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192 lines
8.3 KiB
192 lines
8.3 KiB
namespace Eigen {
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/** \eigenManualPage TutorialArrayClass The Array class and coefficient-wise operations
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This page aims to provide an overview and explanations on how to use
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Eigen's Array class.
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\eigenAutoToc
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\section TutorialArrayClassIntro What is the Array class?
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The Array class provides general-purpose arrays, as opposed to the Matrix class which
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is intended for linear algebra. Furthermore, the Array class provides an easy way to
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perform coefficient-wise operations, which might not have a linear algebraic meaning,
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such as adding a constant to every coefficient in the array or multiplying two arrays coefficient-wise.
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\section TutorialArrayClassTypes Array types
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Array is a class template taking the same template parameters as Matrix.
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As with Matrix, the first three template parameters are mandatory:
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\code
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Array<typename Scalar, int RowsAtCompileTime, int ColsAtCompileTime>
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\endcode
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The last three template parameters are optional. Since this is exactly the same as for Matrix,
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we won't explain it again here and just refer to \ref TutorialMatrixClass.
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Eigen also provides typedefs for some common cases, in a way that is similar to the Matrix typedefs
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but with some slight differences, as the word "array" is used for both 1-dimensional and 2-dimensional arrays.
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We adopt the convention that typedefs of the form ArrayNt stand for 1-dimensional arrays, where N and t are
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the size and the scalar type, as in the Matrix typedefs explained on \ref TutorialMatrixClass "this page". For 2-dimensional arrays, we
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use typedefs of the form ArrayNNt. Some examples are shown in the following table:
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<table class="manual">
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<tr>
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<th>Type </th>
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<th>Typedef </th>
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</tr>
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<tr>
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<td> \code Array<float,Dynamic,1> \endcode </td>
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<td> \code ArrayXf \endcode </td>
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</tr>
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<tr>
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<td> \code Array<float,3,1> \endcode </td>
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<td> \code Array3f \endcode </td>
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</tr>
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<tr>
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<td> \code Array<double,Dynamic,Dynamic> \endcode </td>
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<td> \code ArrayXXd \endcode </td>
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</tr>
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<tr>
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<td> \code Array<double,3,3> \endcode </td>
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<td> \code Array33d \endcode </td>
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</tr>
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</table>
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\section TutorialArrayClassAccess Accessing values inside an Array
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The parenthesis operator is overloaded to provide write and read access to the coefficients of an array, just as with matrices.
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Furthermore, the \c << operator can be used to initialize arrays (via the comma initializer) or to print them.
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr><td>
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\include Tutorial_ArrayClass_accessors.cpp
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</td>
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<td>
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\verbinclude Tutorial_ArrayClass_accessors.out
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</td></tr></table>
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For more information about the comma initializer, see \ref TutorialAdvancedInitialization.
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\section TutorialArrayClassAddSub Addition and subtraction
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Adding and subtracting two arrays is the same as for matrices.
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The operation is valid if both arrays have the same size, and the addition or subtraction is done coefficient-wise.
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Arrays also support expressions of the form <tt>array + scalar</tt> which add a scalar to each coefficient in the array.
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This provides a functionality that is not directly available for Matrix objects.
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr><td>
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\include Tutorial_ArrayClass_addition.cpp
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</td>
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<td>
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\verbinclude Tutorial_ArrayClass_addition.out
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</td></tr></table>
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\section TutorialArrayClassMult Array multiplication
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First of all, of course you can multiply an array by a scalar, this works in the same way as matrices. Where arrays
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are fundamentally different from matrices, is when you multiply two together. Matrices interpret
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multiplication as matrix product and arrays interpret multiplication as coefficient-wise product. Thus, two
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arrays can be multiplied if and only if they have the same dimensions.
