In this paper we present the design and implementation of the Linpack benchmark for the IBM BladeCenter QS22, which incorporates two IBM PowerXCell 8i1 processors. The PowerXCell 8i is a new implementation of the Cell Broadband Engine™2 architecture and contains a set of special-purpose processing cores known as Synergistic Processing Elements (SPEs). The SPEs can be used as computational accelerators to augment the main PowerPC processor. The added computational capability of the SPEs results in a peak double precision floating point capability of 108.8 GFLOPS. We explain how we modified the standard open source implementation of Linpack to accelerate key computational kernels using the SPEs of the PowerXCell 8i processors. We describe in detail the implementation and performance of the computational kernels and also explain how we employed the SPEs for high-speed data movement and reformatting. The result of these modifications is a Linpack benchmark optimized for the IBM PowerXCell 8i processor that achieves 170.7 GFLOPS on a BladeCenter QS22 with 32 GB of DDR2 SDRAM memory. Our implementation of Linpack also supports clusters of QS22s, and was used to achieve a result of 11.1 TFLOPS on a cluster of 84 QS22 blades. We compare our results on a single BladeCenter QS22 with the base Linpack implementation without SPE acceleration to illustrate the benefits of our optimizations.