There is no library to link to. For good performance, add the \c -O2 compile-flag. Note however that this makes it impossible to debug inside Eigen code, as many functions get inlined. In some cases, performance can be further improved by disabling Eigen assertions: use \c -DEIGEN_NO_DEBUG or \c -DNDEBUG to disable them.
On the x86 architecture, the SSE2 instruction set is not enabled by default. Use \c -msse2 to enable it, and Eigen will then automatically enable its vectorized paths. On x86-64 and AltiVec-based architectures, vectorization is enabled by default.
By fixed-size, we mean that the number of rows and columns are fixed at compile-time. In this case, Eigen avoids dynamic memory allocation, and unroll loops when that makes sense. This is useful for very small sizes: typically up to 4x4, sometimes up to 16x16.
By dynamic-size, we mean that the numbers of rows and columns are not fixed at compile-time. In this case, they are stored as runtime variables and the arrays are dynamically allocated.
In Eigen, all kinds of dense matrices and vectors are represented by the template class Matrix. In most cases, you can simply use one of the \ref matrixtypedefs "convenience typedefs".
The template class Matrix takes a number of template parameters, but for now it is enough to understand the 3 first ones (and the others can then be left unspecified):
For dynamic-size, that is in order to left the number of rows or of columns unspecified at compile-time, use the special value Eigen::Dynamic. For example, \c VectorXd is a typedef for \code Matrix<double, Dynamic, 1> \endcode
Eigen also offers a \ref MatrixBaseCommaInitRef "comma initializer syntax" which allows you to set all the coefficients of a matrix to specific values:
In short, all arithmetic operators can be used right away as in the following example. Note however that arithmetic operators are only given their usual meaning from mathematics tradition. For other operations, such as taking the coefficient-wise product of two vectors, see the discussion of \link Cwise .cwise() \endlink below. Anyway, here is an example demonstrating basic arithmetic operators:
<span class="note">\b Side \b note: If you think that the \c .cwise() syntax is too verbose for your own taste and prefer to have non-conventional mathematical operators directly available, then feel free to extend MatrixBase as described \ref ExtendingMatrixBase "here".</span>
So far, we saw the notation \code mat1*mat2 \endcode for matrix product, and \code mat1.cwise()*mat2 \endcode for coefficient-wise product. What about other kinds of products, which in some other libraries also use arithmetic operators? In Eigen, they are accessed as follows -- note that here we are anticipating on further sections, for convenience.
Also note that maxCoeff and minCoeff can takes optional arguments returning the coordinates of the respective min/max coeff: \link MatrixBase::maxCoeff(int*,int*) const maxCoeff(int* i, int* j) \endlink, \link MatrixBase::minCoeff(int*,int*) const minCoeff(int* i, int* j) \endlink.
<span class="note">\b Side \b note: The all() and any() functions are especially useful in combinaison with coeff-wise comparison operators (\ref CwiseAll "example").</span>
Eigen determines automatically, for each sub-expression, whether to evaluate it into a temporary variable. Indeed, in certain cases it is better to evaluate immediately a sub-expression into a temporary variable, while in other cases it is better to avoid that.
A traditional math library without expression templates always evaluates all sub-expressions into temporaries. So with this code,
\code vec1 = vec2 + vec3; \endcode
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, instead of immediately. However, most other expression-templates-based libraries <i>always</i> choose lazy evaluation. There are two problems with that: first, lazy evaluation is not always a good choice for performance; second, lazy evaluation can be very dangerous, for example with matrix products: doing <tt>matrix = matrix*matrix</tt> gives a wrong result if the matrix product is lazy-evaluated, because of the way matrix product works.
For these reasons, Eigen has intelligent compile-time mechanisms to determine automatically when to use lazy evaluation, and when on the contrary it should evaluate immediately into a temporary variable.
So in the basic example,
\code matrix1 = matrix2 + matrix3; \endcode
Eigen chooses lazy evaluation. Thus the arrays are traversed only once, producing optimized code. If you really want to force immediate evaluation, use \link MatrixBase::eval() eval() \endlink:
Eigen chooses lazy evaluation at every stage in that example, which is clearly the correct choice. In fact, lazy evaluation is the "default choice" and Eigen will choose it except in a few circumstances.
<b>The first circumstance</b> in which Eigen chooses immediate evaluation, 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 importat example of such an expression is the \link Product matrix product expression \endlink. For example, when you do
\code matrix = matrix * matrix; \endcode
Eigen first evaluates <tt>matrix * matrix</tt> into a temporary matrix, and then copies it into the original \c matrix. 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.
What if you know what you are doing and want to force lazy evaluation? Then use \link MatrixBase::lazy() .lazy() \endlink instead. Here is an example:
Here, since we know that matrix2 is not the same matrix as matrix1, we know that lazy evaluation is not dangerous, so we may force lazy evaluation. Concretely, the effect of lazy() here is to remove the evaluate-before-assigning \link flags flag \endlink and also the evaluate-before-nesting \link flags flag \endlink which we now discuss.
<b>The second circumstance</b> in which Eigen chooses immediate evaluation, 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 importat example of such an expression is the \link Product matrix product expression \endlink. For example, when you do
the product <tt>matrix3 * matrix4</tt> gets evaluated immediately into a temporary matrix. Indeed, experiments showed that it is often beneficial for performance to evaluate immediately matrix products when they are nested into bigger expressions.
Again, \link MatrixBase::lazy() .lazy() \endlink can be used to force lazy evaluation here.
<b>The third circumstance</b> in which Eigen chooses immediate evaluation, 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 coefficienct of the expression <tt>matrix3 + matrix4</tt> is going to be used several times in the matrix product. Instead of computing the sum everytime, it is much better to compute it once and store it in a temporary variable. Eigen understands this and evaluates <tt>matrix3 + matrix4</tt> into a temporary variable before evaluating the product.
