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106 lines
4.5 KiB
Plaintext
106 lines
4.5 KiB
Plaintext
namespace Eigen {
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/** \page TopicAliasing Aliasing
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In Eigen, aliasing refers to assignment statement in which the same matrix (or array or vector) appears on the
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left and on the right of the assignment operators. Statements like <tt>mat = 2 * mat;</tt> or <tt>mat =
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mat.transpose();</tt> exhibit aliasing. The aliasing in the first example is harmless, but the aliasing in the
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second example leads to unexpected results. This page explains what aliasing is, when it is harmful, and what
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to do about it.
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<b>Table of contents</b>
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- \ref TopicAliasingExamples
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- \ref TopicAliasingSolution
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- \ref TopicAliasingCwise
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- \ref TopicAliasingMatrixMult
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- \ref TopicAliasingSummary
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\section TopicAliasingExamples Examples
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Here is a simple example exhibiting aliasing:
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<table class="tutorial_code"><tr><td>
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Example: \include TopicAliasing_block.cpp
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</td>
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<td>
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Output: \verbinclude TopicAliasing_block.out
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</td></tr></table>
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The output is not what one would expect. The problem is the assignment
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\code
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mat.bottomRightCorner(2,2) = mat.topLeftCorner(2,2);
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\endcode
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After the assignment, the (2,2) entry in the bottom right corner should have the value of \c mat(1,1) before
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the assignment, which is 5. However, the output shows that \c mat(2,2) is actually 1. The problem is that
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Eigen uses lazy evaluation (see \ref TopicEigenExpressionTemplates) for <tt>mat.topLeftCorner(2,2)</tt>. The
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result is similar to
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\code
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mat(1,1) = mat(0,0);
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mat(1,2) = mat(0,1);
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mat(2,1) = mat(1,0);
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mat(2,2) = mat(1,1);
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\endcode
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Thus, \c mat(2,2) is assigned the \e new value of \c mat(1,1) instead of the old value. The issue here is that
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the blocks <tt>mat.bottomRightCorner(2,2)</tt> and <tt>mat.topLeftCorner(2,2)</tt> overlap, because both
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contain the coefficient <tt>mat(1,1)</tt> at the centre of the 3-by-3 matrix \c mat . The next section
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explains how to solve this problem by calling \link DenseBase::eval() eval()\endlink.
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Note that if \c mat were a bigger, then the blocks would not overlap, and there would be no aliasing
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problem. This means that in general aliasing cannot be detected at compile time. However, Eigen does detect
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some instances of aliasing, albeit at run time. The following example exhibiting aliasing was mentioned in
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\ref TutorialMatrixArithmetic :
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<table class="tutorial_code"><tr><td>
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Example: \include tut_arithmetic_transpose_aliasing.cpp
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</td>
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<td>
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Output: \verbinclude tut_arithmetic_transpose_aliasing.out
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</td></tr></table>
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Again, the output shows the aliasing issue. However, by default Eigen uses a run-time assertion to detect this
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and exits with a message like
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\verbatim
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void Eigen::DenseBase<Derived>::checkTransposeAliasing(const OtherDerived&) const
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[with OtherDerived = Eigen::Transpose<Eigen::Matrix<int, 2, 2, 0, 2, 2> >, Derived = Eigen::Matrix<int, 2, 2, 0, 2, 2>]:
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Assertion `(!ei_check_transpose_aliasing_selector<Scalar,ei_blas_traits<Derived>::IsTransposed,OtherDerived>::run(ei_extract_data(derived()), other))
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&& "aliasing detected during tranposition, use transposeInPlace() or evaluate the rhs into a temporary using .eval()"' failed.
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\endverbatim
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The user can turn Eigen's run-time assertions like the one to detect this aliasing problem off by defining the
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#EIGEN_NO_DEBUG macro, and the above program was compiled with this macro turned off in order to illustrate the
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aliasing problem. See \ref TopicAssertions for more information about Eigen's run-time assertions.
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\section TopicAliasingSolution Resolving aliasing issues
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Synopsis: xxxInPlace(), eval().
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\section TopicAliasingCwise Aliasing and component-wise operations
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Synopsis: Things like mat = 2 * mat, matA = matA + matB and arr = arr.sin() are safe.
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\section TopicAliasingMatrixMult Aliasing and matrix multiplication
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Synopsis: %Matrix multiplication assumes aliasing by default. Use noalias() to improve performance if there is
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no aliasing.
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\section TopicAliasingSummary Summary
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Aliasing occurs when the same matrix or array coefficients appear both on the left- and the right-hand side of
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an assignment operator.
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- Aliasing is harmless with coefficient-wise computations; this includes scalar multiplication and matrix or
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array addition.
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- When you multiply two matrices, Eigen assumes that aliasing occurs. If you know that there is no aliasing,
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then you can use \link MatrixBase::noalias() noalias()\endlink.
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- In all other situations, Eigen assumes that there is no aliasing issue and thus gives the wrong result if
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aliasing does in fact occur. To prevent this, you have to use \link DenseBase::eval() eval() \endlink or
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one of the xxxInPlace() functions.
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*/
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}
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