libtrilinos-stokhos-dev 12.4.2-2 / usr / include / trilinos / Stokhos_LTBSparse3Tensor.hpp

// @HEADER
// ***********************************************************************
// 
//                           Stokhos Package
//                 Copyright (2009) Sandia Corporation
// 
// Under terms of Contract DE-AC04-94AL85000, there is a non-exclusive
// license for use of this work by or on behalf of the U.S. Government.
// 
// Redistribution and use in source and binary forms, with or without
// modification, are permitted provided that the following conditions are
// met:
//
// 1. Redistributions of source code must retain the above copyright
// notice, this list of conditions and the following disclaimer.
//
// 2. Redistributions in binary form must reproduce the above copyright
// notice, this list of conditions and the following disclaimer in the
// documentation and/or other materials provided with the distribution.
//
// 3. Neither the name of the Corporation nor the names of the
// contributors may be used to endorse or promote products derived from
// this software without specific prior written permission.
//
// THIS SOFTWARE IS PROVIDED BY SANDIA CORPORATION "AS IS" AND ANY
// EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
// IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
// PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL SANDIA CORPORATION OR THE
// CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
// EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
// PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
// PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
// LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
// NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
// SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
//
// Questions? Contact Eric T. Phipps (etphipp@sandia.gov).
// 
// ***********************************************************************
// @HEADER

#ifndef STOKHOS_LTB_SPARSE_3_TENSOR_HPP
#define STOKHOS_LTB_SPARSE_3_TENSOR_HPP

#include <ostream>

#include "Stokhos_TotalOrderBasis.hpp"
#include "Stokhos_OrthogPolyApprox.hpp"

namespace Stokhos {

  /*!
   * \brief Data structure storing a sparse 3-tensor C(i,j,k) in a
   * a tree-based format for lexicographically ordered product bases.
   */
  template <typename ordinal_type, typename value_type>
  class LTBSparse3Tensor {
  public:

    //! Node type used in constructing the tree
    struct CijkNode {
      Teuchos::Array< Teuchos::RCP<CijkNode> > children;
      Teuchos::Array<value_type> values;
      ordinal_type p_i, p_j, p_k;
      ordinal_type i_begin, j_begin, k_begin;
      ordinal_type i_size, j_size, k_size;
      ordinal_type my_num_entries, total_num_entries;
      ordinal_type total_num_leafs;
      bool is_leaf, parent_j_equals_k;
    };

    //! Constructor
    LTBSparse3Tensor(const bool symm) : is_symmetric(symm) {}

    //! Destructor
    ~LTBSparse3Tensor() {}

    //! Set the head node
    void setHeadNode(const Teuchos::RCP<CijkNode>& h) { head = h; }

    //! Get the head node
    Teuchos::RCP<const CijkNode> getHeadNode() const { return head; }

    //! Print tensor
    void print(std::ostream& os) const {}

    //! Get Cijk value for a given i, j, k indices
    value_type getValue(ordinal_type i, ordinal_type j, ordinal_type k) const
      {}

    //! Return number of non-zero entries
    ordinal_type num_entries() const {
      if (head != Teuchos::null)
        return head->total_num_entries;
      return 0;
    }

    //! Return number of nodes
    ordinal_type num_leafs() const {
      if (head != Teuchos::null)
        return head->total_num_leafs;
      return 0;
    }

    //! Return if symmetric
    bool symmetric() const { return is_symmetric; }

  private:

    // Prohibit copying
    LTBSparse3Tensor(const LTBSparse3Tensor&);

    // Prohibit Assignment
    LTBSparse3Tensor& operator=(const LTBSparse3Tensor& b);

  protected:

    Teuchos::RCP<CijkNode> head;
    bool is_symmetric;

