1 namespace Eigen { 2 3 /** \page TutorialReductionsVisitorsBroadcasting Tutorial page 7 - Reductions, visitors and broadcasting 4 \ingroup Tutorial 5 6 \li \b Previous: \ref TutorialLinearAlgebra 7 \li \b Next: \ref TutorialGeometry 8 9 This tutorial explains Eigen's reductions, visitors and broadcasting and how they are used with 10 \link MatrixBase matrices \endlink and \link ArrayBase arrays \endlink. 11 12 \b Table \b of \b contents 13 - \ref TutorialReductionsVisitorsBroadcastingReductions 14 - \ref TutorialReductionsVisitorsBroadcastingReductionsNorm 15 - \ref TutorialReductionsVisitorsBroadcastingReductionsBool 16 - \ref TutorialReductionsVisitorsBroadcastingReductionsUserdefined 17 - \ref TutorialReductionsVisitorsBroadcastingVisitors 18 - \ref TutorialReductionsVisitorsBroadcastingPartialReductions 19 - \ref TutorialReductionsVisitorsBroadcastingPartialReductionsCombined 20 - \ref TutorialReductionsVisitorsBroadcastingBroadcasting 21 - \ref TutorialReductionsVisitorsBroadcastingBroadcastingCombined 22 23 24 \section TutorialReductionsVisitorsBroadcastingReductions Reductions 25 In Eigen, a reduction is a function taking a matrix or array, and returning a single 26 scalar value. One of the most used reductions is \link DenseBase::sum() .sum() \endlink, 27 returning the sum of all the coefficients inside a given matrix or array. 28 29 <table class="example"> 30 <tr><th>Example:</th><th>Output:</th></tr> 31 <tr><td> 32 \include tut_arithmetic_redux_basic.cpp 33 </td> 34 <td> 35 \verbinclude tut_arithmetic_redux_basic.out 36 </td></tr></table> 37 38 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>. 39 40 41 \subsection TutorialReductionsVisitorsBroadcastingReductionsNorm Norm computations 42 43 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. 44 45 Eigen also provides the \link MatrixBase::norm() norm() \endlink method, which returns the square root of \link MatrixBase::squaredNorm() squaredNorm() \endlink. 46 47 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. 48 49 If you want other \f$\ell^p\f$ norms, use the \link MatrixBase::lpNorm() lpNnorm<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. 50 51 The following example demonstrates these methods. 52 53 <table class="example"> 54 <tr><th>Example:</th><th>Output:</th></tr> 55 <tr><td> 56 \include Tutorial_ReductionsVisitorsBroadcasting_reductions_norm.cpp 57 </td> 58 <td> 59 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_reductions_norm.out 60 </td></tr></table> 61 62 \subsection TutorialReductionsVisitorsBroadcastingReductionsBool Boolean reductions 63 64 The following reductions operate on boolean values: 65 - \link DenseBase::all() all() \endlink returns \b true if all of the coefficients in a given Matrix or Array evaluate to \b true . 66 - \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 . 67 - \link DenseBase::count() count() \endlink returns the number of coefficients in a given Matrix or Array that evaluate to \b true. 68 69 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: 70 71 <table class="example"> 72 <tr><th>Example:</th><th>Output:</th></tr> 73 <tr><td> 74 \include Tutorial_ReductionsVisitorsBroadcasting_reductions_bool.cpp 75 </td> 76 <td> 77 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_reductions_bool.out 78 </td></tr></table> 79 80 \subsection TutorialReductionsVisitorsBroadcastingReductionsUserdefined User defined reductions 81 82 TODO 83 84 In the meantime you can have a look at the DenseBase::redux() function. 85 86 \section TutorialReductionsVisitorsBroadcastingVisitors Visitors 87 Visitors are useful when one wants to obtain the location of a coefficient inside 88 a Matrix or Array. The simplest examples are 89 \link MatrixBase::maxCoeff() maxCoeff(&x,&y) \endlink and 90 \link MatrixBase::minCoeff() minCoeff(&x,&y)\endlink, which can be used to find 91 the location of the greatest or smallest coefficient in a Matrix or 92 Array. 