NVIDIA GPU Calculator to calculate largest eigenvalue of a huge matrix using shift-inverse method

Author: Damodar Rajbhandari (2023-Jan-01 - Last Update: 2023-Feb-21)

Usage

# Shifted-inverse power method using cuSolver
make
./maxeigenvalue mtxs/L11.mtx

# Power method using cuSparse and thrust
make mainpower
./maxeigenvaluepower mtxs/L11.mtx

# Compile Shifted-inverse power method, power method, and Spectra library
make all

Software Dependencies

• nvcc compiler supporting c++14 to compile the codebase
• cuSparse (in CUDA Toolkit) for sparse matrix operations
• cuSolver (in CUDA Toolkit) to find largest eigenvalue
• thrust (in CUDA Toolkit) for parallel data-structure & algorithms
• NVIDIA Nsight for vscode extension (Optional) for debugging and profiling purposes

Hardware Requirements

• Code ran on NVIDIA GeForce RTX 2080 Ti and CUDA Version 11.7. Check yours using nvidia-smi -q command.

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CPU Result Comparison

• Used Spectra version 1.0.1 on the top of Eigen3 version 3.4.0
• Code ran on Intel(R) Core(TM) i9-10980XE CPU @ 3.00GHz with 125GB RAM and Arch GNU/Linux x86-64 with Linux kernel: 5.15.41-1-lts. Check yours using these commands: lscpu to get CPU details, free -g -h -t to get RAM details, and cat /etc/os-release OS details.

Usage

make mainspectra
./maxeigenvaluespectra mtxs/dL22.mtx

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Performance Comparison of GPU Power Method Vs Spectra Library

• Please install hyperfine. It is a command-line benchmarking tool.
  • It can be installed via conda from the conda-forge channel:

    conda install -c conda-forge hyperfine
    

Here are the results:

• Using power method on GPU (unoptimized code, without using tolerance for convergence. See: computeMaxEigenvaluePowerMethod)

  hyperfine './maxeigenvaluepower mtxs/dL22.mtx'
  

  • Results:

    Benchmark 1: ./maxeigenvaluepower mtxs/dL22.mtx
      Time (mean ± σ):     14.282 s ±  0.043 s    [User: 12.608 s, System: 1.569 s]
      Range (min … max):   14.241 s … 14.373 s    10 runs
    

• Using power method on GPU (optimized code, using tolerance for convergence. See computeMaxEigenvaluePowerMethodOptimized)

  hyperfine './maxeigenvaluepower mtxs/dL22.mtx'
  

  • Results:

    Benchmark 1: ./maxeigenvaluepower mtxs/dL22.mtx
      Time (mean ± σ):     13.782 s ±  0.038 s    [User: 12.112 s, System: 1.569 s]
      Range (min … max):   13.726 s … 13.873 s    10 runs
    

• Using Spectra library on CPU

  hyperfine './maxeigenvaluespectra mtxs/dL22.mtx'
  

  • Results:

    Benchmark 1: ./maxeigenvaluespectra mtxs/dL22.mtx
      Time (mean ± σ):      2.485 s ±  0.012 s    [User: 2.478 s, System: 0.007 s]
      Range (min … max):    2.466 s …  2.506 s    10 runs
    

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Notes

• maxeigenvalue which is based on cusolverSpScsreigvs doesnot work for larger matrices. For example: mtx/dL22.mtx.
• maxeigenvaluepower doesnot work for mtx/dL00.mtx or mtx/L00.mtx if you use computeMaxEigenvaluePowerMethod but for others in mtx/ directory; it works fine. This is because we set the initial vector x_i sets to 1.0 for all its elements. This initial vector may gives rise to an orthogonal vector with eigenvector for some matrices. Ideally choosing a random vector such that its norm is 1 and entries is mostly non-zero (because $Ax = 0$ if $x$ is $0$) allows the chance to decrease that our vector is orthogonal to the eigenvector. It is done in computeMaxEigenvaluePowerMethodOptimized function.