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr><td>
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\include Tutorial_ArrayClass_mult.cpp
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</td>
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<td>
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\verbinclude Tutorial_ArrayClass_mult.out
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</td></tr></table>
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\section TutorialArrayClassCwiseOther Other coefficient-wise operations
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The Array class defines other coefficient-wise operations besides the addition, subtraction and multiplication
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operators described above. For example, the \link ArrayBase::abs() .abs() \endlink method takes the absolute
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value of each coefficient, while \link ArrayBase::sqrt() .sqrt() \endlink computes the square root of the
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coefficients. If you have two arrays of the same size, you can call \link ArrayBase::min(const Eigen::ArrayBase<OtherDerived>&) const .min(.) \endlink to
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construct the array whose coefficients are the minimum of the corresponding coefficients of the two given
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arrays. These operations are illustrated in the following example.
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr><td>
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\include Tutorial_ArrayClass_cwise_other.cpp
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</td>
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<td>
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\verbinclude Tutorial_ArrayClass_cwise_other.out
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</td></tr></table>
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More coefficient-wise operations can be found in the \ref QuickRefPage.
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\section TutorialArrayClassConvert Converting between array and matrix expressions
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When should you use objects of the Matrix class and when should you use objects of the Array class? You cannot
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apply Matrix operations on arrays, or Array operations on matrices. Thus, if you need to do linear algebraic
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operations such as matrix multiplication, then you should use matrices; if you need to do coefficient-wise
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operations, then you should use arrays. However, sometimes it is not that simple, but you need to use both
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Matrix and Array operations. In that case, you need to convert a matrix to an array or reversely. This gives
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access to all operations regardless of the choice of declaring objects as arrays or as matrices.
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\link MatrixBase Matrix expressions \endlink have an \link MatrixBase::array() .array() \endlink method that
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'converts' them into \link ArrayBase array expressions\endlink, so that coefficient-wise operations
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can be applied easily. Conversely, \link ArrayBase array expressions \endlink
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have a \link ArrayBase::matrix() .matrix() \endlink method. As with all Eigen expression abstractions,
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this doesn't have any runtime cost (provided that you let your compiler optimize).
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Both \link MatrixBase::array() .array() \endlink and \link ArrayBase::matrix() .matrix() \endlink
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can be used as rvalues and as lvalues.
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Mixing matrices and arrays in an expression is forbidden with Eigen. For instance, you cannot add a matrix and
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array directly; the operands of a \c + operator should either both be matrices or both be arrays. However,
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it is easy to convert from one to the other with \link MatrixBase::array() .array() \endlink and
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\link ArrayBase::matrix() .matrix()\endlink. The exception to this rule is the assignment operator: it is
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allowed to assign a matrix expression to an array variable, or to assign an array expression to a matrix
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variable.
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The following example shows how to use array operations on a Matrix object by employing the
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\link MatrixBase::array() .array() \endlink method. For example, the statement
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<tt>result = m.array() * n.array()</tt> takes two matrices \c m and \c n, converts them both to an array, uses
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* to multiply them coefficient-wise and assigns the result to the matrix variable \c result (this is legal
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because Eigen allows assigning array expressions to matrix variables).
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As a matter of fact, this usage case is so common that Eigen provides a \link MatrixBase::cwiseProduct() const
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.cwiseProduct(.) \endlink method for matrices to compute the coefficient-wise product. This is also shown in
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the example program.
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr><td>
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\include Tutorial_ArrayClass_interop_matrix.cpp
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</td>
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<td>
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\verbinclude Tutorial_ArrayClass_interop_matrix.out
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</td></tr></table>
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Similarly, if \c array1 and \c array2 are arrays, then the expression <tt>array1.matrix() * array2.matrix()</tt>
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computes their matrix product.
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Here is a more advanced example. The expression <tt>(m.array() + 4).matrix() * m</tt> adds 4 to every
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coefficient in the matrix \c m and then computes the matrix product of the result with \c m. Similarly, the
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expression <tt>(m.array() * n.array()).matrix() * m</tt> computes the coefficient-wise product of the matrices
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\c m and \c n and then the matrix product of the result with \c m.
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<table class="example">
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<tr><th>Example:</th><th>Output:</th></tr>
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<tr><td>
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\include Tutorial_ArrayClass_interop.cpp
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</td>
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<td>
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\verbinclude Tutorial_ArrayClass_interop.out
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</td></tr></table>
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*/
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}
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