Eigen
3.3.9

This page aims to provide an overview and explanations on how to use Eigen's Array class.
The Array class provides generalpurpose arrays, as opposed to the Matrix class which is intended for linear algebra. Furthermore, the Array class provides an easy way to perform coefficientwise operations, which might not have a linear algebraic meaning, such as adding a constant to every coefficient in the array or multiplying two arrays coefficientwise.
Array is a class template taking the same template parameters as Matrix. As with Matrix, the first three template parameters are mandatory:
The last three template parameters are optional. Since this is exactly the same as for Matrix, we won't explain it again here and just refer to The Matrix class.
Eigen also provides typedefs for some common cases, in a way that is similar to the Matrix typedefs but with some slight differences, as the word "array" is used for both 1dimensional and 2dimensional arrays. We adopt the convention that typedefs of the form ArrayNt stand for 1dimensional arrays, where N and t are the size and the scalar type, as in the Matrix typedefs explained on this page. For 2dimensional arrays, we use typedefs of the form ArrayNNt. Some examples are shown in the following table:
Type  Typedef 

Array<float,Dynamic,1>
 ArrayXf

Array<float,3,1>
 Array3f

Array<double,Dynamic,Dynamic>
 ArrayXXd

Array<double,3,3>
 Array33d

The parenthesis operator is overloaded to provide write and read access to the coefficients of an array, just as with matrices. Furthermore, the <<
operator can be used to initialize arrays (via the comma initializer) or to print them.
Example:  Output: 

#include <Eigen/Dense>
#include <iostream>
using namespace Eigen;
using namespace std;
int main()
{
ArrayXXf m(2,2);
// assign some values coefficient by coefficient
m(0,0) = 1.0; m(0,1) = 2.0;
m(1,0) = 3.0; m(1,1) = m(0,1) + m(1,0);
// print values to standard output
cout << m << endl << endl;
// using the commainitializer is also allowed
m << 1.0,2.0,
3.0,4.0;
// print values to standard output
cout << m << endl;
}
 1 2 3 5 1 2 3 4 
For more information about the comma initializer, see Advanced initialization.
Adding and subtracting two arrays is the same as for matrices. The operation is valid if both arrays have the same size, and the addition or subtraction is done coefficientwise.
Arrays also support expressions of the form array + scalar
which add a scalar to each coefficient in the array. This provides a functionality that is not directly available for Matrix objects.
Example:  Output: 

#include <Eigen/Dense>
#include <iostream>
using namespace Eigen;
using namespace std;
int main()
{
ArrayXXf a(3,3);
ArrayXXf b(3,3);
a << 1,2,3,
4,5,6,
7,8,9;
b << 1,2,3,
1,2,3,
1,2,3;
// Adding two arrays
cout << "a + b = " << endl << a + b << endl << endl;
// Subtracting a scalar from an array
cout << "a  2 = " << endl << a  2 << endl;
}
 a + b = 2 4 6 5 7 9 8 10 12 a  2 = 1 0 1 2 3 4 5 6 7 
First of all, of course you can multiply an array by a scalar, this works in the same way as matrices. Where arrays are fundamentally different from matrices, is when you multiply two together. Matrices interpret multiplication as matrix product and arrays interpret multiplication as coefficientwise product. Thus, two arrays can be multiplied if and only if they have the same dimensions.
Example:  Output: 

a * b = 5 12 21 32 
The Array class defines other coefficientwise operations besides the addition, subtraction and multiplication operators described above. For example, the .abs() method takes the absolute value of each coefficient, while .sqrt() computes the square root of the coefficients. If you have two arrays of the same size, you can call .min(.) to construct the array whose coefficients are the minimum of the corresponding coefficients of the two given arrays. These operations are illustrated in the following example.
Example:  Output: 