  }; // class LTBSparse3Tensor

  /*! \relates LTBSparse3Tensor
   * Print triple product tensor to output stream
   */
  template <typename ordinal_type, typename value_type>
  std::ostream&
  operator << (std::ostream& os,
               const LTBSparse3Tensor<ordinal_type, value_type>& Cijk) {
    Cijk.print(os);
    return os;
  }

  template <typename ordinal_type>
  struct LexicographicTreeBasisNode {
    Teuchos::Array< Teuchos::RCP<LexicographicTreeBasisNode> > children;
    Teuchos::Array< Stokhos::MultiIndex<ordinal_type> > terms;
    ordinal_type index_begin;
    ordinal_type block_size;

    // Default constructor
    LexicographicTreeBasisNode() :
      children(), terms(), index_begin(0), block_size(0) {}

    // Copy constructor
    LexicographicTreeBasisNode(const LexicographicTreeBasisNode& node) :
      children(node.children.size()), terms(node.terms),
      index_begin(node.index_begin), block_size(node.block_size) {
      for (ordinal_type i=0; i<children.size(); ++i)
        children[i] =
          Teuchos::rcp(new LexicographicTreeBasisNode(*(node->children[i])));
    }

    // Assignment operator
    LexicographicTreeBasisNode&
    operator=(const LexicographicTreeBasisNode& node) {
      if (this != &node) {
        children.resize(node.children.size());
        for (ordinal_type i=0; i<children.size(); ++i)
          children[i] =
            Teuchos::rcp(new LexicographicTreeBasisNode(*(node->children[i])));
        terms = node.terms;
        index_begin = node.index_begin;
        block_size = node.block_size;
      }
      return *this;
    }
  };

  template <typename ordinal_type>
  Teuchos::RCP< LexicographicTreeBasisNode<ordinal_type> >
  build_lexicographic_basis_tree(
    const Teuchos::ArrayView<const ordinal_type>& basis_orders,
    const ordinal_type total_order,
    const ordinal_type index_begin = ordinal_type(0),
    const ordinal_type order_sum = ordinal_type(0),
    const Stokhos::MultiIndex<ordinal_type>& term_prefix =
    Stokhos::MultiIndex<ordinal_type>()) {

    typedef LexicographicTreeBasisNode<ordinal_type> node_type;

    ordinal_type my_d = basis_orders.size();
    ordinal_type prev_d = term_prefix.dimension();
    ordinal_type p = basis_orders[0];
    ordinal_type my_p = std::min(total_order-order_sum, p);

    Teuchos::RCP<node_type> node = Teuchos::rcp(new node_type);
    node->index_begin = index_begin;
    node->terms.resize(my_p+1);
    for (ordinal_type i=0; i<=my_p; ++i) {
      node->terms[i].resize(prev_d+1);
      for (ordinal_type j=0; j<prev_d; ++j)
        node->terms[i][j] = term_prefix[j];
      node->terms[i][prev_d] = i;
    }

    // Base case for dimension = 1
    if (my_d == 1) {
      node->block_size = my_p+1;
    }

    // General case -- call recursively stripping off first dimension
    else {
      Teuchos::ArrayView<const ordinal_type> bo =
        Teuchos::arrayView(&basis_orders[1],my_d-1);
      node->children.resize(my_p+1);
      node->index_begin = index_begin;
      node->block_size = 0;
      for (ordinal_type i=0; i<=my_p; ++i) {
        Teuchos::RCP<node_type> child = build_lexicographic_basis_tree(
          bo, total_order, index_begin+node->block_size, order_sum+i,
          node->terms[i]);
        node->children[i] = child;
        node->block_size += child->block_size;
      }
    }

    return node;
  }

  // An approach for building a sparse 3-tensor only for lexicographically
  // ordered total order basis using a tree-based format
  template <typename ordinal_type,
            typename value_type>
  Teuchos::RCP< LTBSparse3Tensor<ordinal_type, value_type> >
  computeTripleProductTensorLTB(
    const TotalOrderBasis<ordinal_type, value_type,LexographicLess<MultiIndex<ordinal_type> > >& product_basis,
    bool symmetric = false) {
#ifdef STOKHOS_TEUCHOS_TIME_MONITOR
    TEUCHOS_FUNC_TIME_MONITOR("Stokhos: Total Triple-Product Tensor Time");
#endif
    using Teuchos::RCP;
    using Teuchos::rcp;
    using Teuchos::Array;
    using Teuchos::ArrayView;

    const Array< RCP<const OneDOrthogPolyBasis<ordinal_type, value_type> > >& bases = product_basis.getCoordinateBases();
    ordinal_type d = bases.size();
    ordinal_type p = product_basis.order();
    Array<ordinal_type> basis_orders(d);
    for (int i=0; i<d; ++i)
      basis_orders[i] = bases[i]->order();
    ArrayView<const ordinal_type> basis_orders_view = basis_orders();

