1 namespace Eigen { 2 3 /** \eigenManualPage TutorialArrayClass The Array class and coefficient-wise operations 4 5 This page aims to provide an overview and explanations on how to use 6 Eigen's Array class. 7 8 \eigenAutoToc 9 10 \section TutorialArrayClassIntro What is the Array class? 11 12 The Array class provides general-purpose arrays, as opposed to the Matrix class which 13 is intended for linear algebra. Furthermore, the Array class provides an easy way to 14 perform coefficient-wise operations, which might not have a linear algebraic meaning, 15 such as adding a constant to every coefficient in the array or multiplying two arrays coefficient-wise. 16 17 18 \section TutorialArrayClassTypes Array types 19 Array is a class template taking the same template parameters as Matrix. 20 As with Matrix, the first three template parameters are mandatory: 21 \code 22 Array<typename Scalar, int RowsAtCompileTime, int ColsAtCompileTime> 23 \endcode 24 The last three template parameters are optional. Since this is exactly the same as for Matrix, 25 we won't explain it again here and just refer to \ref TutorialMatrixClass. 26 27 Eigen also provides typedefs for some common cases, in a way that is similar to the Matrix typedefs 28 but with some slight differences, as the word "array" is used for both 1-dimensional and 2-dimensional arrays. 29 We adopt the convention that typedefs of the form ArrayNt stand for 1-dimensional arrays, where N and t are 30 the size and the scalar type, as in the Matrix typedefs explained on \ref TutorialMatrixClass "this page". For 2-dimensional arrays, we 31 use typedefs of the form ArrayNNt. Some examples are shown in the following table: 32 33 <table class="manual"> 34 <tr> 35 <th>Type </th> 36 <th>Typedef </th> 37 </tr> 38 <tr> 39 <td> \code Array<float,Dynamic,1> \endcode </td> 40 <td> \code ArrayXf \endcode </td> 41 </tr> 42 <tr> 43 <td> \code Array<float,3,1> \endcode </td> 44 <td> \code Array3f \endcode </td> 45 </tr> 46 <tr> 47 <td> \code Array<double,Dynamic,Dynamic> \endcode </td> 48 <td> \code ArrayXXd \endcode </td> 49 </tr> 50 <tr> 51 <td> \code Array<double,3,3> \endcode </td> 52 <td> \code Array33d \endcode </td> 53 </tr> 54 </table> 55 56 57 \section TutorialArrayClassAccess Accessing values inside an Array 58 59 The parenthesis operator is overloaded to provide write and read access to the coefficients of an array, just as with matrices. 60 Furthermore, the \c << operator can be used to initialize arrays (via the comma initializer) or to print them. 61 62 <table class="example"> 63 <tr><th>Example:</th><th>Output:</th></tr> 64 <tr><td> 65 \include Tutorial_ArrayClass_accessors.cpp 66 </td> 67 <td> 68 \verbinclude Tutorial_ArrayClass_accessors.out 69 </td></tr></table> 70 71 For more information about the comma initializer, see \ref TutorialAdvancedInitialization. 72 73 74 \section TutorialArrayClassAddSub Addition and subtraction 75 76 Adding and subtracting two arrays is the same as for matrices. 77 The operation is valid if both arrays have the same size, and the addition or subtraction is done coefficient-wise. 78 79 Arrays also support expressions of the form <tt>array + scalar</tt> which add a scalar to each coefficient in the array. 80 This provides a functionality that is not directly available for Matrix objects. 81 82 <table class="example"> 83 <tr><th>Example:</th><th>Output:</th></tr> 84 <tr><td> 85 \include Tutorial_ArrayClass_addition.cpp 86 </td> 87 <td> 88 \verbinclude Tutorial_ArrayClass_addition.out 89 </td></tr></table> 90 91 92 \section TutorialArrayClassMult Array multiplication 93 94 First of all, of course you can multiply an array by a scalar, this works in the same way as matrices. Where arrays 95 are fundamentally different from matrices, is when you multiply two together. Matrices interpret 96 multiplication as matrix product and arrays interpret multiplication as coefficient-wise product. Thus, two 97 arrays can be multiplied if and only if they have the same dimensions. 98 99 <table class="example"> 100 <tr><th>Example:</th><th>Output:</th></tr> 101 <tr><td> 102 \include Tutorial_ArrayClass_mult.cpp 103 </td> 104 <td> 105 \verbinclude Tutorial_ArrayClass_mult.out 106 </td></tr></table> 107 108 109 \section TutorialArrayClassCwiseOther Other coefficient-wise operations 110 111 The Array class defines other coefficient-wise operations besides the addition, subtraction and multiplication 112 operators described above. For example, the \link ArrayBase::abs() .abs() \endlink method takes the absolute 113 value of each coefficient, while \link ArrayBase::sqrt() .sqrt() \endlink computes the square root of the 114 coefficients. If you have two arrays of the same size, you can call \link ArrayBase::min(const Eigen::ArrayBase<OtherDerived>&) const .min(.) \endlink to 115 construct the array whose coefficients are the minimum of the corresponding coefficients of the two given 116 arrays. These operations are illustrated in the following example. 