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Block operations

This page explains the essentials of block operations. A block is a rectangular part of a matrix or array. Blocks expressions can be used both as rvalues and as lvalues. As usual with Eigen expressions, this abstraction has zero runtime cost provided that you let your compiler optimize.

Using block operations

The most general block operation in Eigen is called .block() . There are two versions, whose syntax is as follows:

Block operation Version constructing a
dynamic-size block expression 		Version constructing a
fixed-size block expression
Block of size (p,q), starting at (i,j) 
matrix.block(i,j,p,q);

matrix.block<p,q>(i,j);

As always in Eigen, indices start at 0.

Both versions can be used on fixed-size and dynamic-size matrices and arrays. These two expressions are semantically equivalent. The only difference is that the fixed-size version will typically give you faster code if the block size is small, but requires this size to be known at compile time.

The following program uses the dynamic-size and fixed-size versions to print the values of several blocks inside a matrix.

Example: Output: 
#include <Eigen/Dense>
#include <iostream>
 
using namespace std;
 
int main()
{
  Eigen::MatrixXf m(4,4);
  m <<  1, 2, 3, 4,
        5, 6, 7, 8,
        9,10,11,12,
       13,14,15,16;
  cout << "Block in the middle" << endl;
  cout << m.block<2,2>(1,1) << endl << endl;
  for (int i = 1; i <= 3; ++i)
  {
    cout << "Block of size " << i << "x" << i << endl;
    cout << m.block(0,0,i,i) << endl << endl;
  }
}
 
Block in the middle
 6  7
10 11

Block of size 1x1
1

Block of size 2x2
1 2
5 6

Block of size 3x3
 1  2  3
 5  6  7
 9 10 11

In the above example the .block() function was employed as a rvalue, i.e. it was only read from. However, blocks can also be used as lvalues, meaning that you can assign to a block.

This is illustrated in the following example. This example also demonstrates blocks in arrays, which works exactly like the above-demonstrated blocks in matrices.

Example: Output: 
#include <Eigen/Dense>
#include <iostream>
 
using namespace std;
using namespace Eigen;
 
int main()
{
  Array22f m;
  m << 1,2,
       3,4;
  Array44f a = Array44f::Constant(0.6);
  cout << "Here is the array a:" << endl << a << endl << endl;
  a.block<2,2>(1,1) = m;
  cout << "Here is now a with m copied into its central 2x2 block:" << endl << a << endl << endl;
  a.block(0,0,2,3) = a.block(2,1,2,3);
  cout << "Here is now a with bottom-right 2x3 block copied into top-left 2x3 block:" << endl << a << endl << endl;
}
 
Here is the array a:
0.6 0.6 0.6 0.6
0.6 0.6 0.6 0.6
0.6 0.6 0.6 0.6
0.6 0.6 0.6 0.6

Here is now a with m copied into its central 2x2 block:
0.6 0.6 0.6 0.6
0.6   1   2 0.6
0.6   3   4 0.6
0.6 0.6 0.6 0.6

Here is now a with bottom-right 2x3 block copied into top-left 2x3 block:
  3   4 0.6 0.6
0.6 0.6 0.6 0.6
0.6   3   4 0.6
0.6 0.6 0.6 0.6

While the .block() method can be used for any block operation, there are other methods for special cases, providing more specialized API and/or better performance. On the topic of performance, all what matters is that you give Eigen as much information as possible at compile time. For example, if your block is a single whole column in a matrix, using the specialized .col() function described below lets Eigen know that, which can give it optimization opportunities.

The rest of this page describes these specialized methods.

Columns and rows

Individual columns and rows are special cases of blocks. Eigen provides methods to easily address them: .col() and .row().

Block operation Method
ith row * 
matrix.row(i);
jth column * 
matrix.col(j);

The argument for col() and row() is the index of the column or row to be accessed. As always in Eigen, indices start at 0.

