Optimized matrix multiplication in C

What Every Programmer Should Know About Memory (pdf link) by Ulrich Drepper has a lot of good ideas about memory efficiency, but in particular, he uses matrix multiplication as an example of how knowing about memory and using that knowledge can speed this process. Look at appendix A.1 in his paper, and read through section 6.2.1. Table 6.2 in the paper shows that he could get his running time to be 10% from a naive implementation's time for a 1000x1000 matrix.

Granted, his final code is pretty hairy and uses a lot of system-specific stuff and compile-time tuning, but still, if you really need speed, reading that paper and reading his implementation is definitely worth it.

Getting this right is non-trivial. Using an existing BLAS library is highly recommended.

Should you really be inclined to roll your own matrix multiplication, loop tiling is an optimization that is of particular importance for large matrices. The tiling should be tuned to the cache size to ensure that the cache is not being continually thrashed, which will occur with a naive implementation. I once measured a 12x performance difference tiling a matrix multiply with matrix sizes picked to consume multiples of my cache (circa '97 so the cache was probably small).

Loop tiling algorithms assume that a contiguous linear array of elements is used, as opposed to rows or columns of pointers. With such a storage choice, your indexing scheme determines which dimension changes fastest, and you are free to decide whether row or column access will have the best cache performance.

There's a lot of literature on the subject. The following references, especially the Banerjee books, may be helpful:

[Ban93] Banerjee, Utpal, Loop Transformations for Restructuring Compilers: the Foundations, Kluwer Academic Publishers, Norwell, MA, 1993.

[Ban94] Banerjee, Utpal, Loop Parallelization, Kluwer Academic Publishers, Norwell, MA, 1994.

[BGS93] Bacon, David F., Susan L. Graham, and Oliver Sharp, Compiler Transformations for High-Performance Computing, Computer Science Division, University of California, Berkeley, Calif., Technical Report No UCB/CSD-93-781.

[LRW91] Lam, Monica S., Edward E. Rothberg, and Michael E Wolf. The Cache Performance and Optimizations of Blocked Algorithms, In 4th International Conference on Architectural Support for Programming Languages, held in Santa Clara, Calif., April, 1991, 63-74.

[LW91] Lam, Monica S., and Michael E Wolf. A Loop Transformation Theory and an Algorithm to Maximize Parallelism, In IEEE Transactions on Parallel and Distributed Systems, 1991, 2(4):452-471.

[PW86] Padua, David A., and Michael J. Wolfe, Advanced Compiler Optimizations for Supercomputers, In Communications of the ACM, 29(12):1184-1201, 1986.

[Wolfe89] Wolfe, Michael J. Optimizing Supercompilers for Supercomputers, The MIT Press, Cambridge, MA, 1989.

[Wolfe96] Wolfe, Michael J., High Performance Compilers for Parallel Computing, Addison-Wesley, CA, 1996.

ATTENTION: You have a BUG in your second implementation

for (f = 0; f < i; f++) {
    for (co = 0; co < i; co++) {
        MatrixB[f][co] = MatrixB[co][f];
    }
}

When you do f=0, c=1

        MatrixB[0][1] = MatrixB[1][0];

you overwrite MatrixB[0][1] and lose that value! When the loop gets to f=1, c=0

        MatrixB[1][0] = MatrixB[0][1];

the value copied is the same that was already there.

If the matrix is not large enough or you don't repeat the operations a high number of times you won't see appreciable differences.

If the matrix is, say, 1,000x1,000 you will begin to see improvements, but I would say that if it is below 100x100 you should not worry about it.

Also, any 'improvement' may be of the order of milliseconds, unless yoy are either working with extremely large matrices or repeating the operation thousands of times.

Finally, if you change the computer you are using for a faster one the differences will be even narrower!