update doc

1 files changed, 95 insertions, 2 deletions

diff --git a/doc/I02_HiPerformance.dox b/doc/I02_HiPerformance.dox
index 012e7d71b..8f23b5d19 100644
--- a/doc/I02_HiPerformance.dox
+++ b/doc/I02_HiPerformance.dox
@@ -3,6 +3,99 @@ namespace Eigen {
 
 /** \page HiPerformance Advanced - Using Eigen with high performance
 
+In general achieving good performance with Eigen does no require any special effort:
+simply write your expressions in the most high level way. This is especially true
+for small fixed size matrices. For large matrices, however, it might useful to
+take some care when writing your expressions in order to minimize useless evaluations
+and optimize the performance.
+In this page we will give a brief overview of the Eigen's internal mechanism to simplify
+and evaluate complex expressions, and discuss the current limitations.
+In particular we will focus on expressions matching level 2 and 3 BLAS routines, i.e,
+all kind of matrix products and triangular solvers.
+
+Indeed, in Eigen we have implemented a set of highly optimized routines which are very similar
+to BLAS's ones. Unlike BLAS, those routines are made available to user via a high level and
+natural API. Each of these routines can perform in a single evaluation a wide variety of expressions.
+Given an expression, the challenge is then to map it to a minimal set of primitives.
+As explained latter, this mechanism has some limitations, and knowing them will allow
+you to write faster code by making your expressions more Eigen friendly.
+
+\section GEMM General Matrix-Matrix product (GEMM)
+
+Let's start with the most common primitive: the matrix product of general dense matrices.
+In the BLAS world this corresponds to the GEMM routine. Our equivalent primitive can
+perform the following operation:
+\f$ C += \alpha op1(A) * op2(B) \f$
+where A, B, and C are column and/or row major matrices (or sub-matrices),
+alpha is a scalar value, and op1, op2 can be transpose, adjoint, conjugate, or the identity.
+When Eigen detects a matrix product, it analyzes both sides of the product to extract a
+unique scalar factor alpha, and for each side its effective storage (order and shape) and conjugate state.
+More precisely each side is simplified by iteratively removing trivial expressions such as scalar multiple,
+negate and conjugate. Transpose and Block expressions are not evaluated and only modify the storage order
+and shape. All other expressions are immediately evaluated.
+For instance, the following expression:
+\code m1 -= (s1 * m2.adjoint() * (-(s3*m3).conjugate()*s2)).lazy()  \endcode
+is automatically simplified to:
+\code m1 += (s1*s2*conj(s3)) * m2.adjoint() * m3.conjugate() \endcode
+which exactly matches our GEMM routine.
+
+\subsection GEMM_Limitations Limitations
+Unfortunately, this simplification mechanism is not perfect yet and not all expressions which could be
+handled by a single GEMM-like call are correctly detected.
+<table class="tutorial_code">
+<tr>
+<td>Not optimal expression</td>
+<td>Evaluated as</td>
+<td>Optimal version (single evaluation)</td>
+<td>Comments</td>
+</tr>
+<tr>
+<td>\code m1 += m2 * m3; \endcode</td>
+<td>\code temp = m2 * m3; m1 += temp; \endcode</td>
+<td>\code m1 += (m2 * m3).lazy(); \endcode</td>
+<td>Use .lazy() to tell Eigen the result and right-hand-sides do not alias.</td>
+</tr>
+<tr>
+<td>\code m1 += (s1 * (m2 * m3)).lazy(); \endcode</td>
+<td>\code temp = (m2 * m3).lazy(); m1 += s1 * temp; \endcode</td>
+<td>\code m1 += (s1 * m2 * m3).lazy(); \endcode</td>
+<td>This is because m2 * m3 is immediately evaluated by the scalar product. <br>
+    Make sure the matrix product is the top most expression.</td>
+</tr>
+<tr>
+<td>\code m1 = m1 + m2 * m3; \endcode</td>
+<td>\code temp = (m2 * m3).lazy(); m1 = m1 + temp; \endcode</td>
+<td>\code m1 += (m2 * m3).lazy(); \endcode</td>
+<td>Here there is no way to detect at compile time that the two m1 are the same,
+    and so the matrix product will be immediately evaluated.</td>
+</tr>
+<tr>
+<td>\code m1 += ((s1 * m2).transpose() * m3).lazy(); \endcode</td>
+<td>\code temp = (s1*m2).transpose(); m1 = (temp * m3).lazy(); \endcode</td>
+<td>\code m1 += (s1 * m2.transpose() * m3).lazy(); \endcode</td>
+<td>This is because our expression analyzer stops at the first expression which cannot
+    be converted to a scalar multiple of a conjugate and therefore the nested scalar
+    multiple cannot be properly extracted.</td>
+</tr>
+<tr>
+<td>\code m1 += (m2.conjugate().transpose() * m3).lazy(); \endcode</td>
+<td>\code temp = m2.conjugate().transpose(); m1 += (temp * m3).lazy(); \endcode</td>
+<td>\code m1 += (m2.adjoint() * m3).lazy(); \endcode</td>
+<td>Same reason. Use .adjoint() or .transpose().conjugate()</td>
+</tr>
+<tr>
+<td>\code m1 += ((s1*m2).block(....) * m3).lazy(); \endcode</td>
+<td>\code temp = (s1*m2).block(....); m1 += (temp * m3).lazy(); \endcode</td>
+<td>\code m1 += (s1 * m2.block(....) * m3).lazy(); \endcode</td>
+<td>Same reason.</td>
+</tr>
+</table>
+
+Of course all these remarks hold for all other kind of products that we will describe in the following paragraphs.
+
+
+
+
 <table class="tutorial_code">
 <tr>
 <td>BLAS equivalent routine</td>
@@ -20,7 +113,7 @@ namespace Eigen {
 <td>GEMM</td>
 <td>m1 += s1 * m2.adjoint() * m3</td>
 <td>m1 += (s1 * m2).adjoint() * m3</td>
-<td>This is because our expression analyser stops at the first transpose expression and cannot extract the nested scalar multiple.</td>
+<td>This is because our expression analyzer stops at the first transpose expression and cannot extract the nested scalar multiple.</td>
 </tr>
 <tr>
 <td>GEMM</td>
@@ -36,7 +129,7 @@ namespace Eigen {
 </tr>
 <tr>
 <td>SYR</td>
-<td>m.sefadjointView<LowerTriangular>().rankUpdate(v,s)</td>
+<td>m.seductive<LowerTriangular>().rankUpdate(v,s)</td>
 <td></td>
 <td>Computes m += s * v * v.adjoint()</td>
 </tr>