Fixed-size vectorizable Eigen objects

The goal of this page is to explain what we mean by "fixed-size vectorizable".

Summary

An Eigen object is called "fixed-size vectorizable" if it has fixed size and that size is a multiple of 16 bytes.

Examples include:

• Eigen::Vector2d
• Eigen::Vector4d
• Eigen::Vector4f
• Eigen::Matrix2d
• Eigen::Matrix2f
• Eigen::Matrix4d
• Eigen::Matrix4f
• Eigen::Affine3d
• Eigen::Affine3f
• Eigen::Quaterniond
• Eigen::Quaternionf

General explanation of this assertion

First, "fixed-size" should be clear: an Eigen object has fixed size if its number of rows and its number of columns are fixed at compile-time. So for example Matrix3f has fixed size, but MatrixXf doesn't (the opposite of fixed-size is dynamic-size).

The array of coefficients of a fixed-size Eigen object is a plain "static array", it is not dynamically allocated. For example, the data behind a Matrix4f is just a "float array[16]".

Fixed-size objects are typically very small, which means that we want to handle them with zero runtime overhead -- both in terms of memory usage and of speed.

Now, vectorization (both SSE and AltiVec) works with 128-bit packets. Moreover, for performance reasons, these packets need to be have 128-bit alignment.

So it turns out that the only way that fixed-size Eigen objects can be vectorized, is if their size is a multiple of 128 bits, or 16 bytes. Eigen will then request 16-byte alignment for these objects, and henceforth rely on these objects being aligned so no runtime check for alignment is performed.