# float128 in linear algebra

Is there any paper or research concerning float128 arithmetics applied to linear algebra problems(e.g. iterative solvers, decompositions etc.)?
How much benefit is really there in comparison with double? How much slower is it?

I've discovered this toolbox: http://www.advanpix.com/ Unfortunately it is commercial, but results if they are true, look impressive.

• As far as I know, you would need an application with horrible conditioning to warrant the use of float128. I'm not sure if that helped..... Mar 11 '12 at 13:44
• I guess any matrix that has condition number close to $\epsilon^{-1}$, where $\epsilon$ is the smallest significant floating point number can be better treated using more precise arithmetics. Mar 11 '12 at 14:04
• Here, one could use sym() when declaring the matrix and have any precision but it is horribly slow. Mar 11 '12 at 14:24

The quad-precision type __float128 is available in PETSc. Since most sparse linear algebra methods are limited by memory bandwidth instead of floating point performance, it is often only about half the speed of double precision. It is especially useful when using matrix-free finite differencing (frequently used with Jacobian-free Newton-Krylov methods) or exploring whether a nearly-singular operator is actually singular. For most problems, it is possible and preferable to use a well-scaled and conditioning-friendly discretization so that quad precision is not necessary for production runs.