The impact of multicore on math software and exploiting single precision computing to obtain double precision results

Recent versions of microprocessors exhibit performance characteristics for 32 bit floating point arithmetic (single precision) that is substantially higher than 64 bit floating point arithmetic (double precision). Examples include the Intel's Pentium IV and M processors, AMD's Opteron architectures, the IBM's Cell processor and various GPUs. When working in single precision, floating point operations can be performed up to two times faster on the Pentium and up to ten times faster on the Cell over double precision. The motivation for this work is to exploit single precision operations whenever possible and resort to double precision at critical stages while attempting to provide the full double precision results. The results described here are fairly general and can be applied to various problems in linear algebra such as solving large sparse systems, using direct or iterative methods and some eigenvalue problems. There are limitations to the success of this process, such as when the conditioning of the problem exceeds the reciprocal of the accuracy of the single precision computations. In that case the double precision algorithm should be used.