Squashed 'third_party/eigen/' changes from 61d72f6..cf794d3
Change-Id: I9b814151b01f49af6337a8605d0c42a3a1ed4c72
git-subtree-dir: third_party/eigen
git-subtree-split: cf794d3b741a6278df169e58461f8529f43bce5d
diff --git a/doc/TutorialReductionsVisitorsBroadcasting.dox b/doc/TutorialReductionsVisitorsBroadcasting.dox
index 992cf6f..f5322b4 100644
--- a/doc/TutorialReductionsVisitorsBroadcasting.dox
+++ b/doc/TutorialReductionsVisitorsBroadcasting.dox
@@ -32,7 +32,7 @@
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 @@
\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 @@
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 @@
\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 @@
\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 @@
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 @@
\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