {
            try {
                ensure_tree_state_respecting_fence(a, TreeState::redundant, fence);
                ensure_tree_state_respecting_fence(b, TreeState::redundant, fence);
            } catch (...) {
                print("could not respect fence in mul");
                make_redundant(world, a, false);
                make_redundant(world, b, false);
                world.gop.fence();
            }
        }
        std::vector< Function<TENSOR_RESULT_TYPE(T,R),NDIM> > q(a.size());
        for (unsigned int i=0; i<a.size(); ++i) {
            q[i] = mul(a[i], b[i], false, false);
        }
        if (fence) world.gop.fence();
        return q;
    }
 
 
    /// multiply a high-dimensional function with a low-dimensional function
 
    /// @param[in]  f   NDIM function of NDIM dimensions
    /// @param[in]  g   LDIM function of LDIM
    /// @param[in]  v   dimension indices of f to multiply
    /// @return     h[i](0,1,2,3) = f(0,1,2,3) * g[i](1,2,3) for v={1,2,3}
    template<typename T, std::size_t NDIM, std::size_t LDIM>
    std::vector<Function<T,NDIM> > partial_mul(const Function<T,NDIM> f, const std::vector<Function<T,LDIM> > g,
                                 const int particle) {
 
        World& world=f.world();
        std::vector<Function<T,NDIM> > result(g.size());
        for (auto& r : result) r.set_impl(f, false);
 
        FunctionImpl<T,NDIM>* fimpl=f.get_impl().get();
//        fimpl->make_redundant(false);
        fimpl->change_tree_state(redundant,false);
        make_redundant(world,g,false);
        world.gop.fence();
 
        for (std::size_t i=0; i<result.size(); ++i) {
            FunctionImpl<T,LDIM>* gimpl=g[i].get_impl().get();
            result[i].get_impl()->multiply(fimpl,gimpl,particle);     // stupid naming inconsistency
        }
        world.gop.fence();
 
        fimpl->undo_redundant(false);
        for (auto& ig : g) ig.get_impl()->undo_redundant(false);
        world.gop.fence();
        return result;
    }
 
    template<typename T, std::size_t NDIM, std::size_t LDIM>
    std::vector<Function<T,NDIM> > multiply(const Function<T,NDIM> f, const std::vector<Function<T,LDIM> > g,
                              const std::tuple<int,int,int> v) {
        return partial_mul<T,NDIM,LDIM>(f,g,std::array<int,3>({std::get<0>(v),std::get<1>(v),std::get<2>(v)}));
    }
 
 
/// Computes the square of a vector of functions --- q[i] = v[i]**2
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    square(World& world,
           const std::vector< Function<T,NDIM> >& v,
           bool fence=true) {
        return mul<T,T,NDIM>(world, v, v, fence);
//         std::vector< Function<T,NDIM> > vsq(v.size());
//         for (unsigned int i=0; i<v.size(); ++i) {
//             vsq[i] = square(v[i], false);
//         }
//         if (fence) world.gop.fence();
//         return vsq;
    }
 
 
    /// Computes the square of a vector of functions --- q[i] = abs(v[i])**2
    template <typename T, std::size_t NDIM>
    std::vector< Function<typename Tensor<T>::scalar_type,NDIM> >
    abssq(World& world,
           const std::vector< Function<T,NDIM> >& v,
           bool fence=true) {
        typedef typename Tensor<T>::scalar_type scalartype;
        reconstruct(world,v);
        std::vector<Function<scalartype,NDIM> > result(v.size());
        for (size_t i=0; i<v.size(); ++i) result[i]=abs_square(v[i],false);
        if (fence) world.gop.fence();
        return result;
    }
 
 
    /// Sets the threshold in a vector of functions
    template <typename T, std::size_t NDIM>
    void set_thresh(World& world, std::vector< Function<T,NDIM> >& v, double thresh, bool fence=true) {
        for (unsigned int j=0; j<v.size(); ++j) {
            v[j].set_thresh(thresh,false);
        }
        if (fence) world.gop.fence();
    }
 
    /// Returns the complex conjugate of the vector of functions
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    conj(World& world,
         const std::vector< Function<T,NDIM> >& v,
         bool fence=true) {
        PROFILE_BLOCK(Vconj);
        std::vector< Function<T,NDIM> > r = copy(world, v); // Currently don't have oop conj
        for (unsigned int i=0; i<v.size(); ++i) {
            r[i].conj(false);
        }
        if (fence) world.gop.fence();
        return r;
    }
 
    /// Returns a deep copy of a vector of functions
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<R,NDIM> > convert(World& world,
            const std::vector< Function<T,NDIM> >& v, bool fence=true) {
        PROFILE_BLOCK(Vcopy);
        std::vector< Function<R,NDIM> > r(v.size());
        for (unsigned int i=0; i<v.size(); ++i) {
            r[i] = convert<T,R,NDIM>(v[i], false);
        }
        if (fence) world.gop.fence();
        return r;
    }
 
 
    /// Returns a deep copy of a vector of functions
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    copy(World& world,
         const std::vector< Function<T,NDIM> >& v,
         bool fence=true) {
        PROFILE_BLOCK(Vcopy);
        std::vector< Function<T,NDIM> > r(v.size());
        for (unsigned int i=0; i<v.size(); ++i) {
            r[i] = copy(v[i], false);
        }
        if (fence) world.gop.fence();
        return r;
    }
 
 
    /// Returns a deep copy of a vector of functions
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    copy(const std::vector< Function<T,NDIM> >& v, bool fence=true) {
        PROFILE_BLOCK(Vcopy);
        std::vector< Function<T,NDIM> > r(v.size());
        if (v.size()>0) r=copy(v.front().world(),v,fence);
        return r;
    }
 
    /// Returns a vector of `n` deep copies of a function
    template <typename T, std::size_t NDIM>
    std::vector< Function<T,NDIM> >
    copy_n(World& world,
         const Function<T,NDIM>& v,
         const unsigned int n,
         bool fence=true) {
        PROFILE_BLOCK(Vcopy1);
        std::vector< Function<T,NDIM> > r(n);
        for (unsigned int i=0; i<n; ++i) {
            r[i] = copy(v, false);
        }
        if (fence) world.gop.fence();
        return r;
    }
 
