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-namespace Eigen {
-
-/** \eigenManualPage TutorialReductionsVisitorsBroadcasting Reductions, visitors and broadcasting
-
-This page explains Eigen's reductions, visitors and broadcasting and how they are used with
-\link MatrixBase matrices \endlink and \link ArrayBase arrays \endlink.
-
-\eigenAutoToc
-
-\section TutorialReductionsVisitorsBroadcastingReductions Reductions
-In Eigen, a reduction is a function taking a matrix or array, and returning a single
-scalar value. One of the most used reductions is \link DenseBase::sum() .sum() \endlink,
-returning the sum of all the coefficients inside a given matrix or array.
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include tut_arithmetic_redux_basic.cpp
-</td>
-<td>
-\verbinclude tut_arithmetic_redux_basic.out
-</td></tr></table>
-
-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>.
-
-
-\subsection TutorialReductionsVisitorsBroadcastingReductionsNorm Norm computations
-
-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.
-
-Eigen also provides the \link MatrixBase::norm() norm() \endlink method, which returns the square root of \link MatrixBase::squaredNorm() squaredNorm() \endlink.
-
-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 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.
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include Tutorial_ReductionsVisitorsBroadcasting_reductions_norm.cpp
-</td>
-<td>
-\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:
-  - \link DenseBase::all() all() \endlink returns \b true if all of the coefficients in a given Matrix or Array evaluate to \b true .
-  - \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 .
-  - \link DenseBase::count() count() \endlink returns the number of coefficients in a given Matrix or Array that evaluate to  \b true.
-
-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:
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include Tutorial_ReductionsVisitorsBroadcasting_reductions_bool.cpp
-</td>
-<td>
-\verbinclude Tutorial_ReductionsVisitorsBroadcasting_reductions_bool.out
-</td></tr></table>
-
-\subsection TutorialReductionsVisitorsBroadcastingReductionsUserdefined User defined reductions
-
-TODO
-
-In the meantime you can have a look at the DenseBase::redux() function.
-
-\section TutorialReductionsVisitorsBroadcastingVisitors Visitors
-Visitors are useful when one wants to obtain the location of a coefficient inside 
-a Matrix or Array. The simplest examples are 
-\link MatrixBase::maxCoeff() maxCoeff(&x,&y) \endlink and 
-\link MatrixBase::minCoeff() minCoeff(&x,&y)\endlink, which can be used to find
-the location of the greatest or smallest coefficient in a Matrix or 
-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 Eigen::Index Index \endlink, as shown below:
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include Tutorial_ReductionsVisitorsBroadcasting_visitors.cpp
-</td>
-<td>
-\verbinclude Tutorial_ReductionsVisitorsBroadcasting_visitors.out
-</td></tr></table>
-
-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 
-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:
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include Tutorial_ReductionsVisitorsBroadcasting_colwise.cpp
-</td>
-<td>
-\verbinclude Tutorial_ReductionsVisitorsBroadcasting_colwise.out
-</td></tr></table>
-
-The same operation can be performed row-wise:
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include Tutorial_ReductionsVisitorsBroadcasting_rowwise.cpp
-</td>
-<td>
-\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>
-
-\subsection TutorialReductionsVisitorsBroadcastingPartialReductionsCombined Combining partial reductions with other operations
-It is also possible to use the result of a partial reduction to do further processing.
-Here is another example that finds the column whose sum of elements is the maximum
- within a matrix. With column-wise partial reductions this can be coded as:
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include Tutorial_ReductionsVisitorsBroadcasting_maxnorm.cpp
-</td>
-<td>
-\verbinclude Tutorial_ReductionsVisitorsBroadcasting_maxnorm.out
-</td></tr></table>
-
-The previous example applies the \link DenseBase::sum() sum() \endlink reduction on each column
-though the \link DenseBase::colwise() colwise() \endlink visitor, obtaining a new matrix whose
-size is 1x4.