93 94 The arguments passed to a visitor are pointers to the variables where the 95 row and column position are to be stored. These variables should be of type 96 \link DenseBase::Index Index \endlink, as shown below: 97 98 <table class="example"> 99 <tr><th>Example:</th><th>Output:</th></tr> 100 <tr><td> 101 \include Tutorial_ReductionsVisitorsBroadcasting_visitors.cpp 102 </td> 103 <td> 104 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_visitors.out 105 </td></tr></table> 106 107 Note that both functions also return the value of the minimum or maximum coefficient if needed, 108 as if it was a typical reduction operation. 109 110 \section TutorialReductionsVisitorsBroadcastingPartialReductions Partial reductions 111 Partial reductions are reductions that can operate column- or row-wise on a Matrix or 112 Array, applying the reduction operation on each column or row and 113 returning a column or row-vector with the corresponding values. Partial reductions are applied 114 with \link DenseBase::colwise() colwise() \endlink or \link DenseBase::rowwise() rowwise() \endlink. 115 116 A simple example is obtaining the maximum of the elements 117 in each column in a given matrix, storing the result in a row-vector: 118 119 <table class="example"> 120 <tr><th>Example:</th><th>Output:</th></tr> 121 <tr><td> 122 \include Tutorial_ReductionsVisitorsBroadcasting_colwise.cpp 123 </td> 124 <td> 125 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_colwise.out 126 </td></tr></table> 127 128 The same operation can be performed row-wise: 129 130 <table class="example"> 131 <tr><th>Example:</th><th>Output:</th></tr> 132 <tr><td> 133 \include Tutorial_ReductionsVisitorsBroadcasting_rowwise.cpp 134 </td> 135 <td> 136 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_rowwise.out 137 </td></tr></table> 138 139 <b>Note that column-wise operations return a 'row-vector' while row-wise operations 140 return a 'column-vector'</b> 141 142 \subsection TutorialReductionsVisitorsBroadcastingPartialReductionsCombined Combining partial reductions with other operations 143 It is also possible to use the result of a partial reduction to do further processing. 144 Here is another example that finds the column whose sum of elements is the maximum 145 within a matrix. With column-wise partial reductions this can be coded as: 146 147 <table class="example"> 148 <tr><th>Example:</th><th>Output:</th></tr> 149 <tr><td> 150 \include Tutorial_ReductionsVisitorsBroadcasting_maxnorm.cpp 151 </td> 152 <td> 153 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_maxnorm.out 154 </td></tr></table> 155 156 The previous example applies the \link DenseBase::sum() sum() \endlink reduction on each column 157 though the \link DenseBase::colwise() colwise() \endlink visitor, obtaining a new matrix whose 158 size is 1x4. 159 160 Therefore, if 161 \f[ 162 \mbox{m} = \begin{bmatrix} 1 & 2 & 6 & 9 \\ 163 3 & 1 & 7 & 2 \end{bmatrix} 164 \f] 165 166 then 167 168 \f[ 169 \mbox{m.colwise().sum()} = \begin{bmatrix} 4 & 3 & 13 & 11 \end{bmatrix} 170 \f] 171 172 The \link DenseBase::maxCoeff() maxCoeff() \endlink reduction is finally applied 173 to obtain the column index where the maximum sum is found, 174 which is the column index 2 (third column) in this case. 175 176 177 \section TutorialReductionsVisitorsBroadcastingBroadcasting Broadcasting 178 The concept behind broadcasting is similar to partial reductions, with the difference that broadcasting 179 constructs an expression where a vector (column or row) is interpreted as a matrix by replicating it in 180 one direction. 181 182 A simple example is to add a certain column-vector to each column in a matrix. 