Dense matrix and array manipulation » Slicing and Indexing module

Overview

All the aforementioned operations are handled through the generic DenseBase::operator()(const RowIndices&, const ColIndices&) method. Each argument can be:

• An integer indexing a single row or column, including symbolic indices.
• The symbol Eigen::all representing the whole set of respective rows or columns in increasing order.
• An ArithmeticSequence as constructed by the Eigen::seq, Eigen::seqN, or Eigen::lastN functions.
• Any 1D vector/array of integers including Eigen's vector/array, expressions, std::vector, std::array, as well as plain C arrays: int[N].

More generally, it can accepts any object exposing the following two member functions:

<integral type> operator[](<integral type>) const;
<integral type> size() const;

where <integral type> stands for any integer type compatible with Eigen::Index (i.e. std::ptrdiff_t).

Basic slicing

Taking a set of rows, columns, or elements, uniformly spaced within a matrix or vector is achieved through the Eigen::seq or Eigen::seqN functions where "seq" stands for arithmetic sequence. Their signatures are summarized below:

seq(firstIdx,lastIdx)

seq(2,5) <=> {2,3,4,5}

seq(firstIdx,lastIdx,incr)

seq(2,8,2) <=> {2,4,6,8}

seqN(firstIdx,size)

seqN(2,5) <=> {2,3,4,5,6}

seqN(firstIdx,size,incr)

seqN(2,3,3) <=> {2,5,8}

The firstIdx and lastIdx parameters can also be defined with the help of the Eigen::last symbol representing the index of the last row, column or element of the underlying matrix/vector once the arithmetic sequence is passed to it through operator(). Here are some examples for a 2D array/matrix A and a 1D array/vector v.

A(seq(i,last), seqN(0,n))

A.bottomLeftCorner(A.rows()-i,n)

A(seqN(i,m), seqN(i,n)

A.block(i,j,m,n)

A(seq(i0,i1), seq(j0,j1)

A.block(i0,j0,i1-i0+1,j1-j0+1)

A(all, seq(0,last,2))

A(seqN(1,n,2), all)

A(all, last-1)

A.col(A.cols()-2)

A(last/2,all)

A.row((A.rows()-1)/2)

v(seq(i,last))

v.tail(v.size()-i)

v(seq(last+1-n,last))

v.tail(n)

As seen in the last exemple, referencing the last n elements (or rows/columns) is a bit cumbersome to write. This becomes even more tricky and error prone with a non-default increment. Here comes Eigen::lastN(size), and Eigen::lastN(size,incr):

v(lastN(n))

v.tail(n)

v(lastN(m), lastN(n))

A.bottomRightCorner(m,n)

v(lastN(m), lastN(n))

A.bottomRightCorner(m,n)

A(all, lastN(n,3))

Compile time size and increment

In terms of performance, Eigen and the compiler can take advantage of compile-time size and increment. To this end, you can enforce compile-time parameters using Eigen::fix<val>. Such compile-time value can be combined with the Eigen::last symbol: v(seq(last-fix<7>, last-fix<2>)) In this example Eigen knowns at compile-time that the returned expression has 6 elements. It is equivalent to: v(seqN(last-7, fix<6>))

We can revisit the even columns of A example as follows: A(all, seq(0,last,fix<2>))

Reverse order

Row/column indices can also be enumerated in decreasing order using a negative increment. For instance, one over two columns of A from the column 20 to 10: A(all, seq(20, 10, fix<-2>)) The last n rows starting from the last one: A(seqN(last, n, fix<-1>), all) You can also use the ArithmeticSequence::reverse() method to reverse its order. The previous example can thus also be written as: A(lastN(n).reverse(), all)

Array of indices

The generic operator() can also takes as input an arbitrary list of row or column indices stored as either an ArrayXi, a std::vector<int>, std::array<int,N>, etc.

std::vector<int> ind{4,2,5,5,3};
MatrixXi A = MatrixXi::Random(4,6);
cout << "Initial matrix A:\n" << A << "\n\n";
cout << "A(all,ind):\n" << A(all,ind) << "\n\n";

Initial matrix A:
  7   9  -5  -3   3 -10
 -2  -6   1   0   5  -5
  6  -3   0   9  -8  -8
  6   6   3   9   2   6

A(all,ind):
  3  -5 -10 -10  -3
  5   1  -5  -5   0
 -8   0  -8  -8   9
  2   3   6   6   9

You can also directly pass a static array:

#if EIGEN_HAS_STATIC_ARRAY_TEMPLATE
MatrixXi A = MatrixXi::Random(4,6);
cout << "Initial matrix A:\n" << A << "\n\n";
cout << "A(all,{4,2,5,5,3}):\n" << A(all,{4,2,5,5,3}) << "\n\n";
#endif

Initial matrix A:
  7   9  -5  -3   3 -10
 -2  -6   1   0   5  -5
  6  -3   0   9  -8  -8
  6   6   3   9   2   6

A(all,{4,2,5,5,3}):
  3  -5 -10 -10  -3
  5   1  -5  -5   0
 -8   0  -8  -8   9
  2   3   6   6   9

or expressions:

ArrayXi ind(5); ind<<4,2,5,5,3;
MatrixXi A = MatrixXi::Random(4,6);
cout << "Initial matrix A:\n" << A << "\n\n";
cout << "A(all,ind-1):\n" << A(all,ind-1) << "\n\n";

Initial matrix A:
  7   9  -5  -3   3 -10
 -2  -6   1   0   5  -5
  6  -3   0   9  -8  -8
  6   6   3   9   2   6

A(all,ind-1):
-3  9  3  3 -5
 0 -6  5  5  1
 9 -3 -8 -8  0
 9  6  2  2  3

When passing an object with a compile-time size such as Array4i, std::array<int,N>, or a static array, then the returned expression also exhibit compile-time dimensions.

Custom index list

More generally, operator() can accept as inputs any object ind of type T compatible with:

Index s = ind.size(); or Index s = size(ind);
Index i;
i = ind[i];

This means you can easily build your own fancy sequence generator and pass it to operator(). Here is an exemple enlarging a given matrix while padding the additional first rows and columns through repetition:

struct pad {
  Index size() const { return out_size; }
  Index operator[] (Index i) const { return std::max<Index>(0,i-(out_size-in_size)); }
  Index in_size, out_size;
};

Matrix3i A;
A.reshaped() = VectorXi::LinSpaced(9,1,9);
cout << "Initial matrix A:\n" << A << "\n\n";
MatrixXi B(5,5);
B = A(pad{3,5}, pad{3,5});
cout << "A(pad{3,N}, pad{3,N}):\n" << B << "\n\n";

Initial matrix A:
1 4 7
2 5 8
3 6 9

A(pad{3,N}, pad{3,N}):
1 1 1 4 7
1 1 1 4 7
1 1 1 4 7
2 2 2 5 8
3 3 3 6 9