    /// Create a new copy of the function with different distribution and optional
    /// fence
 
    /// Works in either basis.  Different distributions imply
    /// asynchronous communication and the optional fence is
    /// collective.
    //
    /// Returns a deep copy of a vector of functions
 
    template <typename T, std::size_t NDIM>
    std::vector<Function<T, NDIM>> copy(World& world,
                                        const std::vector<Function<T, NDIM>>& v,
                                        const std::shared_ptr<WorldDCPmapInterface<Key<NDIM>>>& pmap,
                                        bool fence = true) {
      PROFILE_BLOCK(Vcopy);
      std::vector<Function<T, NDIM>> r(v.size());
      for (unsigned int i = 0; i < v.size(); ++i) {
        r[i] = copy(v[i], pmap, false);
      }
      if (fence) world.gop.fence();
      return r;
    }
 
    /// owner[j] = j % nranks. For redistribute_to_batches.
    inline std::vector<ProcessID> assign_round_robin(std::size_t nfunc, int nranks) {
        MADNESS_CHECK(nranks > 0);
        std::vector<ProcessID> owner(nfunc);
        for (std::size_t j = 0; j < nfunc; ++j) owner[j] = ProcessID(j % std::size_t(nranks));
        return owner;
    }
 
    /// Cost-balanced assignment: descending cost, each function to the least-loaded
    /// rank (LPT greedy). Deterministic for a replicated cost[] (stable sort, index
    /// tie-break), so owner[] agrees across ranks. For redistribute_to_batches.
    /// @param[in] cost   per-function cost proxy (>= 0), identical on every rank
    inline std::vector<ProcessID> assign_cost_aware(const std::vector<double>& cost, int nranks) {
        MADNESS_CHECK(nranks > 0);
        const std::size_t nfunc = cost.size();
        std::vector<ProcessID> owner(nfunc);
        std::vector<std::size_t> order(nfunc);
        for (std::size_t j = 0; j < nfunc; ++j) order[j] = j;
        // descending cost; ascending index breaks ties -> stable and reproducible
        std::stable_sort(order.begin(), order.end(),
                         [&](std::size_t a, std::size_t b) { return cost[a] > cost[b]; });
        std::vector<double> load(std::size_t(nranks), 0.0);
        for (std::size_t k = 0; k < nfunc; ++k) {
            int best = 0;                                    // least-loaded rank; smallest
            for (int r = 1; r < nranks; ++r)                 // rank index breaks ties
                if (load[std::size_t(r)] < load[std::size_t(best)]) best = r;
            const std::size_t j = order[k];
            owner[j] = ProcessID(best);
            load[std::size_t(best)] += std::max(1.0, cost[j]);  // floor: zero-cost funcs still rotate
        }
        return owner;
    }
 
    /// Global coefficient count per function -- one reduction, identical on every rank,
    /// so safe for a deterministic assignment. A proxy for convolution cost; does not
    /// predict result-tree refinement.
    template <typename T, std::size_t NDIM>
    std::vector<double> function_costs(World& world, const std::vector<Function<T, NDIM>>& v) {
        std::vector<double> cost(v.size(), 0.0);
        for (std::size_t j = 0; j < v.size(); ++j) cost[j] = double(v[j].size_local());
        if (!v.empty()) world.gop.sum(cost.data(), cost.size());
        return cost;
    }
 
    /// Move each v[j] so its whole tree lives on rank owner[j], via the coalesced
    /// WorldContainer transport (bulk AMs, erase-after-copy: streams, no 2x transient).
    /// State-preserving. Each function gets its OWN single-owner pmap (Key<NDIM> is
    /// function-agnostic), and the pmap outlives the call -- until redistributed again,
    /// every operation on v[j] runs on rank owner[j] alone.
    ///
    /// @param[in,out] v       functions to localize (moved in place)
    /// @param[in] owner       destination rank per function; MUST be identical on every
    ///                        rank (checked collectively)
    /// @param[in] cap_bytes   soft cap per message (0 => ~1 MiB, sized for the default
    ///                        MAD_BUFFER_SIZE; lower it if that buffer was shrunk)
    /// @param[in] rotate      stagger destinations to reduce incast
    template <typename T, std::size_t NDIM>
    void redistribute_to_batches(World& world,
                                 std::vector<Function<T, NDIM>>& v,
                                 const std::vector<ProcessID>& owner,
                                 std::size_t cap_bytes = 0,
                                 bool rotate = true) {
        MADNESS_CHECK(owner.size() == v.size());
        if (v.empty()) return;
 
        // owner[] must agree across ranks -- divergence silently corrupts ownership
        {
            long h = 0;
            for (std::size_t j = 0; j < owner.size(); ++j) h += long(owner[j]) * long(j + 1);
            long hmax = h, hmin = h;
            world.gop.max(hmax);
            world.gop.min(hmin);
            MADNESS_CHECK(hmax == hmin);
        }
 
        // chunk cap in #boxes; box size bounded by the functions' own k, not
        // FunctionDefaults (v may carry its own k)
        if (cap_bytes == 0) cap_bytes = 1024 * 1024; // ~1 MiB, under the default RMI buffer
        long kmax = 1;
        for (const auto& f : v) kmax = std::max(kmax, long(f.k()));
        std::size_t box_bytes = sizeof(T);
        for (std::size_t d = 0; d < NDIM; ++d) box_bytes *= std::size_t(2 * kmax);
        const std::size_t cap_boxes = std::max<std::size_t>(1, cap_bytes / box_bytes);
 
        // fence, phase1, fence, phase2, fence: the middle fence is REQUIRED -- phase1
        // iterates the ConcurrentHashMap and needs a quiescent window (see worlddc.h)
        world.gop.fence();
        for (std::size_t j = 0; j < v.size(); ++j) {
            auto pmap = std::shared_ptr<WorldDCPmapInterface<Key<NDIM>>>(
                new WorldDCSingleOwnerPmap<Key<NDIM>>(owner[j]));
            v[j].get_impl()->get_coeffs().redistribute_coalesced_phase1(pmap);
        }
        world.gop.fence();
        for (std::size_t j = 0; j < v.size(); ++j)
            v[j].get_impl()->get_coeffs().redistribute_coalesced_phase2(cap_boxes, rotate);
        world.gop.fence();
    }
 
