1 namespace Eigen { 2 3 /** \eigenManualPage TutorialReductionsVisitorsBroadcasting Reductions, visitors and broadcasting 4 5 This page explains Eigen's reductions, visitors and broadcasting and how they are used with 6 \link MatrixBase matrices \endlink and \link ArrayBase arrays \endlink. 7 8 \eigenAutoToc 9 10 \section TutorialReductionsVisitorsBroadcastingReductions Reductions 11 In Eigen, a reduction is a function taking a matrix or array, and returning a single 12 scalar value. One of the most used reductions is \link DenseBase::sum() .sum() \endlink, 13 returning the sum of all the coefficients inside a given matrix or array. 14 15 <table class="example"> 16 <tr><th>Example:</th><th>Output:</th></tr> 17 <tr><td> 18 \include tut_arithmetic_redux_basic.cpp 19 </td> 20 <td> 21 \verbinclude tut_arithmetic_redux_basic.out 22 </td></tr></table> 23 24 The \em trace of a matrix, as returned by the function \c trace(), is the sum of the diagonal coefficients and can equivalently be computed <tt>a.diagonal().sum()</tt>. 25 26 27 \subsection TutorialReductionsVisitorsBroadcastingReductionsNorm Norm computations 28 29 The (Euclidean a.k.a. \f$\ell^2\f$) squared norm of a vector can be obtained \link MatrixBase::squaredNorm() squaredNorm() \endlink. It is equal to the dot product of the vector by itself, and equivalently to the sum of squared absolute values of its coefficients. 30 31 Eigen also provides the \link MatrixBase::norm() norm() \endlink method, which returns the square root of \link MatrixBase::squaredNorm() squaredNorm() \endlink. 32 33 These operations can also operate on matrices; in that case, a n-by-p matrix is seen as a vector of size (n*p), so for example the \link MatrixBase::norm() norm() \endlink method returns the "Frobenius" or "Hilbert-Schmidt" norm. We refrain from speaking of the \f$\ell^2\f$ norm of a matrix because that can mean different things. 34 35 If you want other coefficient-wise \f$\ell^p\f$ norms, use the \link MatrixBase::lpNorm lpNorm<p>() \endlink method. The template parameter \a p can take the special value \a Infinity if you want the \f$\ell^\infty\f$ norm, which is the maximum of the absolute values of the coefficients. 36 37 The following example demonstrates these methods. 38 39 <table class="example"> 40 <tr><th>Example:</th><th>Output:</th></tr> 41 <tr><td> 42 \include Tutorial_ReductionsVisitorsBroadcasting_reductions_norm.cpp 43 </td> 44 <td> 45 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_reductions_norm.out 46 </td></tr></table> 47 48 \b Operator \b norm: The 1-norm and \f$\infty\f$-norm <a href="https://en.wikipedia.org/wiki/Operator_norm">matrix operator norms</a> can easily be computed as follows: 49 <table class="example"> 50 <tr><th>Example:</th><th>Output:</th></tr> 51 <tr><td> 52 \include Tutorial_ReductionsVisitorsBroadcasting_reductions_operatornorm.cpp 53 </td> 54 <td> 55 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_reductions_operatornorm.out 56 </td></tr></table> 57 See below for more explanations on the syntax of these expressions. 58 59 \subsection TutorialReductionsVisitorsBroadcastingReductionsBool Boolean reductions 60 61 The following reductions operate on boolean values: 62 - \link DenseBase::all() all() \endlink returns \b true if all of the coefficients in a given Matrix or Array evaluate to \b true . 63 - \link DenseBase::any() any() \endlink returns \b true if at least one of the coefficients in a given Matrix or Array evaluates to \b true . 64 - \link DenseBase::count() count() \endlink returns the number of coefficients in a given Matrix or Array that evaluate to \b true. 65 66 These are typically used in conjunction with the coefficient-wise comparison and equality operators provided by Array. For instance, <tt>array > 0</tt> is an %Array of the same size as \c array , with \b true at those positions where the corresponding coefficient of \c array is positive. Thus, <tt>(array > 0).all()</tt> tests whether all coefficients of \c array are positive. This can be seen in the following example: 67 68 <table class="example"> 69 <tr><th>Example:</th><th>Output:</th></tr> 70 <tr><td> 71 \include Tutorial_ReductionsVisitorsBroadcasting_reductions_bool.cpp 72 </td> 73 <td> 74 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_reductions_bool.out 75 </td></tr></table> 76 77 \subsection TutorialReductionsVisitorsBroadcastingReductionsUserdefined User defined reductions 78 79 TODO 80 81 In the meantime you can have a look at the DenseBase::redux() function. 