nest-open-source / nest-cam / v366 / eigen / 9d75fdbcd2b5a2fdab03a859ddefda95ce1217ed / . / doc / TutorialArrayClass.dox

namespace Eigen { | |

/** \eigenManualPage TutorialArrayClass The Array class and coefficient-wise operations | |

This page aims to provide an overview and explanations on how to use | |

Eigen's Array class. | |

\eigenAutoToc | |

\section TutorialArrayClassIntro What is the Array class? | |

The Array class provides general-purpose arrays, as opposed to the Matrix class which | |

is intended for linear algebra. Furthermore, the Array class provides an easy way to | |

perform coefficient-wise operations, which might not have a linear algebraic meaning, | |

such as adding a constant to every coefficient in the array or multiplying two arrays coefficient-wise. | |

\section TutorialArrayClassTypes Array types | |

Array is a class template taking the same template parameters as Matrix. | |

As with Matrix, the first three template parameters are mandatory: | |

\code | |

Array<typename Scalar, int RowsAtCompileTime, int ColsAtCompileTime> | |

\endcode | |

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 \ref TutorialMatrixClass. | |

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 1-dimensional and 2-dimensional arrays. | |

We adopt the convention that typedefs of the form ArrayNt stand for 1-dimensional arrays, where N and t are | |

the size and the scalar type, as in the Matrix typedefs explained on \ref TutorialMatrixClass "this page". For 2-dimensional arrays, we | |

use typedefs of the form ArrayNNt. Some examples are shown in the following table: | |

<table class="manual"> | |

<tr> | |

<th>Type </th> | |

<th>Typedef </th> | |

</tr> | |

<tr> | |

<td> \code Array<float,Dynamic,1> \endcode </td> | |

<td> \code ArrayXf \endcode </td> | |

</tr> | |

<tr> | |

<td> \code Array<float,3,1> \endcode </td> | |

<td> \code Array3f \endcode </td> | |

</tr> | |

<tr> | |

<td> \code Array<double,Dynamic,Dynamic> \endcode </td> | |

<td> \code ArrayXXd \endcode </td> | |

</tr> | |

<tr> | |

<td> \code Array<double,3,3> \endcode </td> | |

<td> \code Array33d \endcode </td> | |

</tr> | |

</table> | |

\section TutorialArrayClassAccess Accessing values inside an Array | |

The parenthesis operator is overloaded to provide write and read access to the coefficients of an array, just as with matrices. | |

Furthermore, the \c << operator can be used to initialize arrays (via the comma initializer) or to print them. | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr><td> | |

\include Tutorial_ArrayClass_accessors.cpp | |

</td> | |

<td> | |

\verbinclude Tutorial_ArrayClass_accessors.out | |

</td></tr></table> | |

For more information about the comma initializer, see \ref TutorialAdvancedInitialization. | |

\section TutorialArrayClassAddSub Addition and subtraction | |

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 coefficient-wise. | |

Arrays also support expressions of the form <tt>array + scalar</tt> which add a scalar to each coefficient in the array. | |

This provides a functionality that is not directly available for Matrix objects. | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr><td> | |

\include Tutorial_ArrayClass_addition.cpp | |

</td> | |

<td> | |

\verbinclude Tutorial_ArrayClass_addition.out | |

</td></tr></table> | |

\section TutorialArrayClassMult Array multiplication | |

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 coefficient-wise product. Thus, two | |

arrays can be multiplied if and only if they have the same dimensions. | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr><td> | |

\include Tutorial_ArrayClass_mult.cpp | |

</td> | |

<td> | |

\verbinclude Tutorial_ArrayClass_mult.out | |

</td></tr></table> | |

\section TutorialArrayClassCwiseOther Other coefficient-wise operations | |

The Array class defines other coefficient-wise operations besides the addition, subtraction and multiplication | |

operators described above. For example, the \link ArrayBase::abs() .abs() \endlink method takes the absolute | |

value of each coefficient, while \link ArrayBase::sqrt() .sqrt() \endlink computes the square root of the | |

coefficients. If you have two arrays of the same size, you can call \link ArrayBase::min(const Eigen::ArrayBase<OtherDerived>&) const .min(.) \endlink 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. | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr><td> | |

\include Tutorial_ArrayClass_cwise_other.cpp | |

</td> | |

<td> | |

\verbinclude Tutorial_ArrayClass_cwise_other.out | |

</td></tr></table> | |

More coefficient-wise operations can be found in the \ref QuickRefPage. | |

\section TutorialArrayClassConvert Converting between array and matrix expressions | |

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 coefficient-wise | |

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. | |

\link MatrixBase Matrix expressions \endlink have an \link MatrixBase::array() .array() \endlink method that | |

'converts' them into \link ArrayBase array expressions\endlink, so that coefficient-wise operations | |

can be applied easily. Conversely, \link ArrayBase array expressions \endlink | |

have a \link ArrayBase::matrix() .matrix() \endlink method. As with all Eigen expression abstractions, | |

this doesn't have any runtime cost (provided that you let your compiler optimize). | |

Both \link MatrixBase::array() .array() \endlink and \link ArrayBase::matrix() .matrix() \endlink | |

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 \c + operator should either both be matrices or both be arrays. However, | |

it is easy to convert from one to the other with \link MatrixBase::array() .array() \endlink and | |

\link ArrayBase::matrix() .matrix()\endlink. 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 | |

\link MatrixBase::array() .array() \endlink method. For example, the statement | |

<tt>result = m.array() * n.array()</tt> takes two matrices \c m and \c n, converts them both to an array, uses | |

* to multiply them coefficient-wise and assigns the result to the matrix variable \c 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 \link MatrixBase::cwiseProduct const | |

.cwiseProduct(.) \endlink method for matrices to compute the coefficient-wise product. This is also shown in | |

the example program. | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr><td> | |

\include Tutorial_ArrayClass_interop_matrix.cpp | |

</td> | |

<td> | |

\verbinclude Tutorial_ArrayClass_interop_matrix.out | |

</td></tr></table> | |

Similarly, if \c array1 and \c array2 are arrays, then the expression <tt>array1.matrix() * array2.matrix()</tt> | |

computes their matrix product. | |

Here is a more advanced example. The expression <tt>(m.array() + 4).matrix() * m</tt> adds 4 to every | |

coefficient in the matrix \c m and then computes the matrix product of the result with \c m. Similarly, the | |

expression <tt>(m.array() * n.array()).matrix() * m</tt> computes the coefficient-wise product of the matrices | |

\c m and \c n and then the matrix product of the result with \c m. | |

<table class="example"> | |

<tr><th>Example:</th><th>Output:</th></tr> | |

<tr><td> | |

\include Tutorial_ArrayClass_interop.cpp | |

</td> | |

<td> | |

\verbinclude Tutorial_ArrayClass_interop.out | |

</td></tr></table> | |

*/ | |

} |