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

namespace Eigen {

/** \eigenManualPage TopicAliasing Aliasing

In %Eigen, aliasing refers to assignment statement in which the same matrix (or array or vector) appears on the
left and on the right of the assignment operators. Statements like <tt>mat = 2 * mat;</tt> or <tt>mat =
mat.transpose();</tt> exhibit aliasing. The aliasing in the first example is harmless, but the aliasing in the
second example leads to unexpected results. This page explains what aliasing is, when it is harmful, and what
to do about it.

\eigenAutoToc


\section TopicAliasingExamples Examples

Here is a simple example exhibiting aliasing:

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

The output is not what one would expect. The problem is the assignment
\code
mat.bottomRightCorner(2,2) = mat.topLeftCorner(2,2);
\endcode
This assignment exhibits aliasing: the coefficient \c mat(1,1) appears both in the block
<tt>mat.bottomRightCorner(2,2)</tt> on the left-hand side of the assignment and the block
<tt>mat.topLeftCorner(2,2)</tt> on the right-hand side. After the assignment, the (2,2) entry in the bottom
right corner should have the value of \c mat(1,1) before the assignment, which is 5. However, the output shows
that \c mat(2,2) is actually 1. The problem is that %Eigen uses lazy evaluation (see 
\ref TopicEigenExpressionTemplates) for <tt>mat.topLeftCorner(2,2)</tt>. The result is similar to
\code
mat(1,1) = mat(0,0);
mat(1,2) = mat(0,1);
mat(2,1) = mat(1,0);
mat(2,2) = mat(1,1);
\endcode
Thus, \c mat(2,2) is assigned the \e new value of \c mat(1,1) instead of the old value. The next section
explains how to solve this problem by calling \link DenseBase::eval() eval()\endlink.

Aliasing occurs more naturally when trying to shrink a matrix. For example, the expressions <tt>vec =
vec.head(n)</tt> and <tt>mat = mat.block(i,j,r,c)</tt> exhibit aliasing.

In general, aliasing cannot be detected at compile time: if \c mat in the first example were a bit bigger,
then the blocks would not overlap, and there would be no aliasing problem. However, %Eigen does detect some
instances of aliasing, albeit at run time.  The following example exhibiting aliasing was mentioned in \ref
TutorialMatrixArithmetic :

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

Again, the output shows the aliasing issue. However, by default %Eigen uses a run-time assertion to detect this
and exits with a message like

\verbatim
void Eigen::DenseBase<Derived>::checkTransposeAliasing(const OtherDerived&) const 
[with OtherDerived = Eigen::Transpose<Eigen::Matrix<int, 2, 2, 0, 2, 2> >, Derived = Eigen::Matrix<int, 2, 2, 0, 2, 2>]: 
Assertion `(!internal::check_transpose_aliasing_selector<Scalar,internal::blas_traits<Derived>::IsTransposed,OtherDerived>::run(internal::extract_data(derived()), other)) 
&& "aliasing detected during transposition, use transposeInPlace() or evaluate the rhs into a temporary using .eval()"' failed.
\endverbatim

The user can turn %Eigen's run-time assertions like the one to detect this aliasing problem off by defining the
EIGEN_NO_DEBUG macro, and the above program was compiled with this macro turned off in order to illustrate the
aliasing problem. See \ref TopicAssertions for more information about %Eigen's run-time assertions.


\section TopicAliasingSolution Resolving aliasing issues

If you understand the cause of the aliasing issue, then it is obvious what must happen to solve it: %Eigen has
to evaluate the right-hand side fully into a temporary matrix/array and then assign it to the left-hand
side. The function \link DenseBase::eval() eval() \endlink does precisely that.

For example, here is the corrected version of the first example above:

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

Now, \c mat(2,2) equals 5 after the assignment, as it should be.

The same solution also works for the second example, with the transpose: simply replace the line 
<tt>a = a.transpose();</tt> with <tt>a = a.transpose().eval();</tt>. However, in this common case there is a
better solution. %Eigen provides the special-purpose function 
\link DenseBase::transposeInPlace() transposeInPlace() \endlink which replaces a matrix by its transpose. 
This is shown below:

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

If an xxxInPlace() function is available, then it is best to use it, because it indicates more clearly what you
are doing. This may also allow %Eigen to optimize more aggressively. These are some of the xxxInPlace()
functions provided: 

<table class="manual">
<tr><th>Original function</th><th>In-place function</th></tr>
<tr> <td> MatrixBase::adjoint() </td> <td> MatrixBase::adjointInPlace() </td> </tr>
<tr class="alt"> <td> DenseBase::reverse() </td> <td> DenseBase::reverseInPlace() </td> </tr>
<tr> <td> LDLT::solve() </td> <td> LDLT::solveInPlace() </td> </tr>
<tr class="alt"> <td> LLT::solve() </td> <td> LLT::solveInPlace() </td> </tr>
<tr> <td> TriangularView::solve() </td> <td> TriangularView::solveInPlace() </td> </tr>
<tr class="alt"> <td> DenseBase::transpose() </td> <td> DenseBase::transposeInPlace() </td> </tr>
</table>

In the special case where a matrix or vector is shrunk using an expression like <tt>vec = vec.head(n)</tt>,
you can use \link PlainObjectBase::conservativeResize() conservativeResize() \endlink.