    /// Returns new vector of functions --- q[i] = a[i] + b[i]
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    add(World& world,
        const std::vector< Function<T,NDIM> >& a,
        const std::vector< Function<R,NDIM> >& b,
        bool fence=true) {
        PROFILE_BLOCK(Vadd);
        MADNESS_ASSERT(a.size() == b.size());
        compress(world, a);
        compress(world, b);
 
        std::vector< Function<TENSOR_RESULT_TYPE(T,R),NDIM> > r(a.size());
        for (unsigned int i=0; i<a.size(); ++i) {
            r[i] = add(a[i], b[i], false);
        }
        if (fence) world.gop.fence();
        return r;
    }
 
     /// Returns new vector of functions --- q[i] = a + b[i]
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    add(World& world,
        const Function<T,NDIM> & a,
        const std::vector< Function<R,NDIM> >& b,
        bool fence=true) {
        PROFILE_BLOCK(Vadd1);
        a.compress();
        compress(world, b);
 
        std::vector< Function<TENSOR_RESULT_TYPE(T,R),NDIM> > r(b.size());
        for (unsigned int i=0; i<b.size(); ++i) {
            r[i] = add(a, b[i], false);
        }
        if (fence) world.gop.fence();
        return r;
    }
    template <typename T, typename R, std::size_t NDIM>
    inline std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    add(World& world,
        const std::vector< Function<R,NDIM> >& b,
        const Function<T,NDIM> & a,
        bool fence=true) {
        return add(world, a, b, fence);
    }
 
    /// Returns new vector of functions --- q[i] = a[i] - b[i]
    template <typename T, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    sub(World& world,
        const std::vector< Function<T,NDIM> >& a,
        const std::vector< Function<R,NDIM> >& b,
        bool fence=true) {
        PROFILE_BLOCK(Vsub);
        MADNESS_ASSERT(a.size() == b.size());
        compress(world, a);
        compress(world, b);
 
        std::vector< Function<TENSOR_RESULT_TYPE(T,R),NDIM> > r(a.size());
        for (unsigned int i=0; i<a.size(); ++i) {
            r[i] = sub(a[i], b[i], false);
        }
        if (fence) world.gop.fence();
        return r;
    }
 
    /// Returns new function --- q = sum_i f[i]
    template <typename T, std::size_t NDIM>
    Function<T, NDIM> sum(World& world, const std::vector<Function<T,NDIM> >& f,
        bool fence=true) {
 
        compress(world, f);
        Function<T,NDIM> r=FunctionFactory<T,NDIM>(world).compressed();
 
        for (unsigned int i=0; i<f.size(); ++i) r.gaxpy(1.0,f[i],1.0,false);
        if (fence) world.gop.fence();
        return r;
    }
 
    template <typename T, std::size_t NDIM>
    DistributedMatrix<T> matrix_dot(const DistributedMatrixDistribution& d,
                                      const std::vector<Function<T, NDIM>>& f,
                                      const std::vector<Function<T, NDIM>>& g,
                                      bool sym=false)
    {
        PROFILE_FUNC;
        DistributedMatrix<T> A(d);
        const int64_t n = A.coldim();
        const int64_t m = A.rowdim();
        MADNESS_ASSERT(int64_t(f.size()) == n && int64_t(g.size()) == m);
 
        // Assume we can always create an ichunk*jchunk matrix locally
        const int ichunk = 1000;
        const int jchunk = 1000; // 1000*1000*8 = 8 MBytes
        for (int64_t ilo = 0; ilo < n; ilo += ichunk) {
            int64_t ihi = std::min(ilo + ichunk, n);
            std::vector<Function<T, NDIM>> ivec(f.begin() + ilo, f.begin() + ihi);
            for (int64_t jlo = 0; jlo < m; jlo += jchunk) {
                int64_t jhi = std::min(jlo + jchunk, m);
                std::vector<Function<T, NDIM>> jvec(g.begin() + jlo, g.begin() + jhi);
 
                Tensor<T> P = matrix_dot(A.get_world(), ivec, jvec, sym);
                A.copy_from_replicated_patch(ilo, ihi - 1, jlo, jhi - 1, P);
            }
        }
        return A;
    }
 
    /// Computes the matrix dot product of two function vectors - q(i,j) = dot(f[i],g[j])
 
    /// For complex types symmetric is interpreted as Hermitian.
    ///
    /// The current parallel loop is non-optimal but functional.
    template <typename T, typename R, std::size_t NDIM>
    Tensor<TENSOR_RESULT_TYPE(T, R)> matrix_dot(World& world,
                                                   const std::vector<Function<T, NDIM>>& f,
                                                   const std::vector<Function<R, NDIM>>& g,
                                                   bool sym=false)
    {
        world.gop.fence();
        compress(world, f);
        // if ((void*)(&f) != (void*)(&g)) compress(world, g);
        compress(world, g);
 
        std::vector<const FunctionImpl<T, NDIM>*> left(f.size());
        std::vector<const FunctionImpl<R, NDIM>*> right(g.size());
        for (unsigned int i = 0; i < f.size(); i++) left[i] = f[i].get_impl().get();
        for (unsigned int i = 0; i < g.size(); i++) right[i] = g[i].get_impl().get();
 
        Tensor<TENSOR_RESULT_TYPE(T, R)> r = FunctionImpl<T, NDIM>::dot_local(left, right, sym);
 
        world.gop.fence();
        world.gop.sum(r.ptr(), f.size() * g.size());
 
        return r;
    }
 
    /// Computes the matrix dot product of two function vectors - q(i,j) = dot(f[i],g[j])
 