    // Create 1-D triple products
    Array< RCP<Sparse3Tensor<ordinal_type,value_type> > > Cijk_1d(d);
    for (ordinal_type i=0; i<d; i++) {
      Cijk_1d[i] =
        bases[i]->computeSparseTripleProductTensor(bases[i]->order()+1);
    }
    ArrayView<const RCP<Sparse3Tensor<ordinal_type,value_type> > > Cijk_1d_view
      = Cijk_1d();

    // Create lexicographic tree basis
    Teuchos::RCP< LexicographicTreeBasisNode<ordinal_type> > ltb =
      build_lexicographic_basis_tree(basis_orders_view, p);

    // Current implementation is recursive on the dimension d
    typedef LTBSparse3Tensor<ordinal_type, value_type> Cijk_type;
    RCP<Cijk_type> Cijk = rcp(new Cijk_type(symmetric));
    RCP<typename Cijk_type::CijkNode> head =
      computeCijkLTBNode(
        basis_orders_view, Cijk_1d_view, ltb, ltb, ltb, p, symmetric);
    Cijk->setHeadNode(head);

    return Cijk;
  }

  template <typename ordinal_type,
            typename value_type>
  Teuchos::RCP<typename LTBSparse3Tensor<ordinal_type, value_type>::CijkNode>
  computeCijkLTBNode(
    const Teuchos::ArrayView<const ordinal_type>& basis_orders,
    const Teuchos::ArrayView<const Teuchos::RCP<Sparse3Tensor<ordinal_type,value_type> > >& Cijk_1d,
    const Teuchos::RCP<LexicographicTreeBasisNode<ordinal_type> >& i_ltb,
    const Teuchos::RCP<LexicographicTreeBasisNode<ordinal_type> >& j_ltb,
    const Teuchos::RCP<LexicographicTreeBasisNode<ordinal_type> >& k_ltb,
    const ordinal_type total_order,
    const bool symmetric,
    const ordinal_type sum_i = ordinal_type(0),
    const ordinal_type sum_j = ordinal_type(0),
    const ordinal_type sum_k = ordinal_type(0),
    const value_type cijk_base = value_type(1)) {

    using Teuchos::RCP;
    using Teuchos::rcp;
    using Teuchos::ArrayView;
    using Teuchos::arrayView;
    typedef typename LTBSparse3Tensor<ordinal_type, value_type>::CijkNode node_type;
    typedef ProductBasisUtils::Cijk_1D_Iterator<ordinal_type> Cijk_Iterator;

    ordinal_type my_d = basis_orders.size();
    ordinal_type p = basis_orders[0];
    ordinal_type p_i = std::min(total_order-sum_i, p);
    ordinal_type p_j = std::min(total_order-sum_j, p);
    ordinal_type p_k = std::min(total_order-sum_k, p);
    Cijk_Iterator cijk_iterator(p_i, p_j, p_k, symmetric);

    RCP<node_type> node = rcp(new node_type);
    node->p_i = p_i;
    node->p_j = p_j;
    node->p_k = p_k;
    node->i_begin = i_ltb->index_begin;
    node->j_begin = j_ltb->index_begin;
    node->k_begin = k_ltb->index_begin;
    node->i_size = i_ltb->block_size;
    node->j_size = j_ltb->block_size;
    node->k_size = k_ltb->block_size;

