Solving Large Full Sets of Linear Equations in a Paged Virtual Store

Abstract

The problem of solving large full sets of linear equations on a computer with a paged virtual memory is considered and a block column algorithm proposed. Details of software design are considered and results of experimental runs on five different computer systems are reported.

The LINPACK Benchmark: past, present and future

“…If the vectors and matrices involved are of order n, then the original BLAS (Level 1) includes operations that are of order O(n), the extended or Level 2 BLAS provides operations of order O(n 2 ), and the latest BLAS provides operations of order O(n 3 ) (hence the use of the term Level 3 BLAS). There is a long history of block algorithms: early algorithms utilized a small main memory, with tape or disk as secondary storage [12][13][14][15][16][17]. More recently, several researchers have demonstrated the effectiveness of block algorithms on a variety of modern computer architectures with vector-processing or parallel-processing capabilities [9,10,13,[17][18][19][20][21][22].…”

“…There is a long history of block algorithms: early algorithms utilized a small main memory, with tape or disk as secondary storage [12][13][14][15][16][17]. More recently, several researchers have demonstrated the effectiveness of block algorithms on a variety of modern computer architectures with vector-processing or parallel-processing capabilities [9,10,13,[17][18][19][20][21][22]. Additionally, full blocks (and hence the multiplication of full matrices) might appear as a subproblem when handling large sparse systems of equations [14,[23][24][25].…”

Solving large dense systems of linear equations on systems with virtual memory and with cache

“…The 10 of the original algorithm is of order n 3 • It can easily be seen that the IO is reduced by a factor q (cf. Du Croz et al [7]) as opposed to the original algorithm, but it is still of order rl3 !…”

“…In [7,14,15] one can find an analysis of blocked Gaussian elimination. To keep this kind of analysis simple, one has to make some simplifications.…”

Parallel Algorithms for Dense Linear Algebra Computations

“…Calahan [149]. The block form of Version 4 was also considered in a virtual memory setting by Du Croz et al in [50] and used as a model of a block LU factorization in the BLAS3 standard proposal by Dongarra et al [39]. The curves in Fig.…”