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template: titleslide

Using Eigen

Chris Richardson, Rupert Nash

[email protected], [email protected]

CC-BY

What is Eigen3 and why use it?

  • C++ library for matrix arithmetic
  • “Header only” implementation: no libraries to compile and install (easy)
  • Provides some “missing” features needed for scientific computing in C++
  • Contains optimisations to get good performance out of ARM & Intel processors
  • Easy to use interface
  • Support for dense and sparse matrices, vectors and “arrays”
  • Some support for ‘solvers’ (A.x = b)
  • Download from https://eigen.tuxfamily.org or e.g. apt install libeigen3-dev
  • If you know Python, it is a bit like a NumPy for C++

Basics

#include <Eigen/Dense>

int main()
{
   Eigen::Matrix<double, 10, 10> A;
   A.setZero();
   A(9, 0) = 1.234;
   std::cout << A << std::endl;
   return 0;
}

This is pretty similar to double A[10][10];

Dynamic size

int n = 64;
int m = 65;
Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic> A(n, m);

A.resize(20, 20);

std::cout << “Size is ”;
std::cout << A.rows() << “ x ” << A.cols() << std::endl;

So this is more like a 2D version of std::vector<double>

Convenience aliases

using Eigen::Matrix3d = Eigen::Matrix<double, 3, 3>
using Eigen::Matrix3i = Eigen::Matrix<int, 3, 3>
using Eigen::MatrixXd = Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic>
using Eigen::VectorXd = Eigen::Matrix<double, Eigen::Dynamic, 1>
using Eigen::RowVectorXd = Eigen::Matrix<double, 1, Eigen::Dynamic>

You can do Matrix arithmetic...

Eigen::MatrixXd A(5, 10);
Eigen::MatrixXd B(10, 2);
Eigen::VectorXd vec(10);

Eigen::MatrixXd C = A * B;
Eigen::VectorXd w = A * vec;

Also dot and cross products for vectors, transpose, and usual scalar arithmetic + - * /

Element-wise ops with Arrays

For example:

Eigen::Matrix3d A, B;
// set all values to 2.0
A.array() = 2.0;

// set each element of A to sin() of the same element in B
A.array() = B.array().sin();

Eigen::Array3d W;
W = W * A;  // Error - cannot mix Array with Matrix

View other data as a Matrix or Array

Mix and match with std::vector or any contiguous layout

It is easy to “overlay” existing memory with an Eigen Array or Matrix:

std::vector<double> a(1000);

Eigen::Map<Eigen::Matrix<double, 100, 10>> a_eigen(a.data());

a_eigen(10, 0) = 1.0;

Eigen::Map<Eigen::MatrixXd> a2_eigen(a.data(), 10, 100);

Efficiency: Eigen does lots of checks in debug mode

Turn optimisation on: -O2 etc.

Turn off debug: -DNDEBUG

Walk through example

Solve diffusion equation

$$\frac{\partial T}{\partial t} = k \frac{\partial^2 T}{\partial x^2}$$

in 1D using an explicit method

Each timestep can be solved by using \(T_{n+1} = A T_n\)

  1. Create an initial vector for T
  2. Create a dense matrix for A
  3. Matrix multiply several times
  • Convert to an implicit method: \(A.T_{n+1} = T_n\)
  • Sparse matrices

Type along with me - see exercises/eigen

Diffusion equation (explicit)

Subscript means time, square brackets position

$$T_{n+1}[i] = T_n[i] + \frac{k \Delta t}{\Delta x^2}(T_n[i-1] - 2T_n[i] + T_n[i+1])$$

Our matrix equation is now:

$$T_{n+1} = A.T_{n}$$

Left-hand side is unknown (next time step)

Let: \(\delta = k\Delta t/ \Delta x^2\)

Matrix

A = [ 1-ẟ ẟ    0    0    0 …        ]
    [ ẟ   1-2ẟ ẟ    0    0 …        ]
    [ 0   ẟ    1-2ẟ ẟ    0 …        ]
    [ 0   0    ẟ    1-2ẟ ẟ    0  0 …] 
    [ 0   0    0    ẟ    1-2ẟ ẟ  0 …]

Diffusion equation (implicit)

Subscript means time, square brackets position

$$T_{n+1}[i] - \frac{k \Delta t}{\Delta x^2} (T_{n+1}[i-1] - 2*T_{n+1}[i] + T_{n+1}[i+1]) = T[i]$$

Left-hand side is unknown (next time step) $$A.T_{n+1} = T_n$$

Let: $$\delta = k \Delta t/\Delta x^2$$

??? The matrix A is very similar - just flip the sign of the delta terms