\section TopicAliasingCwise Aliasing and component-wise operations

As explained above, it may be dangerous if the same matrix or array occurs on both the left-hand side and the
right-hand side of an assignment operator, and it is then often necessary to evaluate the right-hand side
explicitly. However, applying component-wise operations (such as matrix addition, scalar multiplication and
array multiplication) is safe. 

The following example has only component-wise operations. Thus, there is no need for \link DenseBase::eval()
eval() \endlink even though the same matrix appears on both sides of the assignments.

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

In general, an assignment is safe if the (i,j) entry of the expression on the right-hand side depends only on
the (i,j) entry of the matrix or array on the left-hand side and not on any other entries. In that case it is
not necessary to evaluate the right-hand side explicitly.


\section TopicAliasingMatrixMult Aliasing and matrix multiplication

Matrix multiplication is the only operation in %Eigen that assumes aliasing by default, <strong>under the
condition that the destination matrix is not resized</strong>.
Thus, if \c matA is a \b squared matrix, then the statement <tt>matA = matA * matA;</tt> is safe.
All other operations in %Eigen assume that there are no aliasing problems,
either because the result is assigned to a different matrix or because it is a component-wise operation.

<table class="example">
<tr><th>Example</th><th>Output</th></tr>
<tr><td>
\include TopicAliasing_mult1.cpp
</td>
<td>
\verbinclude TopicAliasing_mult1.out
</td></tr></table>

However, this comes at a price. When executing the expression <tt>matA = matA * matA</tt>, %Eigen evaluates the
product in a temporary matrix which is assigned to \c matA after the computation. This is fine. But %Eigen does
the same when the product is assigned to a different matrix (e.g., <tt>matB = matA * matA</tt>). In that case,
it is more efficient to evaluate the product directly into \c matB instead of evaluating it first into a
temporary matrix and copying that matrix to \c matB.

The user can indicate with the \link MatrixBase::noalias() noalias()\endlink function that there is no
aliasing, as follows: <tt>matB.noalias() = matA * matA</tt>. This allows %Eigen to evaluate the matrix product
<tt>matA * matA</tt> directly into \c matB.

<table class="example">
<tr><th>Example</th><th>Output</th></tr>
<tr><td>
\include TopicAliasing_mult2.cpp
</td>
<td>
\verbinclude TopicAliasing_mult2.out
</td></tr></table>

Of course, you should not use \c noalias() when there is in fact aliasing taking place. If you do, then you
may get wrong results:

<table class="example">
<tr><th>Example</th><th>Output</th></tr>
<tr><td>
\include TopicAliasing_mult3.cpp
</td>
<td>
\verbinclude TopicAliasing_mult3.out
</td></tr></table>

Moreover, starting in Eigen 3.3, aliasing is \b not assumed if the destination matrix is resized and the product is not directly assigned to the destination.
Therefore, the following example is also wrong:

<table class="example">
<tr><th>Example</th><th>Output</th></tr>
<tr><td>
\include TopicAliasing_mult4.cpp
</td>
<td>
\verbinclude TopicAliasing_mult4.out
</td></tr></table>

As for any aliasing issue, you can resolve it by explicitly evaluating the expression prior to assignment:
<table class="example">
<tr><th>Example</th><th>Output</th></tr>
<tr><td>
\include TopicAliasing_mult5.cpp
</td>
<td>
\verbinclude TopicAliasing_mult5.out
</td></tr></table>

\section TopicAliasingSummary Summary

Aliasing occurs when the same matrix or array coefficients appear both on the left- and the right-hand side of
an assignment operator.
 - Aliasing is harmless with coefficient-wise computations; this includes scalar multiplication and matrix or
   array addition.
 - When you multiply two matrices, %Eigen assumes that aliasing occurs. If you know that there is no aliasing,
   then you can use \link MatrixBase::noalias() noalias()\endlink.
 - In all other situations, %Eigen assumes that there is no aliasing issue and thus gives the wrong result if
   aliasing does in fact occur. To prevent this, you have to use \link DenseBase::eval() eval() \endlink or
   one of the xxxInPlace() functions.

*/
}