A New Array Format for Symmetric and Triangular Matrices

We introduce a new data format for matrices called HFP (Hybrid Full Packed). Full format triangular and symmetric arrays waste half the storage but provide high performance via the use of level 3 BLAS. Packed format arrays fully utilize storage (array space) but provide low performance as there are no level 3 packed BLAS. We combine the good features of packed and full storage using HFP format to obtain high performance using L3 (Level 3) BLAS as HFP is totally full format. Also, HFP format requires exactly the same minimal storage as packed storage. An LAPACK full and/or packed symmetric/triangular routine becomes a single new HFP routine. There are some 125 times two such symmetric/triangular routines and hence some 125 new routines can be produced. All such new routines are trivial to produce as they are merely calls to existing LAPACK routines and Level 3 BLAS. Data format conversion routines between the conventional formats and HFP are briefly discussed as these routines make existing codes compatible with the new HFP format. We use the LAPACK routines for Cholesky factorization and inverse to illustrate this new work and describe its performance. Performance for LAPACK full routines is slightly better while using half the storage and performance is roughly one to seven times faster for LAPACK packed routines while using the same storage. Performance runs were done on the IBM Power 4 using existing LAPACK routines and ESSL level 3 BLAS.