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+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. Thus, if \c matA is a
+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>
+
+
+\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.
+
+*/
+}