    /// For complex types symmetric is interpreted as Hermitian.
    ///
    /// The current parallel loop is non-optimal but functional.
    template <typename T, typename R, std::size_t NDIM>
    Tensor< TENSOR_RESULT_TYPE(T,R) > matrix_dot_old(World& world,
            const std::vector< Function<T,NDIM> >& f,
            const std::vector< Function<R,NDIM> >& g,
            bool sym=false) {
        PROFILE_BLOCK(Vmatrix_dot);
        long n=f.size(), m=g.size();
        Tensor< TENSOR_RESULT_TYPE(T,R) > r(n,m);
        if (sym) MADNESS_ASSERT(n==m);
 
        world.gop.fence();
        compress(world, f);
        if ((void*)(&f) != (void*)(&g)) compress(world, g);
 
        for (long i=0; i<n; ++i) {
            long jtop = m;
            if (sym) jtop = i+1;
            for (long j=0; j<jtop; ++j) {
                if (sym) {
                    r(j,i) = f[i].dot_local(g[j]);
                    if (i != j)
                        r(i,j) = conj(r(j,i));
                } else
                    r(i,j) = f[i].dot_local(g[j]);
            }
         }
 
        world.gop.fence();
        world.gop.sum(r.ptr(),n*m);
 
        return r;
    }
 
    /// Multiplies and sums two vectors of functions r = \sum_i a[i] * b[i]
    template <typename T, typename R, std::size_t NDIM>
    Function<TENSOR_RESULT_TYPE(T,R), NDIM>
    dot_sparse(World& world,
        const std::vector< Function<T,NDIM> >& a,
        const std::vector< Function<R,NDIM> >& b,
        double tol,
        bool fence=true,
        bool do_make_redundant=true) {
        MADNESS_CHECK(a.size()==b.size());
        return sum(world,mul_sparse(world,a,b,tol,/*fence=*/true,do_make_redundant),fence);
    }
 
    /// Multiplies and sums two vectors of functions r = \sum_i a[i] * b[i]; see dot_sparse for screening
    template <typename T, typename R, std::size_t NDIM>
    Function<TENSOR_RESULT_TYPE(T,R), NDIM>
    dot(World& world,
        const std::vector< Function<T,NDIM> >& a,
        const std::vector< Function<R,NDIM> >& b,
        bool fence=true,
        bool do_make_redundant=true) {
        MADNESS_CHECK(a.size()==b.size());
        return sum(world,mul(world,a,b,/*fence=*/true,do_make_redundant),fence);
    }
 
    /// out-of-place gaxpy for two vectors: result[i] = alpha * a[i] + beta * b[i]
    template <typename T, typename Q, typename R, std::size_t NDIM>
    std::vector<Function<TENSOR_RESULT_TYPE(Q,TENSOR_RESULT_TYPE(T,R)),NDIM> >
    gaxpy_oop(Q alpha,
               const std::vector< Function<T,NDIM> >& a,
               Q beta,
               const std::vector< Function<R,NDIM> >& b,
               bool fence=true) {
 
        MADNESS_ASSERT(a.size() == b.size());
        typedef TENSOR_RESULT_TYPE(Q,TENSOR_RESULT_TYPE(T,R)) resultT;
        if (a.size()==0) return std::vector<Function<resultT,NDIM> >();
 
        auto tensor_type = [](const std::vector<Function<T,NDIM>>& v) {
            return v.front().get_impl()->get_tensor_type();
        };
 
        // gaxpy can be done either in reconstructed or in compressed state
        World& world=a[0].world();
        std::vector<Function<resultT,NDIM> > result(a.size());
 
        TreeState operating_state=tensor_type(a)==TT_FULL ? compressed : reconstructed;
        try {
            ensure_tree_state_respecting_fence(a,operating_state,fence);
            ensure_tree_state_respecting_fence(b,operating_state,fence);
        } catch (...) {
            print("could not respect fence in gaxpy");
            change_tree_state(a,operating_state,true);
            change_tree_state(b,operating_state,true);
        }
 
        if (operating_state==compressed) {
            for (unsigned int i=0; i<a.size(); ++i) result[i]=gaxpy_oop(alpha, a[i], beta, b[i], false);
        } else {
            for (unsigned int i=0; i<a.size(); ++i) result[i]=gaxpy_oop_reconstructed(alpha, a[i], beta, b[i], false);
        }
 
        if (fence) world.gop.fence();
        return result;
    }
 
 
    /// out-of-place gaxpy for a vectors and a function: result[i] = alpha * a[i] + beta * b
    template <typename T, typename Q, typename R, std::size_t NDIM>
    std::vector<Function<TENSOR_RESULT_TYPE(Q,TENSOR_RESULT_TYPE(T,R)),NDIM> >
    gaxpy_oop(Q alpha,
               const std::vector< Function<T,NDIM> >& a,
               Q beta,
               const Function<R,NDIM>& b,
               bool fence=true) {
 
        typedef TENSOR_RESULT_TYPE(Q,TENSOR_RESULT_TYPE(T,R)) resultT;
        if (a.size()==0) return std::vector<Function<resultT,NDIM> >();
 
        World& world=a[0].world();
        try {
            ensure_tree_state_respecting_fence(a,compressed,fence);
            // ensure_tree_state_respecting_fence({b},compressed,fence);
            ensure_tree_state_respecting_fence(std::vector<Function<T,NDIM>>({b}),compressed,fence);
        } catch (...) {
            print("could not respect fence in gaxpy_oop");
            compress(world,a);
            b.compress();
        }
        std::vector<Function<resultT,NDIM> > result(a.size());
        for (unsigned int i=0; i<a.size(); ++i) {
            result[i]=gaxpy_oop(alpha, a[i], beta, b, false);
        }
        if (fence) world.gop.fence();
        return result;
    }
 
 
    /// Generalized A*X+Y for vectors of functions ---- a[i] = alpha*a[i] + beta*b[i]
    template <typename T, typename Q, typename R, std::size_t NDIM>
    void gaxpy(Q alpha, std::vector<Function<T,NDIM>>& a, Q beta, const std::vector<Function<R,NDIM>>& b, const bool fence) {
        if (a.size() == 0) return;
        World& world=a.front().world();
        gaxpy(world,alpha,a,beta,b,fence);
    }
 
    /// Generalized A*X+Y for vectors of functions ---- a[i] = alpha*a[i] + beta*b[i]
    template <typename T, typename Q, typename R, std::size_t NDIM>
    void gaxpy(World& world,
               Q alpha,
               std::vector< Function<T,NDIM> >& a,
               Q beta,
               const std::vector< Function<R,NDIM> >& b,
               bool fence=true) {
        PROFILE_BLOCK(Vgaxpy);
        MADNESS_ASSERT(a.size() == b.size());
        if (a.empty()) return;
 
        auto tensor_type = [](const std::vector<Function<T,NDIM>>& v) {
            return v.front().get_impl()->get_tensor_type();
        };
 
