+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>

+

+*/

+

+}