117 118 <table class="example"> 119 <tr><th>Example:</th><th>Output:</th></tr> 120 <tr><td> 121 \include Tutorial_ArrayClass_cwise_other.cpp 122 </td> 123 <td> 124 \verbinclude Tutorial_ArrayClass_cwise_other.out 125 </td></tr></table> 126 127 More coefficient-wise operations can be found in the \ref QuickRefPage. 128 129 130 \section TutorialArrayClassConvert Converting between array and matrix expressions 131 132 When should you use objects of the Matrix class and when should you use objects of the Array class? You cannot 133 apply Matrix operations on arrays, or Array operations on matrices. Thus, if you need to do linear algebraic 134 operations such as matrix multiplication, then you should use matrices; if you need to do coefficient-wise 135 operations, then you should use arrays. However, sometimes it is not that simple, but you need to use both 136 Matrix and Array operations. In that case, you need to convert a matrix to an array or reversely. This gives 137 access to all operations regardless of the choice of declaring objects as arrays or as matrices. 138 139 \link MatrixBase Matrix expressions \endlink have an \link MatrixBase::array() .array() \endlink method that 140 'converts' them into \link ArrayBase array expressions\endlink, so that coefficient-wise operations 141 can be applied easily. Conversely, \link ArrayBase array expressions \endlink 142 have a \link ArrayBase::matrix() .matrix() \endlink method. As with all Eigen expression abstractions, 143 this doesn't have any runtime cost (provided that you let your compiler optimize). 144 Both \link MatrixBase::array() .array() \endlink and \link ArrayBase::matrix() .matrix() \endlink 145 can be used as rvalues and as lvalues. 146 147 Mixing matrices and arrays in an expression is forbidden with Eigen. For instance, you cannot add a matrix and 148 array directly; the operands of a \c + operator should either both be matrices or both be arrays. However, 149 it is easy to convert from one to the other with \link MatrixBase::array() .array() \endlink and 150 \link ArrayBase::matrix() .matrix()\endlink. The exception to this rule is the assignment operator: it is 151 allowed to assign a matrix expression to an array variable, or to assign an array expression to a matrix 152 variable. 153 154 The following example shows how to use array operations on a Matrix object by employing the 155 \link MatrixBase::array() .array() \endlink method. For example, the statement 156 <tt>result = m.array() * n.array()</tt> takes two matrices \c m and \c n, converts them both to an array, uses 157 * to multiply them coefficient-wise and assigns the result to the matrix variable \c result (this is legal 158 because Eigen allows assigning array expressions to matrix variables). 159 160 As a matter of fact, this usage case is so common that Eigen provides a \link MatrixBase::cwiseProduct const 161 .cwiseProduct(.) \endlink method for matrices to compute the coefficient-wise product. This is also shown in 162 the example program. 163 164 <table class="example"> 165 <tr><th>Example:</th><th>Output:</th></tr> 166 <tr><td> 167 \include Tutorial_ArrayClass_interop_matrix.cpp 168 </td> 169 <td> 170 \verbinclude Tutorial_ArrayClass_interop_matrix.out 171 </td></tr></table> 172 173 Similarly, if \c array1 and \c array2 are arrays, then the expression <tt>array1.matrix() * array2.matrix()</tt> 174 computes their matrix product. 175 176 Here is a more advanced example. The expression <tt>(m.array() + 4).matrix() * m</tt> adds 4 to every 177 coefficient in the matrix \c m and then computes the matrix product of the result with \c m. Similarly, the 178 expression <tt>(m.array() * n.array()).matrix() * m</tt> computes the coefficient-wise product of the matrices 179 \c m and \c n and then the matrix product of the result with \c m. 180 181 <table class="example"> 182 <tr><th>Example:</th><th>Output:</th></tr> 183 <tr><td> 184 \include Tutorial_ArrayClass_interop.cpp 185 </td> 186 <td> 187 \verbinclude Tutorial_ArrayClass_interop.out 188 </td></tr></table> 189 190 */ 191 192 } 193