        // gaxpy can be done either in reconstructed or in compressed state
        bool do_in_reconstructed_state=tensor_type(a)!=TT_FULL;
        TreeState operating_state=do_in_reconstructed_state ? reconstructed : compressed;
 
        if (operating_state==compressed) {
            // this is strict: both vectors have to be compressed
            try {
                ensure_tree_state_respecting_fence(a,operating_state,fence);
                ensure_tree_state_respecting_fence(b,operating_state,fence);
            } catch (...) {
                print("could not respect fence in gaxpy");
                change_tree_state(a,operating_state,true);
                change_tree_state(b,operating_state,true);
            }
            MADNESS_CHECK_THROW(get_tree_state(a)==get_tree_state(b),"gaxpy requires same tree state for all functions");
            MADNESS_CHECK_THROW(get_tree_state(a)==operating_state,"gaxpy requires reconstructed/compressed tree state for all functions");
        } else {
            // both vectors can be reconstructed or redundant_after_merge, and they don't have to be the same
            TreeState astate=get_tree_state(a);
            TreeState bstate=get_tree_state(b);
            if (not (astate==reconstructed or astate==redundant_after_merge)) {
                try {
                    ensure_tree_state_respecting_fence(a,operating_state,fence);
                } catch (...) {
                    print("could not respect fence in gaxpy for a");
                    change_tree_state(a,reconstructed,true);
                }
            }
            if (not (bstate==reconstructed or bstate==redundant_after_merge)) {
                try {
                    ensure_tree_state_respecting_fence(b,operating_state,fence);
                } catch (...) {
                    print("could not respect fence in gaxpy for b");
                    change_tree_state(b,reconstructed,true);
                }
            }
        }
 
        // finally do the work
        for (unsigned int i=0; i<a.size(); ++i) {
            a[i].gaxpy(alpha, b[i], beta, false);
        }
        if (fence and (get_tree_state(a)==redundant_after_merge)) {
            for (unsigned int i=0; i<a.size(); ++i) a[i].get_impl()->finalize_sum();
        }
 
        if (fence) world.gop.fence();
    }
 
 
    /// Applies a vector of operators to a vector of functions --- q[i] = apply(op[i],f[i])
    template <typename opT, typename R, std::size_t NDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(typename opT::opT,R), NDIM> >
    apply(World& world,
          const std::vector< std::shared_ptr<opT> >& op,
          const std::vector< Function<R,NDIM> > f) {
 
        PROFILE_BLOCK(Vapplyv);
        MADNESS_ASSERT(f.size()==op.size());
 
        std::vector< Function<R,NDIM> >& ncf = *const_cast< std::vector< Function<R,NDIM> >* >(&f);
 
//        reconstruct(world, f);
        make_nonstandard(world, ncf);
 
        std::vector< Function<TENSOR_RESULT_TYPE(typename opT::opT,R), NDIM> > result(f.size());
        for (unsigned int i=0; i<f.size(); ++i) {
            result[i] = apply_only(*op[i], f[i], false);
            result[i].get_impl()->set_tree_state(nonstandard_after_apply);
        }
 
        world.gop.fence();
 
        standard(world, ncf, false);  // restores promise of logical constness
        reconstruct(result);
        world.gop.fence();
 
        return result;
    }
 
 
    /// Applies an operator to a vector of functions --- q[i] = apply(op,f[i])
    template <typename T, typename R, std::size_t NDIM, std::size_t KDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    apply(const SeparatedConvolution<T,KDIM>& op,
          const std::vector< Function<R,NDIM> > f) {
        return apply(op.get_world(),op,f);
    }
 
 
    /// Applies an operator to a vector of functions --- q[i] = apply(op,f[i])
    template <typename T, typename R, std::size_t NDIM, std::size_t KDIM>
    std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> >
    apply(World& world,
          const SeparatedConvolution<T,KDIM>& op,
          const std::vector< Function<R,NDIM> > f) {
        PROFILE_BLOCK(Vapply);
 
        std::vector< Function<R,NDIM> >& ncf = *const_cast< std::vector< Function<R,NDIM> >* >(&f);
        bool print_timings=(NDIM==6) and (world.rank()==0) and op.print_timings;
 
        double wall0=wall_time();
//        reconstruct(world, f);
        make_nonstandard(world, ncf);
        double wall1=wall_time();
        if (print_timings) printf("timer: %20.20s %8.2fs\n", "make_nonstandard", wall1-wall0);
 
        std::vector< Function<TENSOR_RESULT_TYPE(T,R), NDIM> > result(f.size());
        for (unsigned int i=0; i<f.size(); ++i) {
            result[i] = apply_only(op, f[i], false);
        }
 
        world.gop.fence();
 
        // restores promise of logical constness
        if (op.destructive()) {
            for (auto& ff : ncf) ff.clear(false);
            world.gop.fence();
        } else {
            reconstruct(world,f);
        }
 
        // svd-tensor requires some cleanup after apply
        if (result[0].get_impl()->get_tensor_type()==TT_2D) {
            for (auto& r : result) r.get_impl()->finalize_apply();
        }
 
        if (print_timings) {
            for (auto& r : result) r.get_impl()->print_timer();
            op.print_timer();
        }
        reconstruct(world, result);
 
        return result;
    }
 
    /// Normalizes a vector of functions --- v[i] = v[i].scale(1.0/v[i].norm2())
    template <typename T, std::size_t NDIM>
    void normalize(World& world, std::vector< Function<T,NDIM> >& v, bool fence=true) {
        PROFILE_BLOCK(Vnormalize);
        std::vector<double> nn = norm2s(world, v);
        for (unsigned int i=0; i<v.size(); ++i) v[i].scale(1.0/nn[i],false);
        if (fence) world.gop.fence();
    }
 
    template <typename T, std::size_t NDIM>
    void print_size(World &world, const std::vector<Function<T,NDIM> > &v, const std::string &msg = "vectorfunction" ){
        if(v.empty()){
            if(world.rank()==0) std::cout << "print_size: " << msg << " is empty" << std::endl;
        }else if(v.size()==1){
            v.front().print_size(msg);
        }else{
            for(auto x:v){
                // print("impl",x.get_impl().get());
                x.print_size(msg);
            }
        }
    }
 
