Gaussian Elimination

Gaussian Elimination By Yequn Zhang, Yu Zhang

Contents • Introduction • Problem Analysis • Proposed Algorithm • Evaluation

GaussianElimination • Forward Elimination • Back Substitution

Contents • Introduction • ProblemAnalysis • Proposed Algorithm • Evaluation

Problem Analysis • Data size used by kernels changes continuously • Difficult to find an appropriate block size to avoid divergence • Block-based approach • Assign a certain part of computation running on CPU-leave the irregularity to cpu • Manually make the data size changes with a step of block size • Block number per grid is easy to set

Contents • Introduction • Problem Analysis • ProposedAlgorithm • Evaluation

Forward Elimination • A block-based approach • Try to avoid divergence • Try to use GPU • Try to be fine-grained

K 1 Find Max Row

Now start to eliminate the block of data on cpu cpu Swap

Calculate coefficients

Elimination on CPU

K 1 Calculate Coefficients

K 2 K2 Elimination on CPU

K3 K 3 Swap on GPU

K4 K 4 Elimination on GPU

K5 K 5 Elimination on GPU

Intra-blockloop

Inter-block loop

Last inter-block loopprocessedon CPU

BackSubstitution • Launch kernel when number of coefficients per row exceeds four block size (64*4=256) • A fine-grained way, use a similar way as forward elimination, part on CPU and part on GPU

Contents • Introduction • Problem Analysis • Proposed Algorithm • Evaluation

Block size effect

The contribution of swap and find max row • Is it necessary to implement every part on GPU?

Performance breakdown • Contribution of each part to the total performance,including kernels as well as CPU part

Speedup

Questions ?

Gaussian Elimination