82 83 \section TutorialReductionsVisitorsBroadcastingVisitors Visitors 84 Visitors are useful when one wants to obtain the location of a coefficient inside 85 a Matrix or Array. The simplest examples are 86 \link MatrixBase::maxCoeff() maxCoeff(&x,&y) \endlink and 87 \link MatrixBase::minCoeff() minCoeff(&x,&y)\endlink, which can be used to find 88 the location of the greatest or smallest coefficient in a Matrix or 89 Array. 90 91 The arguments passed to a visitor are pointers to the variables where the 92 row and column position are to be stored. These variables should be of type 93 \link Eigen::Index Index \endlink, as shown below: 94 95 <table class="example"> 96 <tr><th>Example:</th><th>Output:</th></tr> 97 <tr><td> 98 \include Tutorial_ReductionsVisitorsBroadcasting_visitors.cpp 99 </td> 100 <td> 101 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_visitors.out 102 </td></tr></table> 103 104 Both functions also return the value of the minimum or maximum coefficient. 105 106 \section TutorialReductionsVisitorsBroadcastingPartialReductions Partial reductions 107 Partial reductions are reductions that can operate column- or row-wise on a Matrix or 108 Array, applying the reduction operation on each column or row and 109 returning a column or row vector with the corresponding values. Partial reductions are applied 110 with \link DenseBase::colwise() colwise() \endlink or \link DenseBase::rowwise() rowwise() \endlink. 111 112 A simple example is obtaining the maximum of the elements 113 in each column in a given matrix, storing the result in a row vector: 114 115 <table class="example"> 116 <tr><th>Example:</th><th>Output:</th></tr> 117 <tr><td> 118 \include Tutorial_ReductionsVisitorsBroadcasting_colwise.cpp 119 </td> 120 <td> 121 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_colwise.out 122 </td></tr></table> 123 124 The same operation can be performed row-wise: 125 126 <table class="example"> 127 <tr><th>Example:</th><th>Output:</th></tr> 128 <tr><td> 129 \include Tutorial_ReductionsVisitorsBroadcasting_rowwise.cpp 130 </td> 131 <td> 132 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_rowwise.out 133 </td></tr></table> 134 135 <b>Note that column-wise operations return a row vector, while row-wise operations return a column vector.</b> 136 137 \subsection TutorialReductionsVisitorsBroadcastingPartialReductionsCombined Combining partial reductions with other operations 138 It is also possible to use the result of a partial reduction to do further processing. 139 Here is another example that finds the column whose sum of elements is the maximum 140 within a matrix. With column-wise partial reductions this can be coded as: 141 142 <table class="example"> 143 <tr><th>Example:</th><th>Output:</th></tr> 144 <tr><td> 145 \include Tutorial_ReductionsVisitorsBroadcasting_maxnorm.cpp 146 </td> 147 <td> 148 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_maxnorm.out 149 </td></tr></table> 150 151 The previous example applies the \link DenseBase::sum() sum() \endlink reduction on each column 152 though the \link DenseBase::colwise() colwise() \endlink visitor, obtaining a new matrix whose 153 size is 1x4. 154 155 Therefore, if 156 \f[ 157 \mbox{m} = \begin{bmatrix} 1 & 2 & 6 & 9 \\ 158 3 & 1 & 7 & 2 \end{bmatrix} 159 \f] 160 161 then 162 163 \f[ 164 \mbox{m.colwise().sum()} = \begin{bmatrix} 4 & 3 & 13 & 11 \end{bmatrix} 165 \f] 166 167 The \link DenseBase::maxCoeff() maxCoeff() \endlink reduction is finally applied 168 to obtain the column index where the maximum sum is found, 169 which is the column index 2 (third column) in this case. 170 171 172 \section TutorialReductionsVisitorsBroadcastingBroadcasting Broadcasting 173 The concept behind broadcasting is similar to partial reductions, with the difference that broadcasting 174 constructs an expression where a vector (column or row) is interpreted as a matrix by replicating it in 175 one direction. 176 177 A simple example is to add a certain column vector to each column in a matrix. 