    /// return the size of a vector of functions for each rank
    template <typename T, std::size_t NDIM>
    double get_size_local(World& world, const std::vector< Function<T,NDIM> >& v){
        double size=0.0;
        for(auto x:v){
            if (x.is_initialized()) size+=x.size_local();
        }
        const double d=sizeof(T);
        const double fac=1024*1024*1024;
        return size/fac*d;
    }
 
    /// return the size of a function for each rank
    template <typename T, std::size_t NDIM>
    double get_size_local(const Function<T,NDIM>& f){
        return get_size_local(f.world(),std::vector<Function<T,NDIM> >(1,f));
    }
 
 
    // gives back the size in GB
    template <typename T, std::size_t NDIM>
    double get_size(World& world, const std::vector< Function<T,NDIM> >& v){
 
        if (v.empty()) return 0.0;
 
        const double d=sizeof(T);
        const double fac=1024*1024*1024;
 
        double size=0.0;
        for(unsigned int i=0;i<v.size();i++){
            if (v[i].is_initialized()) size+=v[i].size();
        }
 
        return size/fac*d;
 
    }
 
    // gives back the size in GB
    template <typename T, std::size_t NDIM>
    double get_size(const Function<T,NDIM> & f){
        const double d=sizeof(T);
        const double fac=1024*1024*1024;
        double size=f.size();
        return size/fac*d;
    }
 
    /// apply op on the input vector yielding an output vector of functions
 
    /// @param[in]  op   the operator working on vin
    /// @param[in]  vin  vector of input Functions; needs to be refined to common level!
    /// @return vector of output Functions vout = op(vin)
    template <typename T, typename opT, std::size_t NDIM>
    std::vector<Function<T,NDIM> > multi_to_multi_op_values(const opT& op,
            const std::vector< Function<T,NDIM> >& vin,
            const bool fence=true) {
        MADNESS_ASSERT(vin.size()>0);
        MADNESS_ASSERT(vin[0].is_initialized()); // might be changed
        World& world=vin[0].world();
        Function<T,NDIM> dummy;
        dummy.set_impl(vin[0], false);
        std::vector<Function<T,NDIM> > vout=zero_functions<T,NDIM>(world, op.get_result_size());
        for (auto& out : vout) out.set_impl(vin[0],false);
        dummy.multi_to_multi_op_values(op, vin, vout, fence);
        return vout;
    }
 
 
 
 
    // convenience operators
 
    /// result[i] = a[i] + b[i]
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > operator+(const std::vector<Function<T,NDIM> >& lhs,
            const std::vector<Function<T,NDIM>>& rhs) {
        MADNESS_CHECK(lhs.size() == rhs.size());
        return gaxpy_oop(1.0,lhs,1.0,rhs);
    }
 
    /// result[i] = a[i] - b[i]
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > operator-(const std::vector<Function<T,NDIM> >& lhs,
            const std::vector<Function<T,NDIM> >& rhs) {
        MADNESS_CHECK(lhs.size() == rhs.size());
        return gaxpy_oop(1.0,lhs,-1.0,rhs);
    }
 
    /// result[i] = a[i] + b
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > operator+(const std::vector<Function<T,NDIM> >& lhs,
            const Function<T,NDIM>& rhs) {
        // MADNESS_CHECK(lhs.size() == rhs.size()); // no!!
        return gaxpy_oop(1.0,lhs,1.0,rhs);
    }
 
    /// result[i] = a[i] - b
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > operator-(const std::vector<Function<T,NDIM> >& lhs,
            const Function<T,NDIM>& rhs) {
        // MADNESS_CHECK(lhs.size() == rhs.size());  // no
        return gaxpy_oop(1.0,lhs,-1.0,rhs);
    }
 
    /// result[i] = a + b[i]
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > operator+(const Function<T,NDIM>& lhs,
            const std::vector<Function<T,NDIM> >& rhs) {
        // MADNESS_CHECK(lhs.size() == rhs.size());   // no
        return gaxpy_oop(1.0,rhs,1.0,lhs);
    }
 
    /// result[i] = a - b[i]
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > operator-(const Function<T,NDIM>& lhs,
            const std::vector<Function<T,NDIM> >& rhs) {
//         MADNESS_CHECK(lhs.size() == rhs.size());  // no
        return gaxpy_oop(-1.0,rhs,1.0,lhs);
    }
 
 
    template <typename T, typename R, std::size_t NDIM>
    std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> > operator*(const R fac,
            const std::vector<Function<T,NDIM> >& rhs) {
        if (rhs.size()>0) {
            std::vector<Function<T,NDIM> > tmp=copy(rhs[0].world(),rhs);
            scale(tmp[0].world(),tmp,TENSOR_RESULT_TYPE(T,R)(fac));
            return tmp;
        }
        return std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> >();
    }
 
    template <typename T, typename R, std::size_t NDIM>
    std::vector<Function<T,NDIM> > operator*(const std::vector<Function<T,NDIM> >& rhs,
            const R fac) {
        if (rhs.size()>0) {
            std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> > tmp=copy(rhs[0].world(),rhs);
            scale(tmp[0].world(),tmp,TENSOR_RESULT_TYPE(T,R)(fac));
            return tmp;
        }
        return std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> >();
    }
 
    /// multiply a vector of functions with a function: r[i] = v[i] * a
    template <typename T, typename R, std::size_t NDIM>
    std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> > operator*(const Function<T,NDIM>& a,
            const std::vector<Function<R,NDIM> >& v) {
        if (v.size()>0) return mul(v[0].world(),a,v,true);
        return std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> >();
    }
 
 
    /// multiply a vector of functions with a function: r[i] = a * v[i]
    template <typename T, typename R, std::size_t NDIM>
    std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> > operator*(const std::vector<Function<T,NDIM> >& v,
            const Function<R,NDIM>& a) {
        if (v.size()>0) return mul(v[0].world(),a,v,true);
        return std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> >();
    }
 
 
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > operator+=(std::vector<Function<T,NDIM> >& lhs, const std::vector<Function<T,NDIM> >& rhs) {
        MADNESS_CHECK(lhs.size() == rhs.size());
        if (lhs.size() > 0) gaxpy(lhs.front().world(), 1.0, lhs, 1.0, rhs);
    return lhs;
    }
 
