COSC-4P80-Assignment-2/lib/eigen-3.4.0/doc/TutorialAdvancedInitialization.dox

namespace Eigen {

/** \eigenManualPage TutorialAdvancedInitialization Advanced initialization

This page discusses several advanced methods for initializing matrices. It gives more details on the
comma-initializer, which was introduced before. It also explains how to get special matrices such as the
identity matrix and the zero matrix.

\eigenAutoToc

\section TutorialAdvancedInitializationCommaInitializer The comma initializer

Eigen offers a comma initializer syntax which allows the user to easily set all the coefficients of a matrix,
vector or array. Simply list the coefficients, starting at the top-left corner and moving from left to right
and from the top to the bottom. The size of the object needs to be specified beforehand. If you list too few
or too many coefficients, Eigen will complain.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include Tutorial_commainit_01.cpp
</td>
<td>
\verbinclude Tutorial_commainit_01.out
</td></tr></table>

Moreover, the elements of the initialization list may themselves be vectors or matrices. A common use is
to join vectors or matrices together. For example, here is how to join two row vectors together. Remember
that you have to set the size before you can use the comma initializer.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include Tutorial_AdvancedInitialization_Join.cpp
</td>
<td>
\verbinclude Tutorial_AdvancedInitialization_Join.out
</td></tr></table>

We can use the same technique to initialize matrices with a block structure.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include Tutorial_AdvancedInitialization_Block.cpp
</td>
<td>
\verbinclude Tutorial_AdvancedInitialization_Block.out
</td></tr></table>

The comma initializer can also be used to fill block expressions such as <tt>m.row(i)</tt>. Here is a more
complicated way to get the same result as in the first example above:

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include Tutorial_commainit_01b.cpp
</td>
<td>
\verbinclude Tutorial_commainit_01b.out
</td></tr></table>


\section TutorialAdvancedInitializationSpecialMatrices Special matrices and arrays

The Matrix and Array classes have static methods like \link DenseBase::Zero() Zero()\endlink, which can be
used to initialize all coefficients to zero. There are three variants. The first variant takes no arguments
and can only be used for fixed-size objects. If you want to initialize a dynamic-size object to zero, you need
to specify the size. Thus, the second variant requires one argument and can be used for one-dimensional
dynamic-size objects, while the third variant requires two arguments and can be used for two-dimensional
objects. All three variants are illustrated in the following example:

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include Tutorial_AdvancedInitialization_Zero.cpp
</td>
<td>
\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
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,
because "identity matrix" is a linear algebra concept.  The method
\link DenseBase::LinSpaced LinSpaced\endlink(size, low, high) is only available for vectors and
one-dimensional arrays; it yields a vector of the specified size whose coefficients are equally spaced between
\c low and \c high. The method \c LinSpaced() is illustrated in the following example, which prints a table
with angles in degrees, the corresponding angle in radians, and their sine and cosine.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include Tutorial_AdvancedInitialization_LinSpaced.cpp
</td>
<td>
\verbinclude Tutorial_AdvancedInitialization_LinSpaced.out
</td></tr></table>

This example shows that objects like the ones returned by LinSpaced() can be assigned to variables (and
expressions). Eigen defines utility functions like \link DenseBase::setZero() setZero()\endlink, 
\link MatrixBase::setIdentity() \endlink and \link DenseBase::setLinSpaced() \endlink to do this
conveniently. The following example contrasts three ways to construct the matrix
\f$ J = \bigl[ \begin{smallmatrix} O & I \\ I & O \end{smallmatrix} \bigr] \f$: using static methods and
assignment, using static methods and the comma-initializer, or using the setXxx() methods.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include Tutorial_AdvancedInitialization_ThreeWays.cpp
</td>
<td>
\verbinclude Tutorial_AdvancedInitialization_ThreeWays.out
</td></tr></table>

A summary of all pre-defined matrix, vector and array objects can be found in the \ref QuickRefPage.


\section TutorialAdvancedInitializationTemporaryObjects Usage as temporary objects

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 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="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include QuickStart_example2_dynamic.cpp
</td>
<td>
\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 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$.

<table class="example">
<tr><th>Example:</th><th>Output:</th></tr>
<tr><td>
\include Tutorial_AdvancedInitialization_CommaTemporary.cpp
</td>
<td>
\verbinclude Tutorial_AdvancedInitialization_CommaTemporary.out
</td></tr></table>

The \link CommaInitializer::finished() finished() \endlink method is necessary here to get the actual matrix
object once the comma initialization of our temporary submatrix is done.


*/

}