-
-Therefore, if
-\f[
-\mbox{m} = \begin{bmatrix} 1 & 2 & 6 & 9 \\
-                    3 & 1 & 7 & 2 \end{bmatrix}
-\f]
-
-then
-
-\f[
-\mbox{m.colwise().sum()} = \begin{bmatrix} 4 & 3 & 13 & 11 \end{bmatrix}
-\f]
-
-The \link DenseBase::maxCoeff() maxCoeff() \endlink reduction is finally applied 
-to obtain the column index where the maximum sum is found, 
-which is the column index 2 (third column) in this case.
-
-
-\section TutorialReductionsVisitorsBroadcastingBroadcasting Broadcasting
-The concept behind broadcasting is similar to partial reductions, with the difference that broadcasting 
-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. 
-This can be accomplished with:
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple.cpp
-</td>
-<td>
-\verbinclude Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple.out
-</td></tr></table>
-
-We can interpret the instruction <tt>mat.colwise() += v</tt> in two equivalent ways. It adds the vector \c v
-to every column of the matrix. Alternatively, it can be interpreted as repeating the vector \c v four times to
-form a four-by-two matrix which is then added to \c mat:
-\f[
-\begin{bmatrix} 1 & 2 & 6 & 9 \\ 3 & 1 & 7 & 2 \end{bmatrix}
-+ \begin{bmatrix} 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 \end{bmatrix}
-= \begin{bmatrix} 1 & 2 & 6 & 9 \\ 4 & 2 & 8 & 3 \end{bmatrix}.
-\f]
-The operators <tt>-=</tt>, <tt>+</tt> and <tt>-</tt> can also be used column-wise and row-wise. On arrays, we 
-can also use the operators <tt>*=</tt>, <tt>/=</tt>, <tt>*</tt> and <tt>/</tt> to perform coefficient-wise 
-multiplication and division column-wise or row-wise. These operators are not available on matrices because it
-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 
-\c v(1), and so on, then use <tt>mat = mat * v.asDiagonal()</tt>.
-
-It is important to point out that the vector to be added column-wise or row-wise must be of type Vector,
-and cannot be a Matrix. If this is not met then you will get compile-time error. This also means that
-broadcasting operations can only be applied with an object of type Vector, when operating with Matrix.
-The same applies for the Array class, where the equivalent for VectorXf is ArrayXf. As always, you should
-not mix arrays and matrices in the same expression.
-
-To perform the same operation row-wise we can do:
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple_rowwise.cpp
-</td>
-<td>
-\verbinclude Tutorial_ReductionsVisitorsBroadcasting_broadcast_simple_rowwise.out
-</td></tr></table>
-
-\subsection TutorialReductionsVisitorsBroadcastingBroadcastingCombined Combining broadcasting with other operations
-Broadcasting can also be combined with other operations, such as Matrix or Array operations, 
-reductions and partial reductions.
-
-Now that broadcasting, reductions and partial reductions have been introduced, we can dive into a more advanced example that finds
-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,
-computing the squared Euclidean distance with the partial reduction named \link MatrixBase::squaredNorm() squaredNorm() \endlink:
-
-<table class="example">
-<tr><th>Example:</th><th>Output:</th></tr>
-<tr><td>
-\include Tutorial_ReductionsVisitorsBroadcasting_broadcast_1nn.cpp
-</td>
-<td>
-\verbinclude Tutorial_ReductionsVisitorsBroadcasting_broadcast_1nn.out
-</td></tr></table>
-
-The line that does the job is 
-\code
-  (m.colwise() - v).colwise().squaredNorm().minCoeff(&index);
-\endcode
-
-We will go step by step to understand what is happening:
-
-  - <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
-is a new matrix whose size is the same as matrix <tt>m</tt>: \f[
-  \mbox{m.colwise() - v} = 
-  \begin{bmatrix}
-    -1 & 21 & 4 & 7 \\
-     0 & 8  & 4 & -1
-  \end{bmatrix}
-\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[
-  \mbox{(m.colwise() - v).colwise().squaredNorm()} =
-  \begin{bmatrix}
-     1 & 505 & 32 & 50
-  \end{bmatrix}
-\f]
-
-  - 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
-distance.
-
-*/
-
-}