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > operator-=(std::vector<Function<T,NDIM> >& lhs,
            const std::vector<Function<T,NDIM> >& rhs) {
        MADNESS_CHECK(lhs.size() == rhs.size());
        if (lhs.size() > 0) gaxpy(lhs.front().world(), 1.0, lhs, -1.0, rhs);
    return lhs;
    }
 
    /// return the real parts of the vector's function (if complex)
    template <typename T, std::size_t NDIM>
    std::vector<Function<typename Tensor<T>::scalar_type,NDIM> >
    real(const std::vector<Function<T,NDIM> >& v, bool fence=true) {
        std::vector<Function<typename Tensor<T>::scalar_type,NDIM> > result(v.size());
        for (std::size_t i=0; i<v.size(); ++i) result[i]=real(v[i],false);
        if (fence and result.size()>0) result[0].world().gop.fence();
        return result;
    }
 
    /// return the imaginary parts of the vector's function (if complex)
    template <typename T, std::size_t NDIM>
    std::vector<Function<typename Tensor<T>::scalar_type,NDIM> >
    imag(const std::vector<Function<T,NDIM> >& v, bool fence=true) {
        std::vector<Function<typename Tensor<T>::scalar_type,NDIM> > result(v.size());
        for (std::size_t i=0; i<v.size(); ++i) result[i]=imag(v[i],false);
        if (fence and result.size()>0) result[0].world().gop.fence();
        return result;
    }
 
    /// shorthand gradient operator
 
    /// returns the differentiated function f in all NDIM directions
    /// @param[in]  f       the function on which the grad operator works on
    /// @param[in]  refine  refinement before diff'ing makes the result more accurate
    /// @param[in]  fence   fence after completion; if reconstruction is needed always fence
    /// @return     the vector \frac{\partial}{\partial x_i} f
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > grad(const Function<T,NDIM>& f,
            bool refine=false, bool fence=true) {
 
        World& world=f.world();
        f.reconstruct();
        if (refine) f.refine();      // refine to make result more precise
 
        std::vector< std::shared_ptr< Derivative<T,NDIM> > > grad=
                gradient_operator<T,NDIM>(world);
 
        std::vector<Function<T,NDIM> > result(NDIM);
        for (size_t i=0; i<NDIM; ++i) result[i]=apply(*(grad[i]),f,false);
        if (fence) world.gop.fence();
        return result;
    }
 
    // BLM first derivative
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > grad_ble_one(const Function<T,NDIM>& f,
            bool refine=false, bool fence=true) {
 
        World& world=f.world();
        f.reconstruct();
        if (refine) f.refine();      // refine to make result more precise
 
        std::vector< std::shared_ptr< Derivative<T,NDIM> > > grad=
                gradient_operator<T,NDIM>(world);
 
        // Read in new coeff for each operator
        for (unsigned int i=0; i<NDIM; ++i) (*grad[i]).set_ble1();
 
        std::vector<Function<T,NDIM> > result(NDIM);
        for (unsigned int i=0; i<NDIM; ++i) result[i]=apply(*(grad[i]),f,false);
        if (fence) world.gop.fence();
        return result;
    }
 
    // BLM second derivative
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > grad_ble_two(const Function<T,NDIM>& f,
            bool refine=false, bool fence=true) {
 
        World& world=f.world();
        f.reconstruct();
        if (refine) f.refine();      // refine to make result more precise
 
        std::vector< std::shared_ptr< Derivative<T,NDIM> > > grad=
                gradient_operator<T,NDIM>(world);
 
        // Read in new coeff for each operator
        for (unsigned int i=0; i<NDIM; ++i) (*grad[i]).set_ble2();
 
        std::vector<Function<T,NDIM> > result(NDIM);
        for (unsigned int i=0; i<NDIM; ++i) result[i]=apply(*(grad[i]),f,false);
        if (fence) world.gop.fence();
        return result;
    }
 
    // Bspline first derivative
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > grad_bspline_one(const Function<T,NDIM>& f,
            bool refine=false, bool fence=true) {
 
        World& world=f.world();
        f.reconstruct();
        if (refine) f.refine();      // refine to make result more precise
 
        std::vector< std::shared_ptr< Derivative<T,NDIM> > > grad=
                gradient_operator<T,NDIM>(world);
 
        // Read in new coeff for each operator
        for (unsigned int i=0; i<NDIM; ++i) (*grad[i]).set_bspline1();
 
        std::vector<Function<T,NDIM> > result(NDIM);
        for (unsigned int i=0; i<NDIM; ++i) result[i]=apply(*(grad[i]),f,false);
        if (fence) world.gop.fence();
        return result;
    }
 
    // Bpsline second derivative
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > grad_bpsline_two(const Function<T,NDIM>& f,
            bool refine=false, bool fence=true) {
 
        World& world=f.world();
        f.reconstruct();
        if (refine) f.refine();      // refine to make result more precise
 
        std::vector< std::shared_ptr< Derivative<T,NDIM> > > grad=
                gradient_operator<T,NDIM>(world);
 
        // Read in new coeff for each operator
        for (unsigned int i=0; i<NDIM; ++i) (*grad[i]).set_bspline2();
 
        std::vector<Function<T,NDIM> > result(NDIM);
        for (unsigned int i=0; i<NDIM; ++i) result[i]=apply(*(grad[i]),f,false);
        if (fence) world.gop.fence();
        return result;
    }
 
    // Bspline third derivative
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > grad_bspline_three(const Function<T,NDIM>& f,
            bool refine=false, bool fence=true) {
 
        World& world=f.world();
        f.reconstruct();
        if (refine) f.refine();      // refine to make result more precise
 
        std::vector< std::shared_ptr< Derivative<T,NDIM> > > grad=
                gradient_operator<T,NDIM>(world);
 
        // Read in new coeff for each operator
        for (unsigned int i=0; i<NDIM; ++i) (*grad[i]).set_bspline3();
 
        std::vector<Function<T,NDIM> > result(NDIM);
        for (unsigned int i=0; i<NDIM; ++i) result[i]=apply(*(grad[i]),f,false);
        if (fence) world.gop.fence();
        return result;
    }
 
 
 
