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Matrix manipulation via nullary-expressions

The main purpose of the class CwiseNullaryOp is to define procedural matrices such as constant or random matrices as returned by the Ones(), Zero(), Constant(), Identity() and Random() methods. Nevertheless, with some imagination it is possible to accomplish very sophisticated matrix manipulation with minimal efforts such that implementing new expression is rarely needed.

Example 1: circulant matrix

To explore these possibilities let us start with the circulant example of the implementing new expression topic. Let us recall that a circulant matrix is a matrix where each column is the same as the column to the left, except that it is cyclically shifted downwards. For example, here is a 4-by-4 circulant matrix:

\[ \begin{bmatrix} 1 & 8 & 4 & 2 \\ 2 & 1 & 8 & 4 \\ 4 & 2 & 1 & 8 \\ 8 & 4 & 2 & 1 \end{bmatrix} \]

A circulant matrix is uniquely determined by its first column. We wish to write a function makeCirculant which, given the first column, returns an expression representing the circulant matrix.

For this exercise, the return type of makeCirculant will be a CwiseNullaryOp that we need to instantiate with: 1 - a proper circulant_functor storing the input vector and implementing the adequate coefficient accessor operator(i,j) 2 - a template instantiation of class Matrix conveying compile-time information such as the scalar type, sizes, and preferred storage layout.

Calling ArgType the type of the input vector, we can construct the equivalent squared Matrix type as follows:

template<class ArgType>
struct circulant_helper {
  typedef Matrix<typename ArgType::Scalar,
                 ArgType::SizeAtCompileTime,
                 ArgType::SizeAtCompileTime,
                 ColMajor,
                 ArgType::MaxSizeAtCompileTime,
                 ArgType::MaxSizeAtCompileTime> MatrixType;
};

This little helper structure will help us to implement our makeCirculant function as follows:

template <class ArgType>
CwiseNullaryOp<circulant_functor<ArgType>, typename circulant_helper<ArgType>::MatrixType>
makeCirculant(const Eigen::MatrixBase<ArgType>& arg)
{
  typedef typename circulant_helper<ArgType>::MatrixType MatrixType;
  return MatrixType::NullaryExpr(arg.size(), arg.size(), circulant_functor<ArgType>(arg.derived()));
}

As usual, our function takes as argument a MatrixBase (see this page for more details). Then, the CwiseNullaryOp object is constructed through the DenseBase::NullaryExpr static method with the adequate runtime sizes.

Then, we need to implement our circulant_functor, which is a straightforward exercise:

template<class ArgType>
class circulant_functor {
  const ArgType &m_vec;
public:
  circulant_functor(const ArgType& arg) : m_vec(arg) {}
 
  const typename ArgType::Scalar& operator() (Index row, Index col) const {
    Index index = row - col;
    if (index < 0) index += m_vec.size();
    return m_vec(index);
  }
};

We are now all set to try our new feature:

int main()
{
  Eigen::VectorXd vec(4);
  vec << 1, 2, 4, 8;
  Eigen::MatrixXd mat;
  mat = makeCirculant(vec);
  std::cout << mat << std::endl;
}

If all the fragments are combined, the following output is produced, showing that the program works as expected:

1 8 4 2
2 1 8 4
4 2 1 8
8 4 2 1

This implementation of makeCirculant is much simpler than defining a new expression from scratch.

