Clearer, Simpler and More Efficient LAPACK Routines for Symmetric Positive Definite Band Factorization

We describe a minor format change for representing a symmetric band matrix AB using the same array space specified by LAPACK. In LAPACK, band codes operating on the lower part of a symmetric matrix reference matrix element (i, j) as AB_1+i−j,j . The format change we propose allows LAPACK band codes to reference the (i, j) element as AB_i,j . Doing this yields lower band codes that use standard matrix terminology so that they become clearer and hence easier to understand. As a second contribution, we simplify the LAPACK Cholesky Band Factorization routine pbtrf by reducing from six to three the number of subroutine calls one needs to invoke during a right-looking block factorization step. Our new routines perform exactly the same number of floating-point arithmetic operations as the current LAPACK routine PB-TRF . Almost always they deliver higher performance. The experimental results show that this is especially true on SMP platforms where parallelism is obtained via the use level-3 multi-threaded BLAS. We only consider the lower triangular case of the factorization here; the upper triangular case is currently under investigation.