This library provides fast algorithms for computing the exponential of dense matrices in C++.

Class MatrixExponential:

• an optimized version of the scaling-and-squaring algorithm (Higham 2005 "The scaling and squaring method for the matrix exponential revisited") for computing the matrix exponential based on the code found in Eigen3-unsupported;
• options for the user to customize the amount of computation performed by the algorithm to achieve the desired performance-accuracy trade-off (e.g. specifying the maximum number of matrix-matrix multiplications);
• an algorithm for computing the product between the matrix exponential and a given vector/matrix, which is more efficient than computing the matrix exponential and multiplying it times the vector/matrix (assuming the matrix is skinny);
• a balancing algorithm to speed-up the matrix exponential computation;

Class LDSUtility:

• a utility class for computing the evolution of the state of a linear dynamical system, and its first two integrals, all of which are based on the computation of matrix exponentials.

All the C files:

• a C version of the library expokit (Sidje, 1998 "Expokit: A Software Package for Computing Matrix Exponentials", ACM Transactions on Mathematical Software), which is a Fortran library that we translated to C using f2c.

In this file you can find a test comparing all the implemented methods for computing the matrix exponential. We have found that for the matrix sizes we were interested in, our optimized version of the scaling-and-squaring algorithm based on Eigen3-unsupported was the fastest. For example, using a matrix size of 24, these were the computation times we obtained:

Our MatrixExponential class: 0.068 ms
Eigen (from eigen3-unsupported): 0.079 ms
DGPADM (from expokit): 0.160 ms
DGCHBV (from expokit): 0.094 ms

Our MatrixExponential class - computeExpTimesVector: 0.044 ms

1. f2c - used in porting routines from Fortran
2. eigen3 - Linear Algebra library
3. lapack - Linear Algebra library

First of all install f2c :

sudo apt install f2c

Then install eigen3, which can be installed from different sources. We propose here to install it from robotpkg using:

sudo apt install robotpkg-eigen3

To compile expokit you need to provide to pkg-conifg the path both to eigen3.pc and f2c.pc. The location of the former depends on how eigen3 was installed. We are assuming here it was installed from robotpkg.

Regarding f2c, the current package provided by apt does not include the f2c.pc file, but we provide our own that can be found in this repo at pkg-config/f2c.pc.

The lapack dependency should be resolved by the build system. Summing up your PKG_CONFIG_PATH should look like this:

export PKG_CONFIG_PATH=/opt/openrobots/lib/pkgconfig:$PKG_CONFIG_PATH
export PKG_CONFIG_PATH=/opt/openrobots/share/pkgconfig:$PKG_CONFIG_PATH
export PKG_CONFIG_PATH=<path to f2c.pc>:$PKG_CONFIG_PATH

Substitute <path to f2c.pc> with your path, it can be located anywhere.

To cite Expokit-cpp in your academic research, please use the following bibtex line:

@article{Hammoud2022,
author = {Hammoud, Bilal and Olivieri, Luca and Righetti, Ludovic and Carpentier, Justin and {Del Prete}, Andrea},
doi = {10.1007/s11044-022-09818-z},
issn = {1573-272X},
journal = {Multibody System Dynamics},
number = {4},
pages = {443--460},
title = {{Exponential integration for efficient and accurate multibody simulation with stiff viscoelastic contacts}},
url = {https://doi.org/10.1007/s11044-022-09818-z},
volume = {54},
year = {2022}
}