For a 3x3 only symmetric and positive definite linear system, is Cholesky still faster than Householder?

Question

I am trying to solve a linear system Ax=b where A is 3x3 symmetric positive definite.

Though it is low in scale, I will have to repeat it for different As millions of times. So efficiency is still important.

There are many solvers for linear systems (C++, via Eigen). I personally prefer: HouseholderQr().solve(), and llt().solve(), ldlt().solve().

I know that when n is very large, solvers based on Cholesky decomposition are faster than that of Householder's. But for my case when n is only 3, how can I compare their relative efficiency? Is there any formula for the exact float operation analysis?

thanks

Answer 1

Yes, Cholesky will still be faster. It will be about n^3/3 flops. The only reason to use QR would be if your matrix was very ill-conditioned.

If you need to solve these systems a million times and efficiency is important, I'd recommend calling LAPACK directly. You want the DPOSV function.

http://www.netlib.org/lapack/lug/node26.html#1272

http://www.netlib.org/clapack/what/double/dposv.c