Example 2: indexing rows and columns

The goal here is to mimic MatLab's ability to index a matrix through two vectors of indices referencing the rows and columns to be picked respectively, like this:

A =
 7  9 -5 -3
-2 -6  1  0
 6 -3  0  9
 6  6  3  9
 
A([1 2 1], [3 2 1 0 0 2]) =
 0  1 -6 -2 -2  1
 9  0 -3  6  6  0
 0  1 -6 -2 -2  1

To this end, let us first write a nullary-functor storing references to the input matrix and to the two arrays of indices, and implementing the required operator()(i,j):

template<class ArgType, class RowIndexType, class ColIndexType>
class indexing_functor {
  const ArgType &m_arg;
  const RowIndexType &m_rowIndices;
  const ColIndexType &m_colIndices;
public:
  typedef Matrix<typename ArgType::Scalar,
                 RowIndexType::SizeAtCompileTime,
                 ColIndexType::SizeAtCompileTime,
                 ArgType::Flags&RowMajorBit?RowMajor:ColMajor,
                 RowIndexType::MaxSizeAtCompileTime,
                 ColIndexType::MaxSizeAtCompileTime> MatrixType;
 
  indexing_functor(const ArgType& arg, const RowIndexType& row_indices, const ColIndexType& col_indices)
    : m_arg(arg), m_rowIndices(row_indices), m_colIndices(col_indices)
  {}
 
  const typename ArgType::Scalar& operator() (Index row, Index col) const {
    return m_arg(m_rowIndices[row], m_colIndices[col]);
  }
};

Then, let's create an indexing(A,rows,cols) function creating the nullary expression:

template <class ArgType, class RowIndexType, class ColIndexType>
CwiseNullaryOp<indexing_functor<ArgType,RowIndexType,ColIndexType>, typename indexing_functor<ArgType,RowIndexType,ColIndexType>::MatrixType>
mat_indexing(const Eigen::MatrixBase<ArgType>& arg, const RowIndexType& row_indices, const ColIndexType& col_indices)
{
  typedef indexing_functor<ArgType,RowIndexType,ColIndexType> Func;
  typedef typename Func::MatrixType MatrixType;
  return MatrixType::NullaryExpr(row_indices.size(), col_indices.size(), Func(arg.derived(), row_indices, col_indices));
}

Finally, here is an example of how this function can be used:

  Eigen::MatrixXi A = Eigen::MatrixXi::Random(4,4);
  Array3i ri(1,2,1);
  ArrayXi ci(6); ci << 3,2,1,0,0,2;
  Eigen::MatrixXi B = mat_indexing(A, ri, ci);
  std::cout << "A =" << std::endl;
  std::cout << A << std::endl << std::endl;
  std::cout << "A([" << ri.transpose() << "], [" << ci.transpose() << "]) =" << std::endl;
  std::cout << B << std::endl;

This straightforward implementation is already quite powerful as the row or column index arrays can also be expressions to perform offsetting, modulo, striding, reverse, etc.

  B =  mat_indexing(A, ri+1, ci);
  std::cout << "A(ri+1,ci) =" << std::endl;
  std::cout << B << std::endl << std::endl;
#if EIGEN_COMP_CXXVER >= 11
  B =  mat_indexing(A, ArrayXi::LinSpaced(13,0,12).unaryExpr([](int x){return x%4;}), ArrayXi::LinSpaced(4,0,3));
  std::cout << "A(ArrayXi::LinSpaced(13,0,12).unaryExpr([](int x){return x%4;}), ArrayXi::LinSpaced(4,0,3)) =" << std::endl;
  std::cout << B << std::endl << std::endl;
#endif

and the output is:

A(ri+1,ci) =
 9  0 -3  6  6  0
 9  3  6  6  6  3
 9  0 -3  6  6  0
 
A(ArrayXi::LinSpaced(13,0,12).unaryExpr([](int x){return x%4;}), ArrayXi::LinSpaced(4,0,3)) =
 7  9 -5 -3
-2 -6  1  0
 6 -3  0  9
 6  6  3  9
 7  9 -5 -3
-2 -6  1  0
 6 -3  0  9
 6  6  3  9
 7  9 -5 -3
-2 -6  1  0
 6 -3  0  9
 6  6  3  9
 7  9 -5 -3

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Licensed under the MPL2 License.
https://eigen.tuxfamily.org/dox/TopicCustomizing_NullaryExpr.html