    // Base case -- compute the actual cijk values
    if (my_d == 1) {
      node->is_leaf = true;
      bool more = true;
      while (more) {
        value_type cijk =
          Cijk_1d[0]->getValue(cijk_iterator.i,
                               cijk_iterator.j,
                               cijk_iterator.k);
        node->values.push_back(cijk*cijk_base);
        more = cijk_iterator.increment();
      }
      node->my_num_entries = node->values.size();
      node->total_num_entries = node->values.size();
      node->total_num_leafs = 1;
    }

    // General case -- call recursively stripping off first dimension
    else {
      node->is_leaf = false;
      ArrayView<const ordinal_type> bo = arrayView(&basis_orders[1], my_d-1);
      ArrayView<const RCP<Sparse3Tensor<ordinal_type,value_type> > > c1d =
        arrayView(&Cijk_1d[1], my_d-1);
      node->total_num_entries = 0;
      node->total_num_leafs = 0;
      bool more = true;
      while (more) {
        value_type cijk =
          Cijk_1d[0]->getValue(cijk_iterator.i,
                               cijk_iterator.j,
                               cijk_iterator.k);
        RCP<node_type> child =
          computeCijkLTBNode(bo, c1d,
                             i_ltb->children[cijk_iterator.i],
                             j_ltb->children[cijk_iterator.j],
                             k_ltb->children[cijk_iterator.k],
                             total_order, symmetric,
                             sum_i + cijk_iterator.i,
                             sum_j + cijk_iterator.j,
                             sum_k + cijk_iterator.k,
                             cijk_base*cijk);
        node->children.push_back(child);
        node->total_num_entries += child->total_num_entries;
        node->total_num_leafs += child->total_num_leafs;
        more = cijk_iterator.increment();
      }
      node->my_num_entries = node->children.size();
    }

    return node;
  }

  // An approach for building a sparse 3-tensor only for lexicographically
  // ordered total order basis using a tree-based format -- the leaf nodes
  // are replaced by a dense p_i x p_j x p_k block
  template <typename ordinal_type,
            typename value_type>
  Teuchos::RCP< LTBSparse3Tensor<ordinal_type, value_type> >
  computeTripleProductTensorLTBBlockLeaf(
    const TotalOrderBasis<ordinal_type, value_type,LexographicLess<MultiIndex<ordinal_type> > >& product_basis,
    bool symmetric = false) {
#ifdef STOKHOS_TEUCHOS_TIME_MONITOR
    TEUCHOS_FUNC_TIME_MONITOR("Stokhos: Total Triple-Product Tensor Time");
#endif
    using Teuchos::RCP;
    using Teuchos::rcp;
    using Teuchos::Array;
    using Teuchos::ArrayView;

    const Array< RCP<const OneDOrthogPolyBasis<ordinal_type, value_type> > >& bases = product_basis.getCoordinateBases();
    ordinal_type d = bases.size();
    ordinal_type p = product_basis.order();
    Array<ordinal_type> basis_orders(d);
    for (int i=0; i<d; ++i)
      basis_orders[i] = bases[i]->order();
    ArrayView<const ordinal_type> basis_orders_view = basis_orders();

    // Create 1-D triple products
    Array< RCP<Dense3Tensor<ordinal_type,value_type> > > Cijk_1d(d);
    for (ordinal_type i=0; i<d; i++) {
      Cijk_1d[i] = bases[i]->computeTripleProductTensor();
    }
    ArrayView<const RCP<Dense3Tensor<ordinal_type,value_type> > > Cijk_1d_view
      = Cijk_1d();

    // Create lexicographic tree basis
    Teuchos::RCP< LexicographicTreeBasisNode<ordinal_type> > ltb =
      build_lexicographic_basis_tree(basis_orders_view, p);

