1 2 namespace Eigen { 3 4 /** \page TopicWritingEfficientProductExpression Writing efficient matrix product expressions 5 6 In general achieving good performance with Eigen does no require any special effort: 7 simply write your expressions in the most high level way. This is especially true 8 for small fixed size matrices. For large matrices, however, it might be useful to 9 take some care when writing your expressions in order to minimize useless evaluations 10 and optimize the performance. 11 In this page we will give a brief overview of the Eigen's internal mechanism to simplify 12 and evaluate complex product expressions, and discuss the current limitations. 13 In particular we will focus on expressions matching level 2 and 3 BLAS routines, i.e, 14 all kind of matrix products and triangular solvers. 15 16 Indeed, in Eigen we have implemented a set of highly optimized routines which are very similar 17 to BLAS's ones. Unlike BLAS, those routines are made available to user via a high level and 18 natural API. Each of these routines can compute in a single evaluation a wide variety of expressions. 19 Given an expression, the challenge is then to map it to a minimal set of routines. 20 As explained latter, this mechanism has some limitations, and knowing them will allow 21 you to write faster code by making your expressions more Eigen friendly. 22 23 \section GEMM General Matrix-Matrix product (GEMM) 24 25 Let's start with the most common primitive: the matrix product of general dense matrices. 26 In the BLAS world this corresponds to the GEMM routine. Our equivalent primitive can 27 perform the following operation: 28 \f$ C.noalias() += \alpha op1(A) op2(B) \f$ 29 where A, B, and C are column and/or row major matrices (or sub-matrices), 30 alpha is a scalar value, and op1, op2 can be transpose, adjoint, conjugate, or the identity. 31 When Eigen detects a matrix product, it analyzes both sides of the product to extract a 32 unique scalar factor alpha, and for each side, its effective storage order, shape, and conjugation states. 33 More precisely each side is simplified by iteratively removing trivial expressions such as scalar multiple, 34 negation and conjugation. Transpose and Block expressions are not evaluated and they only modify the storage order 35 and shape. All other expressions are immediately evaluated. 36 For instance, the following expression: 37 \code m1.noalias() -= s4 * (s1 * m2.adjoint() * (-(s3*m3).conjugate()*s2)) \endcode 38 is automatically simplified to: 39 \code m1.noalias() += (s1*s2*conj(s3)*s4) * m2.adjoint() * m3.conjugate() \endcode 40 which exactly matches our GEMM routine. 41 42 \subsection GEMM_Limitations Limitations 43 Unfortunately, this simplification mechanism is not perfect yet and not all expressions which could be 44 handled by a single GEMM-like call are correctly detected. 45 <table class="manual" style="width:100%"> 46 <tr> 47 <th>Not optimal expression</th> 48 <th>Evaluated as</th> 49 <th>Optimal version (single evaluation)</th> 50 <th>Comments</th> 51 </tr> 52 <tr> 53 <td>\code 54 m1 += m2 * m3; \endcode</td> 55 <td>\code 56 temp = m2 * m3; 57 m1 += temp; \endcode</td> 58 <td>\code 59 m1.noalias() += m2 * m3; \endcode</td> 60 <td>Use .noalias() to tell Eigen the result and right-hand-sides do not alias. 61 Otherwise the product m2 * m3 is evaluated into a temporary.</td> 62 </tr> 63 <tr class="alt"> 64 <td></td> 65 <td></td> 66 <td>\code 67 m1.noalias() += s1 * (m2 * m3); \endcode</td> 68 <td>This is a special feature of Eigen. Here the product between a scalar 69 and a matrix product does not evaluate the matrix product but instead it 70 returns a matrix product expression tracking the scalar scaling factor. <br> 71 Without this optimization, the matrix product would be evaluated into a 72 temporary as in the next example.</td> 73 </tr> 74 <tr> 75 <td>\code 76 m1.noalias() += (m2 * m3).adjoint(); \endcode</td> 77 <td>\code 78 temp = m2 * m3; 79 m1 += temp.adjoint(); \endcode</td> 80 <td>\code 81 m1.noalias() += m3.adjoint() 82 * m2.adjoint(); \endcode</td> 83 <td>This is because the product expression has the EvalBeforeNesting bit which 84 enforces the evaluation of the product by the Tranpose expression.</td> 85 </tr> 86 <tr class="alt"> 87 <td>\code 88 m1 = m1 + m2 * m3; \endcode</td> 89 <td>\code 90 temp = m2 * m3; 91 m1 = m1 + temp; \endcode</td> 92 <td>\code m1.noalias() += m2 * m3; \endcode</td> 93 <td>Here there is no way to detect at compile time that the two m1 are the same, 94 and so the matrix product will be immediately evaluated.</td> 95 </tr> 96 <tr> 97 <td>\code 98 m1.noalias() = m4 + m2 * m3; \endcode</td> 99 <td>\code 100 temp = m2 * m3; 101 m1 = m4 + temp; \endcode</td> 102 <td>\code 103 m1 = m4; 104 m1.noalias() += m2 * m3; \endcode</td> 105 <td>First of all, here the .noalias() in the first expression is useless because 106 m2*m3 will be evaluated anyway. However, note how this expression can be rewritten 107 so that no temporary is required. (tip: for very small fixed size matrix 108 it is slighlty better to rewrite it like this: m1.noalias() = m2 * m3; m1 += m4;</td> 109 </tr> 110 <tr class="alt"> 111 <td>\code 112 m1.noalias() += (s1*m2).block(..) * m3; \endcode</td> 113 <td>\code 114 temp = (s1*m2).block(..); 115 m1 += temp * m3; \endcode</td> 116 <td>\code 117 m1.noalias() += s1 * m2.block(..) * m3; \endcode</td> 118 <td>This is because our expression analyzer is currently not able to extract trivial 119 expressions nested in a Block expression. Therefore the nested scalar 120 multiple cannot be properly extracted.</td> 121 </tr> 122 </table> 123 124 Of course all these remarks hold for all other kind of products involving triangular or selfadjoint matrices. 125 126 */ 127 128 } 129
