%Eigen determines automatically, for each sub-expression, whether to evaluate it into a temporary variable. Indeed, in certain cases it is better to evaluate a sub-expression into a temporary variable, while in other cases it is better to avoid that.
a traditional library would evaluate \c vec2 + vec3 into a temporary \c vec4 and then copy \c vec4 into \c vec1. This is of course inefficient: the arrays are traversed twice, so there are a lot of useless load/store operations.
Expression-templates-based libraries can avoid evaluating sub-expressions into temporaries, which in many cases results in large speed improvements.
This is called <i>lazy evaluation</i> as an expression is getting evaluated as late as possible.
In %Eigen <b>all expressions are lazy-evaluated</b>.
More precisely, an expression starts to be evaluated once it is assigned to a matrix.
Until then nothing happens beyond constructing the abstract expression tree.
In contrast to most other expression-templates-based libraries, however, <b>%Eigen might choose to evaluate some sub-expressions into temporaries</b>.
There are two reasons for that: first, pure lazy evaluation is not always a good choice for performance; second, pure lazy evaluation can be very dangerous, for example with matrix products: doing <tt>mat = mat*mat</tt> gives a wrong result if the matrix product is directly evaluated within the destination matrix, because of the way matrix product works.
For these reasons, %Eigen has intelligent compile-time mechanisms to determine automatically which sub-expression should be evaluated into a temporary variable.
<b>The first circumstance</b> in which %Eigen chooses to evaluate a sub-expression is when it sees an assignment <tt>a = b;</tt> and the expression \c b has the evaluate-before-assigning \link flags flag\endlink.
The most important example of such an expression is the \link Product matrix product expression\endlink. For example, when you do
\code mat = mat * mat;
\endcode
%Eigen will evaluate <tt>mat * mat</tt> into a temporary matrix, and then copies it into the original \c mat.
This guarantees a correct result as we saw above that lazy evaluation gives wrong results with matrix products.
It also doesn't cost much, as the cost of the matrix product itself is much higher.
Note that this temporary is introduced at evaluation time only, that is, within operator= in this example.
The expression <tt>mat * mat</tt> still return a abstract product type.
What if you know that the result does no alias the operand of the product and want to force lazy evaluation? Then use \link MatrixBase::noalias() .noalias()\endlink instead. Here is an example:
Here, since we know that mat2 is not the same matrix as mat1, we know that lazy evaluation is not dangerous, so we may force lazy evaluation. Concretely, the effect of noalias() here is to bypass the evaluate-before-assigning \link flags flag\endlink.
<b>The second circumstance</b> in which %Eigen chooses to evaluate a sub-expression, is when it sees a nested expression such as <tt>a + b</tt> where \c b is already an expression having the evaluate-before-nesting \link flags flag\endlink.
Again, the most important example of such an expression is the \link Product matrix product expression\endlink.
<b>The third circumstance</b> in which %Eigen chooses to evaluate a sub-expression, is when its cost model shows that the total cost of an operation is reduced if a sub-expression gets evaluated into a temporary.
Indeed, in certain cases, an intermediate result is sufficiently costly to compute and is reused sufficiently many times, that is worth "caching". Here is an example:
Here, provided the matrices have at least 2 rows and 2 columns, each coefficient of the expression <tt>mat3 + mat4</tt> is going to be used several times in the matrix product. Instead of computing the sum every time, it is much better to compute it once and store it in a temporary variable. %Eigen understands this and evaluates <tt>mat3 + mat4</tt> into a temporary variable before evaluating the product.