183 This can be accomplished with: 184 185 <table class="example"> 186 <tr><th>Example:</th><th>Output:</th></tr> 187 <tr><td> 188 \include Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple.cpp 189 </td> 190 <td> 191 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple.out 192 </td></tr></table> 193 194 We can interpret the instruction <tt>mat.colwise() += v</tt> in two equivalent ways. It adds the vector \c v 195 to every column of the matrix. Alternatively, it can be interpreted as repeating the vector \c v four times to 196 form a four-by-two matrix which is then added to \c mat: 197 \f[ 198 \begin{bmatrix} 1 & 2 & 6 & 9 \\ 3 & 1 & 7 & 2 \end{bmatrix} 199 + \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix} 200 = \begin{bmatrix} 1 & 2 & 6 & 9 \\ 4 & 2 & 8 & 3 \end{bmatrix}. 201 \f] 202 The operators <tt>-=</tt>, <tt>+</tt> and <tt>-</tt> can also be used column-wise and row-wise. On arrays, we 203 can also use the operators <tt>*=</tt>, <tt>/=</tt>, <tt>*</tt> and <tt>/</tt> to perform coefficient-wise 204 multiplication and division column-wise or row-wise. These operators are not available on matrices because it 205 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 206 \c v(1), and so on, then use <tt>mat = mat * v.asDiagonal()</tt>. 207 208 It is important to point out that the vector to be added column-wise or row-wise must be of type Vector, 209 and cannot be a Matrix. If this is not met then you will get compile-time error. This also means that 210 broadcasting operations can only be applied with an object of type Vector, when operating with Matrix. 211 The same applies for the Array class, where the equivalent for VectorXf is ArrayXf. As always, you should 212 not mix arrays and matrices in the same expression. 213 214 To perform the same operation row-wise we can do: 215 216 <table class="example"> 217 <tr><th>Example:</th><th>Output:</th></tr> 218 <tr><td> 219 \include Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple_rowwise.cpp 220 </td> 221 <td> 222 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple_rowwise.out 223 </td></tr></table> 224 225 \subsection TutorialReductionsVisitorsBroadcastingBroadcastingCombined Combining broadcasting with other operations 226 Broadcasting can also be combined with other operations, such as Matrix or Array operations, 227 reductions and partial reductions. 228 229 Now that broadcasting, reductions and partial reductions have been introduced, we can dive into a more advanced example that finds 230 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, 231 computing the squared Euclidean distance with the partial reduction named \link MatrixBase::squaredNorm() squaredNorm() \endlink: 232 233 <table class="example"> 234 <tr><th>Example:</th><th>Output:</th></tr> 235 <tr><td> 236 \include Tutorial_ReductionsVisitorsBroadcasting_broadcast_1nn.cpp 237 </td> 238 <td> 239 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_broadcast_1nn.out 240 </td></tr></table> 241 242 The line that does the job is 243 \code 244 (m.colwise() - v).colwise().squaredNorm().minCoeff(&index); 245 \endcode 246 247 We will go step by step to understand what is happening: 248 249 - <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 250 is a new matrix whose size is the same as matrix <tt>m</tt>: \f[ 251 \mbox{m.colwise() - v} = 252 \begin{bmatrix} 253 -1 & 21 & 4 & 7 \\ 254 0 & 8 & 4 & -1 255 \end{bmatrix} 256 \f] 257 258 - <tt>(m.colwise() - v).colwise().squaredNorm()</tt> is a partial reduction, computing the squared norm column-wise. The result of 259 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[ 260 \mbox{(m.colwise() - v).colwise().squaredNorm()} = 261 \begin{bmatrix} 262 1 & 505 & 32 & 50 263 \end{bmatrix} 264 \f] 265 266 - 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 267 distance. 268 269 \li \b Next: \ref TutorialGeometry 270 271 */ 272 273 } 274
