Computing the null space of a large matrix

Question

I'm looking for the fastest algorithm/package i could use to compute the null space of an extremely large (millions of elements, and not necessarily square) matrix. Any language would be alright, preferably something in Python/C/C++/Java. Your help would be greatly appreciated!

Answer 1

The manner to avoid trashing CPU caches greatly depends on how the matrix is stored/loaded/transmitted, a point that you did not address.

There are a few generic recommendations:

1. divide the problem into worker threads addressing contiguous rows per threads

2. increment pointers (in C) to traverse rows and keep the count on a per-thread basis

3. consolidate the per-thread results at the end of all worker threads.

If your matrix cells are made of bits (instead of bytes, ints, or arrays) then you can read words (either 4-byte or 8-byte on 32-bit/64-bit platforms) to speedup the count.

There are too many questions left unanswered in the problem description to give you any further guidance.

Answer 2

In what kind of data structure is your matrix represented? If you use an element list to represent the matrix, ie "column, row, value" tuple for one matrix element, then the solution would be just count the number of the tuples (subtracted by the matrix size)