improvements in pages 5 and 7 of the tutorial.

Eigen/src/Core/Dot.h

@@ -170,7 +170,7 @@ struct ei_lpNorm_selector<Derived, Infinity>
 };
 
 /** \returns the \f$ \ell^p \f$ norm of *this, that is, returns the p-th root of the sum of the p-th powers of the absolute values
-  *          of the coefficients of *this. If \a p is the special value \a Eigen::Infinity, this function returns the \f$ \ell^p\infty \f$
+  *          of the coefficients of *this. If \a p is the special value \a Eigen::Infinity, this function returns the \f$ \ell^\infty \f$
   *          norm, that is the maximum of the absolute values of the coefficients of *this.
   *
   * \sa norm()

doc/C05_TutorialAdvancedInitialization.dox

@@ -67,8 +67,8 @@ Example: \include Tutorial_AdvancedInitialization_Zero.cpp
 Output: \verbinclude Tutorial_AdvancedInitialization_Zero.out
 </td></tr></table>
 
-Similarly, the static method \link DenseBase::Constant() Constant\endlink(value) sets all coefficients to \c
-value. If the size of the object needs to be specified, the additional arguments go before the \c value
+Similarly, the static method \link DenseBase::Constant() Constant\endlink(value) sets all coefficients to \c value.
+If the size of the object needs to be specified, the additional arguments go before the \c value
 argument, as in <tt>MatrixXd::Constant(rows, cols, value)</tt>. The method \link DenseBase::Random() Random()
 \endlink fills the matrix or array with random coefficients. The identity matrix can be obtained by calling
 \link MatrixBase::Identity() Identity()\endlink; this method is only available for Matrix, not for Array,
@@ -102,13 +102,15 @@ Output: \verbinclude Tutorial_AdvancedInitialization_ThreeWays.out
 A summary of all pre-defined matrix, vector and array objects can be found in the \ref QuickRefPage.
 
 
-\section TutorialAdvancedInitializationTemporaryObjects Temporary matrices and arrays
+\section TutorialAdvancedInitializationTemporaryObjects Usage as temporary objects
 
-As shown above, static methods as Zero() and Constant() can be used to initialize to variables at the time of
+As shown above, static methods as Zero() and Constant() can be used to initialize variables at the time of
 declaration or at the right-hand side of an assignment operator. You can think of these methods as returning a
-matrix or array (in fact, they return a so-called \ref TopicEigenExpressionTemplates "expression object" which
-evaluates to a matrix when needed). This matrix can also be used as a temporary object. The second example in
-the \ref GettingStarted guide, which we reproduced here, already illustrates this.
+matrix or array; in fact, they return so-called \ref TopicEigenExpressionTemplates "expression objects" which
+evaluate to a matrix or array when needed, so that this syntax does not incur any overhead.
+
+These expressions can also be used as a temporary object. The second example in
+the \ref GettingStarted guide, which we reproduce here, already illustrates this.
 
 <table class="tutorial_code"><tr><td>
 Example: \include QuickStart_example2_dynamic.cpp
@@ -117,9 +119,10 @@ Example: \include QuickStart_example2_dynamic.cpp
 Output: \verbinclude QuickStart_example2_dynamic.out
 </td></tr></table>
 
-The expression <tt>m + MatrixXf::Constant(3,3,1.2)</tt> constructs the 3-by-3 matrix with all its coefficients
-equal to 1.2 and adds it to \c m ; in other words, it adds 1.2 to all the coefficients of \c m . The
-comma-initializer can also be used to construct temporary objects. The following example constructs a random
+The expression <tt>m + MatrixXf::Constant(3,3,1.2)</tt> constructs the 3-by-3 matrix expression with all its coefficients
+equal to 1.2 plus the corresponding coefficient of \a m.
+
+The comma-initializer, too, can also be used to construct temporary objects. The following example constructs a random
 matrix of size 2-by-3, and then multiplies this matrix on the left with 
 \f$ \bigl[ \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \bigr] \f$.
 

doc/C07_TutorialReductionsVisitorsBroadcasting.dox

@@ -22,9 +22,9 @@ This tutorial explains Eigen's reductions, visitors and broadcasting and how the
 
 
 \section TutorialReductionsVisitorsBroadcastingReductions Reductions
-In Eigen, a reduction is a function that is applied to a certain matrix or array, returning a single 
-value of type scalar. One of the most used reductions is \link DenseBase::sum() .sum() \endlink,
-which returns the addition of all the coefficients inside a given matrix or array.
+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="tutorial_code"><tr><td>
 Example: \include tut_arithmetic_redux_basic.cpp
@@ -33,12 +33,20 @@ Example: \include tut_arithmetic_redux_basic.cpp
 Output: \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 also be computed as efficiently using <tt>a.diagonal().sum()</tt>, as we will see later on.
+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 reductions
-Eigen also provides reductions to obtain the Euclidean norm or squared norm of a vector with \link MatrixBase::norm() norm() \endlink and \link MatrixBase::squaredNorm() squaredNorm() \endlink respectively.
-These operations can also operate on matrices; in that case, they use the Frobenius norm. The following example shows these methods. 
+\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 \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.
+
+The following example demonstrates these methods.
 
 <table class="tutorial_code"><tr><td>
 Example: \include Tutorial_ReductionsVisitorsBroadcasting_reductions_norm.cpp
@@ -48,12 +56,12 @@ Output:
 \verbinclude Tutorial_ReductionsVisitorsBroadcasting_reductions_norm.out
 </td></tr></table>
 
-\subsection TutorialReductionsVisitorsBroadcastingReductionsBool Boolean-like reductions
+\subsection TutorialReductionsVisitorsBroadcastingReductionsBool Boolean reductions
 
-Another interesting type of reductions are the ones that deal with \b true and \b false values:
-  - \link DenseBase::all() all() \endlink returns \b true if all of the coefficients in a given Matrix or Array are \b true .
-  - \link DenseBase::any() any() \endlink returns \b true if at least one of the coefficients in a given Matrix or Array is \b true .
-  - \link DenseBase::count() count() \endlink returns the number of \b true coefficients in a given Matrix or Array.
+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:
 

doc/examples/Tutorial_ReductionsVisitorsBroadcasting_reductions_norm.cpp

@@ -9,20 +9,20 @@ int main()
   VectorXf v(2);
   MatrixXf m(2,2), n(2,2);
   
-  v << 5,
-       10;
+  v << -1,
+       2;
   
-  m << 2,2,
-       3,4;
+  m << 1,-2,
+       -3,4;
 
-  n << 1, 2,
-       32,12;
-  
-  cout << "v.norm() = " << v.norm() << endl;
-  cout << "m.norm() = " << m.norm() << endl;
-  cout << "n.norm() = " << n.norm() << endl;
-  cout << endl;
   cout << "v.squaredNorm() = " << v.squaredNorm() << endl;
+  cout << "v.norm() = " << v.norm() << endl;
+  cout << "v.lpNorm<1>() = " << v.lpNorm<1>() << endl;
+  cout << "v.lpNorm<Infinity>() = " << v.lpNorm<Infinity>() << endl;
+
+  cout << endl;
   cout << "m.squaredNorm() = " << m.squaredNorm() << endl;
-  cout << "n.squaredNorm() = " << n.squaredNorm() << endl;
+  cout << "m.norm() = " << m.norm() << endl;
+  cout << "m.lpNorm<1>() = " << m.lpNorm<1>() << endl;
+  cout << "m.lpNorm<Infinity>() = " << m.lpNorm<Infinity>() << endl;
 }