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Diffstat (limited to 'doc/TutorialReductionsVisitorsBroadcasting.dox')
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1 files changed, 19 insertions, 10 deletions
diff --git a/doc/TutorialReductionsVisitorsBroadcasting.dox b/doc/TutorialReductionsVisitorsBroadcasting.dox index 992cf6f34..f5322b4a6 100644 --- a/doc/TutorialReductionsVisitorsBroadcasting.dox +++ b/doc/TutorialReductionsVisitorsBroadcasting.dox @@ -32,7 +32,7 @@ Eigen also provides the \link MatrixBase::norm() norm() \endlink method, which r 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. -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. +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. The following example demonstrates these methods. @@ -45,6 +45,17 @@ The following example demonstrates these methods. \verbinclude Tutorial_ReductionsVisitorsBroadcasting_reductions_norm.out </td></tr></table> +\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: +<table class="example"> +<tr><th>Example:</th><th>Output:</th></tr> +<tr><td> +\include Tutorial_ReductionsVisitorsBroadcasting_reductions_operatornorm.cpp +</td> +<td> +\verbinclude Tutorial_ReductionsVisitorsBroadcasting_reductions_operatornorm.out +</td></tr></table> +See below for more explanations on the syntax of these expressions. + \subsection TutorialReductionsVisitorsBroadcastingReductionsBool Boolean reductions The following reductions operate on boolean values: @@ -79,7 +90,7 @@ Array. The arguments passed to a visitor are pointers to the variables where the row and column position are to be stored. These variables should be of type -\link DenseBase::Index Index \endlink, as shown below: +\link Eigen::Index Index \endlink, as shown below: <table class="example"> <tr><th>Example:</th><th>Output:</th></tr> @@ -90,17 +101,16 @@ row and column position are to be stored. These variables should be of type \verbinclude Tutorial_ReductionsVisitorsBroadcasting_visitors.out </td></tr></table> -Note that both functions also return the value of the minimum or maximum coefficient if needed, -as if it was a typical reduction operation. +Both functions also return the value of the minimum or maximum coefficient. \section TutorialReductionsVisitorsBroadcastingPartialReductions Partial reductions Partial reductions are reductions that can operate column- or row-wise on a Matrix or Array, applying the reduction operation on each column or row and -returning a column or row-vector with the corresponding values. Partial reductions are applied +returning a column or row vector with the corresponding values. Partial reductions are applied with \link DenseBase::colwise() colwise() \endlink or \link DenseBase::rowwise() rowwise() \endlink. A simple example is obtaining the maximum of the elements -in each column in a given matrix, storing the result in a row-vector: +in each column in a given matrix, storing the result in a row vector: <table class="example"> <tr><th>Example:</th><th>Output:</th></tr> @@ -122,8 +132,7 @@ The same operation can be performed row-wise: \verbinclude Tutorial_ReductionsVisitorsBroadcasting_rowwise.out </td></tr></table> -<b>Note that column-wise operations return a 'row-vector' while row-wise operations -return a 'column-vector'</b> +<b>Note that column-wise operations return a row vector, while row-wise operations return a column vector.</b> \subsection TutorialReductionsVisitorsBroadcastingPartialReductionsCombined Combining partial reductions with other operations It is also possible to use the result of a partial reduction to do further processing. @@ -165,7 +174,7 @@ The concept behind broadcasting is similar to partial reductions, with the diffe constructs an expression where a vector (column or row) is interpreted as a matrix by replicating it in one direction. -A simple example is to add a certain column-vector to each column in a matrix. +A simple example is to add a certain column vector to each column in a matrix. This can be accomplished with: <table class="example"> @@ -242,7 +251,7 @@ is a new matrix whose size is the same as matrix <tt>m</tt>: \f[ \f] - <tt>(m.colwise() - v).colwise().squaredNorm()</tt> is a partial reduction, computing the squared norm column-wise. The result of -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[ +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[ \mbox{(m.colwise() - v).colwise().squaredNorm()} = \begin{bmatrix} 1 & 505 & 32 & 50 |