GPU Acceleration of Dense Matrix and Block Operations for Lanczos Method for Systems over Large Prime Finite Field

Abstract

GPU based acceleration of computations with dense matrices and blocks over large prime finite field are studied. Particular attention is paid to the following algorithms:

• multiplication of rectangular $N \times K$ blocks with $N \gg K;$

• multiplication of $N \times K$ blocks by square $K \times K$ matrices;

• LU-decomposition of matrices.

Several approaches for optimal use of GPU resources are proposed.

Efficiency analysis of implemented algorithms is provided for prime finite field with number of elements about $2^{512},$ $2^{768},$ $2^{1024}$ and GPUs of different computational performance and architecture generations. Numerical experiments prove efficiency of proposed solutions.

From numerical results it follows that GPU usage allows to accelerate block operations and to expand area of almost linear parallel scalability of Lanczos method implementation by INM RAS. Moreover, a sparse system of size about 2 millions, with 82 average nonzero elements per row, over field with about $2^{512}$ elements, on 128 nodes of Lomonosov supercomputer will be solved 2 times faster in case of GPUs used.