178 This can be accomplished with: 179 180 <table class="example"> 181 <tr><th>Example:</th><th>Output:</th></tr> 182 <tr><td> 183 \include Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple.cpp 184 </td> 185 <td> 186 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple.out 187 </td></tr></table> 188 189 We can interpret the instruction <tt>mat.colwise() += v</tt> in two equivalent ways. It adds the vector \c v 190 to every column of the matrix. Alternatively, it can be interpreted as repeating the vector \c v four times to 191 form a four-by-two matrix which is then added to \c mat: 192 \f[ 193 \begin{bmatrix} 1 & 2 & 6 & 9 \\ 3 & 1 & 7 & 2 \end{bmatrix} 194 + \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix} 195 = \begin{bmatrix} 1 & 2 & 6 & 9 \\ 4 & 2 & 8 & 3 \end{bmatrix}. 196 \f] 197 The operators <tt>-=</tt>, <tt>+</tt> and <tt>-</tt> can also be used column-wise and row-wise. On arrays, we 198 can also use the operators <tt>*=</tt>, <tt>/=</tt>, <tt>*</tt> and <tt>/</tt> to perform coefficient-wise 199 multiplication and division column-wise or row-wise. These operators are not available on matrices because it 200 is not clear what they would do. If you want multiply column 0 of a matrix \c mat with \c v(0), column 1 with 201 \c v(1), and so on, then use <tt>mat = mat * v.asDiagonal()</tt>. 202 203 It is important to point out that the vector to be added column-wise or row-wise must be of type Vector, 204 and cannot be a Matrix. If this is not met then you will get compile-time error. This also means that 205 broadcasting operations can only be applied with an object of type Vector, when operating with Matrix. 206 The same applies for the Array class, where the equivalent for VectorXf is ArrayXf. As always, you should 207 not mix arrays and matrices in the same expression. 208 209 To perform the same operation row-wise we can do: 210 211 <table class="example"> 212 <tr><th>Example:</th><th>Output:</th></tr> 213 <tr><td> 214 \include Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple_rowwise.cpp 215 </td> 216 <td> 217 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple_rowwise.out 218 </td></tr></table> 219 220 \subsection TutorialReductionsVisitorsBroadcastingBroadcastingCombined Combining broadcasting with other operations 221 Broadcasting can also be combined with other operations, such as Matrix or Array operations, 222 reductions and partial reductions. 223 224 Now that broadcasting, reductions and partial reductions have been introduced, we can dive into a more advanced example that finds 225 the nearest neighbour of a vector <tt>v</tt> within the columns of matrix <tt>m</tt>. The Euclidean distance will be used in this example, 226 computing the squared Euclidean distance with the partial reduction named \link MatrixBase::squaredNorm() squaredNorm() \endlink: 227 228 <table class="example"> 229 <tr><th>Example:</th><th>Output:</th></tr> 230 <tr><td> 231 \include Tutorial_ReductionsVisitorsBroadcasting_broadcast_1nn.cpp 232 </td> 233 <td> 234 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_broadcast_1nn.out 235 </td></tr></table> 236 237 The line that does the job is 238 \code 239 (m.colwise() - v).colwise().squaredNorm().minCoeff(&index); 240 \endcode 241 242 We will go step by step to understand what is happening: 243 244 - <tt>m.colwise() - v</tt> is a broadcasting operation, subtracting <tt>v</tt> from each column in <tt>m</tt>. The result of this operation 245 is a new matrix whose size is the same as matrix <tt>m</tt>: \f[ 246 \mbox{m.colwise() - v} = 247 \begin{bmatrix} 248 -1 & 21 & 4 & 7 \\ 249 0 & 8 & 4 & -1 250 \end{bmatrix} 251 \f] 252 253 - <tt>(m.colwise() - v).colwise().squaredNorm()</tt> is a partial reduction, computing the squared norm column-wise. The result of 254 this operation is a row vector where each coefficient is the squared Euclidean distance between each column in <tt>m</tt> and <tt>v</tt>: \f[ 255 \mbox{(m.colwise() - v).colwise().squaredNorm()} = 256 \begin{bmatrix} 257 1 & 505 & 32 & 50 258 \end{bmatrix} 259 \f] 260 261 - Finally, <tt>minCoeff(&index)</tt> is used to obtain the index of the column in <tt>m</tt> that is closest to <tt>v</tt> in terms of Euclidean 262 distance. 263 264 */ 265 266 } 267