    /// shorthand div operator
 
    /// returns the dot product of nabla with a vector f
    /// @param[in]  f       the vector of functions on which the div operator works on
    /// @param[in]  refine  refinement before diff'ing makes the result more accurate
    /// @param[in]  fence   fence after completion; currently always fences
    /// @return     the vector \frac{\partial}{\partial x_i} f
    /// TODO: add this to operator fusion
    template <typename T, std::size_t NDIM>
    Function<T,NDIM> div(const std::vector<Function<T,NDIM> >& v,
            bool do_refine=false, bool fence=true) {
 
        MADNESS_ASSERT(v.size()>0);
        World& world=v[0].world();
        reconstruct(world,v);
        if (do_refine) refine(world,v);      // refine to make result more precise
 
        std::vector< std::shared_ptr< Derivative<T,NDIM> > > grad=
                gradient_operator<T,NDIM>(world);
 
        std::vector<Function<T,NDIM> > result(NDIM);
        for (size_t i=0; i<NDIM; ++i) result[i]=apply(*(grad[i]),v[i],false);
        world.gop.fence();
        return sum(world,result,fence);
    }
 
    /// shorthand rot operator
 
    /// returns the cross product of nabla with a vector f
    /// @param[in]  f       the vector of functions on which the rot operator works on
    /// @param[in]  refine  refinement before diff'ing makes the result more accurate
    /// @param[in]  fence   fence after completion; currently always fences
    /// @return     the vector \frac{\partial}{\partial x_i} f
    /// TODO: add this to operator fusion
    template <typename T, std::size_t NDIM>
    std::vector<Function<T,NDIM> > rot(const std::vector<Function<T,NDIM> >& v,
            bool do_refine=false, bool fence=true) {
 
        MADNESS_ASSERT(v.size()==3);
        World& world=v[0].world();
        reconstruct(world,v);
        if (do_refine) refine(world,v);      // refine to make result more precise
 
        std::vector< std::shared_ptr< Derivative<T,NDIM> > > grad=
                gradient_operator<T,NDIM>(world);
 
        std::vector<Function<T,NDIM> > d(NDIM),dd(NDIM);
        d[0]=apply(*(grad[1]),v[2],false);  // Dy z
        d[1]=apply(*(grad[2]),v[0],false);  // Dz x
        d[2]=apply(*(grad[0]),v[1],false);  // Dx y
        dd[0]=apply(*(grad[2]),v[1],false); // Dz y
        dd[1]=apply(*(grad[0]),v[2],false); // Dx z
        dd[2]=apply(*(grad[1]),v[0],false); // Dy x
        world.gop.fence();
 
        compress(world,d,false);
        compress(world,dd,false);
        world.gop.fence();
        d[0].gaxpy(1.0,dd[0],-1.0,false);
        d[1].gaxpy(1.0,dd[1],-1.0,false);
        d[2].gaxpy(1.0,dd[2],-1.0,false);
 
        world.gop.fence();
        reconstruct(d);
        return d;
    }
 
    /// shorthand cross operator
 
    /// returns the cross product of vectors f and g
    /// @param[in]  f       the vector of functions on which the rot operator works on
    /// @param[in]  g       the vector of functions on which the rot operator works on
    /// @param[in]  fence   fence after completion; currently always fences
    /// @return     the vector \frac{\partial}{\partial x_i} f, in redundant state
    /// TODO: add this to operator fusion
    template <typename T, typename R, std::size_t NDIM>
    std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> > cross(const std::vector<Function<T,NDIM> >& f,
            const std::vector<Function<R,NDIM> >& g,
            bool do_refine=false, bool fence=true) {
 
        MADNESS_ASSERT(f.size()==3);
        MADNESS_ASSERT(g.size()==3);
        World& world=f[0].world();
        ensure_tree_state_respecting_fence(f, TreeState::redundant, fence);
        ensure_tree_state_respecting_fence(g, TreeState::redundant, fence);
 
        std::vector<Function<TENSOR_RESULT_TYPE(T,R),NDIM> > d(f.size()),dd(f.size());
 
        d[0]=mul(f[1],g[2],false);
        d[1]=mul(f[2],g[0],false);
        d[2]=mul(f[0],g[1],false);
 
        dd[0]=mul(f[2],g[1],false);
        dd[1]=mul(f[0],g[2],false);
        dd[2]=mul(f[1],g[0],false);
        world.gop.fence();
 
        compress(world,d,false);
        compress(world,dd,false);
        world.gop.fence();
 
        d[0].gaxpy(1.0,dd[0],-1.0,false);
        d[1].gaxpy(1.0,dd[1],-1.0,false);
        d[2].gaxpy(1.0,dd[2],-1.0,false);
        world.gop.fence();
 
        make_redundant(world, d);
        return d;
    }
 
    template<typename T, std::size_t NDIM>
    void load_balance(World& world, std::vector<Function<T,NDIM> >& vf) {
 
        struct LBCost {
            LBCost() = default;
            double operator()(const Key<NDIM>& key, const FunctionNode<T,NDIM>& node) const {
                return node.coeff().size();
            }
        };
 
        LoadBalanceDeux<6> lb(world);
        for (const auto& f : vf) lb.add_tree(f, LBCost());
        FunctionDefaults<6>::redistribute(world, lb.load_balance());
 
    }
 
    /// load a vector of functions
    template<typename T, size_t NDIM>
    void load_function(World& world, std::vector<Function<T,NDIM> >& f,
            const std::string name) {
        if (world.rank()==0) print("loading vector of functions",name);
        archive::ParallelInputArchive<archive::BinaryFstreamInputArchive> ar(world, name.c_str(), 1);
        std::size_t fsize=0;
        ar & fsize;
        f.resize(fsize);
        for (std::size_t i=0; i<fsize; ++i) ar & f[i];
    }
 
    /// save a vector of functions
    template<typename T, size_t NDIM>
    void save_function(const std::vector<Function<T,NDIM> >& f, const std::string name) {
        if (f.size()>0) {
            World& world=f.front().world();
            if (world.rank()==0) print("saving vector of functions",name);
            archive::ParallelOutputArchive<archive::BinaryFstreamOutputArchive> ar(world, name.c_str(), 1);
            std::size_t fsize=f.size();
            ar & fsize;
            for (std::size_t i=0; i<fsize; ++i) ar & f[i];
        }
    }
 
 
}
#endif // MADNESS_MRA_VMRA_H__INCLUDED