
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 += s1 * (m2 * m3).lazy(); \endcode</td>
<td>\code m1 += s1 * m2 * m3; // using a naive product \endcode</td>
<td>\code m1 += (s1 * m2 * m3).lazy(); \endcode</td>
<td>Even though this expression is evaluated without temporary, it is actually even
    worse than the previous case because here the .lazy() enforces Eigen to use a
    naive (and slow) evaluation of the product.</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).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>This is because our expression analyzer is currently not able to extract trivial
    expressions nested in a Block expression. Therefore the nested scalar
    multiple cannot be properly extracted.</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>
<td>Efficient version <br> (compile to a single optimized evaluation)</td>
<td>Less efficient equivalent version <br> (requires multiple evaluations)</td>
<td>comments</td>
</tr>
<tr>
<td>GEMM</td>
<td>m1 = s1 * m2 * m3</td>
<td>m1 = s1 * (m2 * m3)</td>
<td>This is because m2 * m3 is evaluated by the scalar product.</td>
</tr>
<tr>
<td>GEMM</td>
<td>m1 += s1 * m2.adjoint() * m3</td>
<td>m1 += (s1 * m2).adjoint() * m3</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>
<td>m1 += m2.adjoint() * m3</td>
<td>m1 += m2.conjugate().transpose() * m3</td>
<td>For the same reason. Use .adjoint() or .transpose().conjugate()</td>
</tr>
<tr>
<td>GEMM</td>
<td>m1 -= (-(s0*m2).conjugate()*s1) * (s2 * m3.adjoint() * s3)</td>
<td></td>
<td>Note that s0 is automatically conjugated during the simplification of the expression.</td>
</tr>
<tr>
<td>SYR</td>
<td>m.sefadjointView<LowerTriangular>().rankUpdate(v,s)</td>
<td></td>
<td>Computes m += s * v * v.adjoint()</td>
</tr>
<tr>
<td>SYR2</td>
<td>m.sefadjointView<LowerTriangular>().rankUpdate(u,v,s)</td>
<td></td>
<td>Computes m += s * u * v.adjoint() + s * v * u.adjoint()</td>
</tr>
<tr>
<td>SYRK</td>
<td>m1.sefadjointView<UpperTriangular>().rankUpdate(m2.adjoint(),s)</td>
<td></td>
<td>Computes m1 += s * m2.adjoint() * m2</td>
</tr>
<tr>
<td>SYMM/HEMM</td>
<td>m3 -= s1 * m1.sefadjointView<UpperTriangular>() * m2.adjoint()</td>
<td></td>
<td></td>
</tr>
<tr>
<td>SYMM/HEMM</td>
<td>m3 += s1 * m2.transpose() * m1.conjugate().sefadjointView<UpperTriangular>()</td>
<td></td>
<td></td>
</tr>
<tr>
<td>TRMM</td>
<td>m3 -= s1 * m1.triangularView<UnitUpperTriangular>() * m2.adjoint()</td>
<td></td>
<td></td>
</tr>
<tr>
<td>TRSV / TRSM</td>
<td>m1.adjoint().triangularView<UnitLowerTriangular>().solveInPlace(m2)</td>
<td></td>
<td></td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
<tr>
<td></td>
<td></td>
<td></td>
<td></td>
</tr>
</table>

*/

}