    // Current implementation is recursive on the dimension d
    typedef LTBSparse3Tensor<ordinal_type, value_type> Cijk_type;
    RCP<Cijk_type> Cijk = rcp(new Cijk_type(symmetric));
    RCP<typename Cijk_type::CijkNode> head =
      computeCijkLTBNodeBlockLeaf(
        basis_orders_view, Cijk_1d_view, ltb, ltb, ltb, p, symmetric);
    Cijk->setHeadNode(head);

    return Cijk;
  }

  template <typename ordinal_type,
            typename value_type>
  Teuchos::RCP<typename LTBSparse3Tensor<ordinal_type, value_type>::CijkNode>
  computeCijkLTBNodeBlockLeaf(
    const Teuchos::ArrayView<const ordinal_type>& basis_orders,
    const Teuchos::ArrayView<const Teuchos::RCP<Dense3Tensor<ordinal_type,value_type> > >& Cijk_1d,
    const Teuchos::RCP<LexicographicTreeBasisNode<ordinal_type> >& i_ltb,
    const Teuchos::RCP<LexicographicTreeBasisNode<ordinal_type> >& j_ltb,
    const Teuchos::RCP<LexicographicTreeBasisNode<ordinal_type> >& k_ltb,
    const ordinal_type total_order,
    const bool symmetric,
    const ordinal_type sum_i = ordinal_type(0),
    const ordinal_type sum_j = ordinal_type(0),
    const ordinal_type sum_k = ordinal_type(0),
    const value_type cijk_base = value_type(1),
    const bool parent_j_equals_k = true) {

    using Teuchos::RCP;
    using Teuchos::rcp;
    using Teuchos::ArrayView;
    using Teuchos::arrayView;
    typedef typename LTBSparse3Tensor<ordinal_type, value_type>::CijkNode node_type;

    ordinal_type my_d = basis_orders.size();
    ordinal_type p = basis_orders[0];
    ordinal_type p_i = std::min(total_order-sum_i, p);
    ordinal_type p_j = std::min(total_order-sum_j, p);
    ordinal_type p_k = std::min(total_order-sum_k, p);

    RCP<node_type> node = rcp(new node_type);
    node->p_i = p_i;
    node->p_j = p_j;
    node->p_k = p_k;
    node->i_begin = i_ltb->index_begin;
    node->j_begin = j_ltb->index_begin;
    node->k_begin = k_ltb->index_begin;
    node->i_size = i_ltb->block_size;
    node->j_size = j_ltb->block_size;
    node->k_size = k_ltb->block_size;
    node->parent_j_equals_k = parent_j_equals_k;

    // Base case -- compute the actual cijk values as a "brick"
    // Could store values as a "pyramid" using commented out code below
    // However that seems to result in very bad performance, e.g., a lot
    // of register spill in multiply code based on this tensor
    if (my_d == 1) {
      node->is_leaf = true;
      node->values.reserve((p_i+1)*(p_j+1)*(p_k+1));
      for (ordinal_type i=0; i<=p_i; ++i) {
        for (ordinal_type j=0; j<=p_j; ++j) {
          // ordinal_type k0 = parent_j_equals_k ? std::max(j,std::abs(i-j)) :
          //   std::abs(i-j);
          // if (symmetric && (k0%2 != (i+j)%2)) ++k0;
          // const ordinal_type k_end = std::min(p_k,i+j);
          // const ordinal_type k_inc = symmetric ? 2 : 1;
          ordinal_type k0 = parent_j_equals_k ? j : 0;
          if (symmetric && (k0%2 != (i+j)%2)) ++k0;
          const ordinal_type k_end = p_k;
          const ordinal_type k_inc = symmetric ? 2 : 1;
          for (ordinal_type k=k0; k<=k_end; k+=k_inc) {
            value_type cijk = (*Cijk_1d[0])(i, j, k);
            if (j+node->j_begin == k+node->k_begin) cijk *= 0.5;
            node->values.push_back(cijk*cijk_base);
          }
        }
      }
      node->my_num_entries = node->values.size();
      node->total_num_entries = node->values.size();
      node->total_num_leafs = 1;
    }

