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fc9480cbb3
improve doc, and workaround aliasing detection in MatrixBase_eval snippet (not very nice but I don't know how to do it in a better way)
107 lines
5.2 KiB
Plaintext
107 lines
5.2 KiB
Plaintext
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namespace Eigen {
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/** \page HiPerformance Advanced - Using Eigen with high performance
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In general achieving good performance with Eigen does no require any special effort:
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simply write your expressions in the most high level way. This is especially true
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for small fixed size matrices. For large matrices, however, it might useful to
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take some care when writing your expressions in order to minimize useless evaluations
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and optimize the performance.
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In this page we will give a brief overview of the Eigen's internal mechanism to simplify
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and evaluate complex product expressions, and discuss the current limitations.
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In particular we will focus on expressions matching level 2 and 3 BLAS routines, i.e,
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all kind of matrix products and triangular solvers.
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Indeed, in Eigen we have implemented a set of highly optimized routines which are very similar
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to BLAS's ones. Unlike BLAS, those routines are made available to user via a high level and
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natural API. Each of these routines can compute in a single evaluation a wide variety of expressions.
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Given an expression, the challenge is then to map it to a minimal set of primitives.
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As explained latter, this mechanism has some limitations, and knowing them will allow
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you to write faster code by making your expressions more Eigen friendly.
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\section GEMM General Matrix-Matrix product (GEMM)
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Let's start with the most common primitive: the matrix product of general dense matrices.
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In the BLAS world this corresponds to the GEMM routine. Our equivalent primitive can
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perform the following operation:
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\f$ C.noalias() += \alpha op1(A) op2(B) \f$
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where A, B, and C are column and/or row major matrices (or sub-matrices),
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alpha is a scalar value, and op1, op2 can be transpose, adjoint, conjugate, or the identity.
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When Eigen detects a matrix product, it analyzes both sides of the product to extract a
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unique scalar factor alpha, and for each side its effective storage (order and shape) and conjugate state.
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More precisely each side is simplified by iteratively removing trivial expressions such as scalar multiple,
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negate and conjugate. Transpose and Block expressions are not evaluated and they only modify the storage order
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and shape. All other expressions are immediately evaluated.
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For instance, the following expression:
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\code m1.noalias() -= s1 * m2.adjoint() * (-(s3*m3).conjugate()*s2) \endcode
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is automatically simplified to:
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\code m1.noalias() += (s1*s2*conj(s3)) * m2.adjoint() * m3.conjugate() \endcode
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which exactly matches our GEMM routine.
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\subsection GEMM_Limitations Limitations
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Unfortunately, this simplification mechanism is not perfect yet and not all expressions which could be
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handled by a single GEMM-like call are correctly detected.
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<table class="tutorial_code">
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<tr>
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<td>Not optimal expression</td>
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<td>Evaluated as</td>
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<td>Optimal version (single evaluation)</td>
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<td>Comments</td>
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</tr>
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<tr>
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<td>\code m1 += m2 * m3; \endcode</td>
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<td>\code temp = m2 * m3; m1 += temp; \endcode</td>
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<td>\code m1.noalias() += m2 * m3; \endcode</td>
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<td>Use .noalias() to tell Eigen the result and right-hand-sides do not alias.
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Otherwise the product m2 * m3 is evaluated into a temporary.</td>
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</tr>
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<tr>
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<td></td>
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<td></td>
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<td>\code m1.noalias() += s1 * (m2 * m3); \endcode</td>
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<td>This is a special feature of Eigen. Here the product between a scalar
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and a matrix product does not evaluate the matrix product but instead it
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returns a matrix product expression tracking the scalar scaling factor. <br>
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Without this optimization, the matrix product would be evaluated into a
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temporary as in the next example.</td>
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</tr>
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<tr>
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<td>\code m1.noalias() += (m2 * m3).transpose(); \endcode</td>
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<td>\code temp = m2 * m3; m1 += temp.transpose(); \endcode</td>
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<td>\code m1.noalias() += m3.adjoint() * m3.adjoint(); \endcode</td>
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<td>This is because the product expression has the EvalBeforeNesting bit which
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enforces the evaluation of the product by the Tranpose expression.</td>
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</tr>
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<tr>
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<td>\code m1 = m1 + m2 * m3; \endcode</td>
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<td>\code temp = (m2 * m3).lazy(); m1 = m1 + temp; \endcode</td>
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<td>\code m1 += (m2 * m3).lazy(); \endcode</td>
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<td>Here there is no way to detect at compile time that the two m1 are the same,
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and so the matrix product will be immediately evaluated.</td>
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</tr>
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<tr>
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<td>\code m1.noalias() = m4 + m2 * m3; \endcode</td>
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<td>\code temp = m2 * m3; m1 = m4 + temp; \endcode</td>
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<td>\code m1 = m4; m1.noalias() += m2 * m3; \endcode</td>
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<td>First of all, here the .noalias() in the first expression is useless because
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m2*m3 will be evaluated anyway. However, note how this expression can be rewritten
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so that no temporary is evaluated. (tip: for very small fixed size matrix
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it is slighlty better to rewrite it like this: m1.noalias() = m2 * m3; m1 += m4;</td>
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</tr>
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<tr>
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<td>\code m1.noalias() += ((s1*m2).block(....) * m3); \endcode</td>
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<td>\code temp = (s1*m2).block(....); m1 += temp * m3; \endcode</td>
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<td>\code m1.noalias() += s1 * m2.block(....) * m3; \endcode</td>
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<td>This is because our expression analyzer is currently not able to extract trivial
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expressions nested in a Block expression. Therefore the nested scalar
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multiple cannot be properly extracted.</td>
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</tr>
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</table>
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Of course all these remarks hold for all other kind of products involving triangular or selfadjoint matrices.
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*/
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}
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