LU Decomposition

Summary

LU Decomposition is an example of how to use Nu+ features to optimize an algorithm, using multithreaded and vectorial code.

Reusable code decomposition and multithreaded and vectorial optimization methods are shown; they can be applied to every other parallelizable algorithm, from image processing to machine learning ones.

Algorithm Description

LU Decomposition is used in solving systems of linear equations

Algorithm decomposes A in:

• upper triangolar matrix
• lower triangolar matrix

Partial Pivoting

Partial pivoting is an optimization of standard LU Decomposition that aims to reduce numerical instability

• Therefore a matrix is needed to keep track of pivoting operations
• Partial because pivoting is applied on rows only

These matrices have to verify the relation:

Pseudocode

nu+ Optimization

In this section, it is shown how the pseudo-code algorithm can be optimized using nu+ features.

Pivoting phase

The pivoting phase has been made multi-threaded. Every k-th iteration (k ranging from 0 to m-1) of the pivoting operations works on the elements belonging to the k-th column and to rows from k to m-1.