fuchsia / third_party / eigen / 64baccc877717d32db291a400c2d5726402fdeb9 / . / doc / TutorialReductionsVisitorsBroadcasting.dox

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. | |

*/ | |

} |