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author | Gael Guennebaud <g.gael@free.fr> | 2013-01-06 23:57:54 +0100 |
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committer | Gael Guennebaud <g.gael@free.fr> | 2013-01-06 23:57:54 +0100 |
commit | 2abe7d8c6e11a02fc345f6ae464b4b759b092a67 (patch) | |
tree | 32b0a1517731ee705e1d6ee7d8d944740642efbf /doc/TopicAliasing.dox | |
parent | 091a49cad5d3aed47de9cb78c821c5f11b85e5f8 (diff) |
Rename the dox files: the number prefixes are not needed anymore
Diffstat (limited to 'doc/TopicAliasing.dox')
-rw-r--r-- | doc/TopicAliasing.dox | 209 |
1 files changed, 209 insertions, 0 deletions
diff --git a/doc/TopicAliasing.dox b/doc/TopicAliasing.dox new file mode 100644 index 000000000..bd1d329ce --- /dev/null +++ b/doc/TopicAliasing.dox @@ -0,0 +1,209 @@ +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. + +Note that if \c mat were a bigger, then the blocks would not overlap, and there would be no aliasing +problem. This means that in general aliasing cannot be detected at compile time. 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 tranposition, 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> + + +\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 .eval() 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. + +*/ +} |