    // General case -- call recursively stripping off first dimension
    else {
      node->is_leaf = false;
      ArrayView<const ordinal_type> bo = arrayView(&basis_orders[1], my_d-1);
      ArrayView<const RCP<Dense3Tensor<ordinal_type,value_type> > > c1d =
        arrayView(&Cijk_1d[1], my_d-1);
      node->total_num_entries = 0;
      node->total_num_leafs = 0;
      for (ordinal_type i=0; i<=p_i; ++i) {
        for (ordinal_type j=0; j<=p_j; ++j) {
          ordinal_type k0 = parent_j_equals_k ? std::max(j,std::abs(i-j)) :
            std::abs(i-j);
          if (symmetric && (k0%2 != (i+j)%2)) ++k0;
          const ordinal_type k_end = std::min(p_k,i+j);
          const ordinal_type k_inc = symmetric ? 2 : 1;
          for (ordinal_type k=k0; k<=k_end; k+=k_inc) {
            value_type cijk = (*Cijk_1d[0])(i, j, k);
            RCP<node_type> child =
              computeCijkLTBNodeBlockLeaf(bo, c1d,
                                          i_ltb->children[i],
                                          j_ltb->children[j],
                                          k_ltb->children[k],
                                          total_order, symmetric,
                                          sum_i + i,
                                          sum_j + j,
                                          sum_k + k,
                                          cijk_base*cijk,
                                          parent_j_equals_k && j == k);
            node->children.push_back(child);
            node->total_num_entries += child->total_num_entries;
            node->total_num_leafs += child->total_num_leafs;
          }
        }
      }
      node->my_num_entries = node->children.size();
    }

    return node;
  }

  template <typename ordinal_type>
  struct FlatLTBSparse3TensorNode {
    ordinal_type i_begin, j_begin, k_begin;
    ordinal_type p_i, p_j, p_k;
    bool parent_j_equals_k;
  };

  template <typename ordinal_type, typename value_type>
  struct FlatLTBSparse3Tensor {
    typedef Teuchos::Array< FlatLTBSparse3TensorNode<ordinal_type> > node_array_type;
    typedef Teuchos::Array< value_type > cijk_array_type;

    typedef typename node_array_type::iterator node_iterator;
    typedef typename node_array_type::const_iterator node_const_iterator;
    typedef typename cijk_array_type::iterator cijk_iterator;
    typedef typename cijk_array_type::const_iterator cijk_const_iterator;

    node_array_type node;
    cijk_array_type cijk;
    bool symmetric;
  };

  template <typename ordinal_type, typename value_type>
  Teuchos::RCP< FlatLTBSparse3Tensor<ordinal_type,value_type> >
  computeFlatTripleProductTensorLTB(
    const TotalOrderBasis<ordinal_type, value_type,LexographicLess<MultiIndex<ordinal_type> > >& product_basis,
    bool symmetric = false) {
#ifdef STOKHOS_TEUCHOS_TIME_MONITOR
    TEUCHOS_FUNC_TIME_MONITOR("Stokhos: Flat Triple-Product Tensor Time");
#endif
    using Teuchos::RCP;
    using Teuchos::rcp;

    // Build LTB 3 tensor
    typedef LTBSparse3Tensor<ordinal_type, value_type> Cijk_LTB_type;
    RCP<Cijk_LTB_type> ltb_tensor =
      computeTripleProductTensorLTBBlockLeaf(product_basis, symmetric);

    // Create flat LTB 3 tensor
    RCP< FlatLTBSparse3Tensor<ordinal_type,value_type> > flat_tensor =
      rcp(new FlatLTBSparse3Tensor<ordinal_type,value_type>);
    flat_tensor->node.reserve(ltb_tensor->num_leafs());
    flat_tensor->cijk.reserve(ltb_tensor->num_entries());
    flat_tensor->symmetric = symmetric;

    // Fill flat 3 tensor
    typedef typename Cijk_LTB_type::CijkNode node_type;
    Teuchos::Array< Teuchos::RCP<const node_type> > node_stack;
    Teuchos::Array< ordinal_type > index_stack;
    node_stack.push_back(ltb_tensor->getHeadNode());
    index_stack.push_back(0);
    Teuchos::RCP<const node_type> node;
    ordinal_type child_index;
    while (node_stack.size() > 0) {
      node = node_stack.back();
      child_index = index_stack.back();