#include <Eigen/Dense>
#include <iostream>
using namespace Eigen;
using namespace std;
int main()
{
ArrayXf a = ArrayXf::Random(5);
a *= 2;
cout << "a =" << endl
<< a << endl;
cout << "a.abs() =" << endl
<< a.abs() << endl;
cout << "a.abs().sqrt() =" << endl
<< a.abs().sqrt() << endl;
cout << "a.min(a.abs().sqrt()) =" << endl
<< a.min(a.abs().sqrt()) << endl;
}
 a = 2 1.47 1.02 0.165 0.131 a.abs() = 2 1.47 1.02 0.165 0.131 a.abs().sqrt() = 1.41 1.21 1.01 0.407 0.362 a.min(a.abs().sqrt()) = 2 1.47 1.01 0.165 0.131 
More coefficientwise operations can be found in the Quick reference guide.
When should you use objects of the Matrix class and when should you use objects of the Array class? You cannot apply Matrix operations on arrays, or Array operations on matrices. Thus, if you need to do linear algebraic operations such as matrix multiplication, then you should use matrices; if you need to do coefficientwise operations, then you should use arrays. However, sometimes it is not that simple, but you need to use both Matrix and Array operations. In that case, you need to convert a matrix to an array or reversely. This gives access to all operations regardless of the choice of declaring objects as arrays or as matrices.
Matrix expressions have an .array() method that 'converts' them into array expressions, so that coefficientwise operations can be applied easily. Conversely, array expressions have a .matrix() method. As with all Eigen expression abstractions, this doesn't have any runtime cost (provided that you let your compiler optimize). Both .array() and .matrix() can be used as rvalues and as lvalues.
Mixing matrices and arrays in an expression is forbidden with Eigen. For instance, you cannot add a matrix and array directly; the operands of a +
operator should either both be matrices or both be arrays. However, it is easy to convert from one to the other with .array() and .matrix(). The exception to this rule is the assignment operator: it is allowed to assign a matrix expression to an array variable, or to assign an array expression to a matrix variable.
The following example shows how to use array operations on a Matrix object by employing the .array() method. For example, the statement result = m.array() * n.array()
takes two matrices m
and n
, converts them both to an array, uses to multiply them coefficientwise and assigns the result to the matrix variable result
(this is legal because Eigen allows assigning array expressions to matrix variables).
As a matter of fact, this usage case is so common that Eigen provides a const .cwiseProduct(.) method for matrices to compute the coefficientwise product. This is also shown in the example program.
Example:  Output: 

#include <Eigen/Dense>
#include <iostream>
using namespace Eigen;
using namespace std;
int main()
{
MatrixXf m(2,2);
MatrixXf n(2,2);
MatrixXf result(2,2);
m << 1,2,
3,4;
n << 5,6,
7,8;
result = m * n;
cout << " Matrix m*n: " << endl << result << endl << endl;
result = m.array() * n.array();
cout << " Array m*n: " << endl << result << endl << endl;
result = m.cwiseProduct(n);
cout << " With cwiseProduct: " << endl << result << endl << endl;
result = m.array() + 4;
cout << " Array m + 4: " << endl << result << endl << endl;
}
  Matrix m*n:  19 22 43 50  Array m*n:  5 12 21 32  With cwiseProduct:  5 12 21 32  Array m + 4:  5 6 7 8 
Similarly, if array1
and array2
are arrays, then the expression array1.matrix() * array2.matrix()
computes their matrix product.
Here is a more advanced example. The expression (m.array() + 4).matrix() * m
adds 4 to every coefficient in the matrix m
and then computes the matrix product of the result with m
. Similarly, the expression (m.array() * n.array()).matrix() * m
computes the coefficientwise product of the matrices m
and n
and then the matrix product of the result with m
.
Example:  Output: 

#include <Eigen/Dense>
#include <iostream>
using namespace Eigen;
using namespace std;
int main()
{
MatrixXf m(2,2);
MatrixXf n(2,2);
MatrixXf result(2,2);
m << 1,2,
3,4;
n << 5,6,
7,8;
result = (m.array() + 4).matrix() * m;
cout << " Combination 1: " << endl << result << endl << endl;
result = (m.array() * n.array()).matrix() * m;
cout << " Combination 2: " << endl << result << endl << endl;
}
  Combination 1:  23 34 31 46  Combination 2:  41 58 117 170 