Welcome to HODLR3Dlib’s Documentation!

About \(\texttt{HODLR3Dlib}\):

HODLR3D (Hierarchical Off Diagonal Low Rank 3D) is a new class of Hierarchical matrix representation for matrices arising out of N-body problems in 3D.

\(\texttt{HODLR3Dlib}\) is a library to compute fast matrix-vector products using HODLR3D.

It relies on the fact that certain off-diagonal blocks in the matrix can be well-approximated by low-rank matrices.

The domain has been hierarchically partitioned using a uniform oct-tree.

The low-rank approximation of the appropriate blocks is obtained using ACA.

The code is written in C++ and features an easy-to-use interface, where the user provides the following inputs:

• a kernel object which abstracts data of the matrix through a member function getMatrixEntry(int i, int j) which returns the entry at the \(i^{\mathrm{th}}\) row and \(j^{\mathrm{th}}\) column of the matrix.

• locations of nodes in the domain through a std::vector particles

• the vector b to be multiplied to the matrix

It obtains \(A x\) in almost linear complexity.

Doc Contents

Contents:

• Installation and Building
  • Downloading the Source
  • Dependencies
  • Installation
  • Testing
  • Building and Executing
• Tutorial
  • Setting Parameters in CMakeLists.txt
  • Running the program::
  • Creating a Derived Class of kernel:
  • Defining location of particles:
  • Defining vector b:
  • Creating the Instance of HODLR3D:
• Benchmarks
• Reproducibility
  • Numerical rank for different kernels
  • Numerical benchmarks of HODLR3D matrix-vector product in comparison with those of HODLR and \(\mathcal{H}\) matrix-vector products for the kernel \(\frac{1}{r}\)
  • Numerical benchmarks of HODLR3D matrix-vector product in comparison with those of HODLR and \(\mathcal{H}\) matrix-vector products for the kernel \(\frac{1}{r^4}\)
  • Numerical benchmarks of HODLR3D matrix-vector product in comparison with those of HODLR and \(\mathcal{H}\) matrix-vector products for the kernel \(\frac{cos(r)}{r}\)
  • Numerical benchmarks of the HODLR3D accelerated iterative solver for the integral equation in comparison with those of HODLR and \(\mathcal{H}\)
  • Numerical benchmarks of parallel HODLR3D matrix-vector product using MPI
• Various interactions in 3D and their ranks

Other Links

Learn more about \(\texttt{HODLR3Dlib}\) by visiting the

• Code Repository: https://github.com/SAFRAN-LAB/HODLR3D

• Documentation: https://hodlr3d.readthedocs.io/en/latest/