      // Leaf
      if (node->is_leaf) {
        FlatLTBSparse3TensorNode<ordinal_type> leaf;
        leaf.i_begin = node->i_begin;
        leaf.j_begin = node->j_begin;
        leaf.k_begin = node->k_begin;
        leaf.p_i = node->p_i;
        leaf.p_j = node->p_j;
        leaf.p_k = node->p_k;
        leaf.parent_j_equals_k = node->parent_j_equals_k;
        flat_tensor->node.push_back(leaf);
        flat_tensor->cijk.insert(flat_tensor->cijk.end(), 
                                 node->values.begin(),
                                 node->values.end());
        node_stack.pop_back();
        index_stack.pop_back();
      }

      // More children to process -- process them first
      else if (child_index < node->children.size()) {
        ++index_stack.back();
        node = node->children[child_index];
        node_stack.push_back(node);
        index_stack.push_back(0);
      }

      // No more children
      else {
        node_stack.pop_back();
        index_stack.pop_back();
      }

    }

    return flat_tensor;
  }

  template <int max_size, typename ordinal_type, typename value_type>
  void
  flatLTB3TensorMultiply(
    OrthogPolyApprox<ordinal_type,value_type>& c,
    const OrthogPolyApprox<ordinal_type,value_type>& a,
    const OrthogPolyApprox<ordinal_type,value_type>& b,
    const FlatLTBSparse3Tensor<ordinal_type,value_type>& cijk) {
    value_type ab[max_size][max_size];

    // Set coefficients to 0
    c.init(value_type(0));

    // Get pointers to coefficients
    const value_type *ca = a.coeff();
    const value_type *cb = b.coeff();
    value_type *cc = c.coeff();

    typedef FlatLTBSparse3Tensor<ordinal_type,value_type> cijk_type;
    typedef typename cijk_type::node_const_iterator node_iterator;
    typedef typename cijk_type::cijk_const_iterator cijk_iterator;
    node_iterator ni = cijk.node.begin();
    node_iterator ni_end = cijk.node.end();
    cijk_iterator ci = cijk.cijk.begin();
    for (; ni != ni_end; ++ni) {
      value_type *c_block = cc + ni->i_begin;
      const value_type *a_j_block = ca + ni->j_begin;
      const value_type *b_k_block = cb + ni->k_begin;
      const value_type *a_k_block = ca + ni->k_begin;
      const value_type *b_j_block = cb + ni->j_begin;
      const ordinal_type p_i = ni->p_i;
      const ordinal_type p_j = ni->p_j;
      const ordinal_type p_k = ni->p_k;
      for (ordinal_type j=0; j<=p_j; ++j)
        for (ordinal_type k=0; k<=p_k; ++k)
          ab[j][k] = a_j_block[j]*b_k_block[k] + a_k_block[k]*b_j_block[j];
      for (ordinal_type i=0; i<=p_i; ++i) {
        value_type tmp = value_type(0);
        for (ordinal_type j=0; j<=p_j; ++j) {
          // This is for pyramid instead of brick
          // ordinal_type k0 = ni->parent_j_equals_k ? std::max(j,std::abs(i-j)) :
          //   std::abs(i-j);
          // if (cijk.symmetric && (k0%2 != (i+j)%2)) ++k0;
          // const ordinal_type k_end = std::min(p_k,i+j);
          // const ordinal_type k_inc = cijk.symmetric ? 2 : 1;

          ordinal_type k0 = ni->parent_j_equals_k ? j : 0;
          if (cijk.symmetric && (k0%2 != (i+j)%2)) ++k0;
          const ordinal_type k_end = p_k;
          const ordinal_type k_inc = cijk.symmetric ? 2 : 1;
          for (ordinal_type k=k0; k<=k_end; k+=k_inc) {
            tmp += (*ci)*ab[j][k];
            ++ci;
          }
        }
        c_block[i] += tmp;
      }
    }
  }

} // namespace Stokhos

// Include template definitions
//#include "Stokhos_LTBSparse3TensorImp.hpp"

#endif // STOKHOS_LTB_SPARSE_3_TENSOR_HPP