LAPACK Cholesky Routines in Rectangular Full Packed Format

We describe a new data format for storing triangular and symmetric matrices called RFP (Rectangular Full Packed). The standard two dimensional arrays of Fortran and C (also known as full format) that are used to store triangular and symmetric matrices waste half the storage space 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 RFP format to obtain high performance using L3 (Level 3) BLAS as RFP is totally full format. Also, RFP format requires exactly the same minimal storage as packed storage. Each full and/or packed symmetric/triangular routine becomes a single new RFP routine. We present LAPACK routines for Cholesky factorization and inverse computation in RFP format to illustrate this new work and to describe its performance on the Intel, IBM, Itanium, SGI, and Sun platforms. Performance of RPF verses LAPACK full routines is about the same while using half the storage. Performance is roughly one to twenty times faster for LAPACK packed routines while using the same storage. In the performance study only existing LAPACK routines and level 3 BLAS were used. We describe codes to directly input RFP format arrays. These codes are similar to codes for inputting full format arrays. 1 Description of Rectangular Full Packed Format We describe RFP format. It represents a standard packed array as a full 2D array. This means that performance of LAPACK’s [2] packed format routines becomes equal to or better than their full array counterparts. RFP format is a variant of hybrid full packed (HFP) format [1]. RFP format is a rearrangement of a standard full rectangular array SA holding a symmetric / triangular matrix A into a compact full storage rectangular array AR that uses minimal storage NT=N(N+1)/2. Note also that the transpose of the matrix in array AR also represents A. Therefore, Level 3 BLAS can be used on AR or its transpose. In fact, with the equivalent LAPACK algorithm, using array AR on its transpose instead