Example: Output: 
#include <Eigen/Dense>
#include <iostream>
 
using namespace std;
 
int main()
{
  Eigen::MatrixXf m(3,3);
  m << 1,2,3,
       4,5,6,
       7,8,9;
  cout << "Here is the matrix m:" << endl << m << endl;
  cout << "2nd Row: " << m.row(1) << endl;
  m.col(2) += 3 * m.col(0);
  cout << "After adding 3 times the first column into the third column, the matrix m is:\n";
  cout << m << endl;
}

Here is the matrix m:
1 2 3
4 5 6
7 8 9
2nd Row: 4 5 6
After adding 3 times the first column into the third column, the matrix m is:
 1  2  6
 4  5 18
 7  8 30

That example also demonstrates that block expressions (here columns) can be used in arithmetic like any other expression.

Corner-related operations

Eigen also provides special methods for blocks that are flushed against one of the corners or sides of a matrix or array. For instance, .topLeftCorner() can be used to refer to a block in the top-left corner of a matrix.

The different possibilities are summarized in the following table:

Block operation Version constructing a
dynamic-size block expression 		Version constructing a
fixed-size block expression
Top-left p by q block * 
matrix.topLeftCorner(p,q);

matrix.topLeftCorner<p,q>();
Bottom-left p by q block * 
matrix.bottomLeftCorner(p,q);

matrix.bottomLeftCorner<p,q>();
Top-right p by q block * 
matrix.topRightCorner(p,q);

matrix.topRightCorner<p,q>();
Bottom-right p by q block * 
matrix.bottomRightCorner(p,q);

matrix.bottomRightCorner<p,q>();
Block containing the first q rows * 
matrix.topRows(q);

matrix.topRows<q>();
Block containing the last q rows * 
matrix.bottomRows(q);

matrix.bottomRows<q>();
Block containing the first p columns * 
matrix.leftCols(p);

matrix.leftCols<p>();
Block containing the last q columns * 
matrix.rightCols(q);

matrix.rightCols<q>();
Block containing the q columns starting from i * 
matrix.middleCols(i,q);

matrix.middleCols<q>(i);
Block containing the q rows starting from i * 
matrix.middleRows(i,q);

matrix.middleRows<q>(i);

Here is a simple example illustrating the use of the operations presented above:

Example: Output: 
#include <Eigen/Dense>
#include <iostream>
 
using namespace std;
 
int main()
{
  Eigen::Matrix4f m;
  m << 1, 2, 3, 4,
       5, 6, 7, 8,
       9, 10,11,12,
       13,14,15,16;
  cout << "m.leftCols(2) =" << endl << m.leftCols(2) << endl << endl;
  cout << "m.bottomRows<2>() =" << endl << m.bottomRows<2>() << endl << endl;
  m.topLeftCorner(1,3) = m.bottomRightCorner(3,1).transpose();
  cout << "After assignment, m = " << endl << m << endl;
}
 
m.leftCols(2) =
 1  2
 5  6
 9 10
13 14

m.bottomRows<2>() =
 9 10 11 12
13 14 15 16

After assignment, m = 
 8 12 16  4
 5  6  7  8
 9 10 11 12
13 14 15 16

Block operations for vectors

Eigen also provides a set of block operations designed specifically for the special case of vectors and one-dimensional arrays:

Block operation Version constructing a
dynamic-size block expression 		Version constructing a
fixed-size block expression
Block containing the first n elements * 
vector.head(n);

vector.head<n>();
Block containing the last n elements * 
vector.tail(n);

vector.tail<n>();
Block containing n elements, starting at position i * 
vector.segment(i,n);

vector.segment<n>(i);

An example is presented below:

Example: Output: 
#include <Eigen/Dense>
#include <iostream>
 
using namespace std;
 
int main()
{
  Eigen::ArrayXf v(6);
  v << 1, 2, 3, 4, 5, 6;
  cout << "v.head(3) =" << endl << v.head(3) << endl << endl;
  cout << "v.tail<3>() = " << endl << v.tail<3>() << endl << endl;
  v.segment(1,4) *= 2;
  cout << "after 'v.segment(1,4) *= 2', v =" << endl << v << endl;
}
 
v.head(3) =
1
2
3

v.tail<3>() = 
4
5
6

after 'v.segment(1,4) *= 2', v =
 1
 4
 6
 8
10
 6

© Eigen.
Licensed under the MPL2 License.
https://eigen.tuxfamily.org/dox/group__TutorialBlockOperations.html