Eigen

Language: CPP

Data

Eigen was created by Gael Guennebaud and Benoit Jacob to provide a fast, flexible, and easy-to-use library for linear algebra in C++. Its template-based design allows for efficient computations at compile-time and runtime. Eigen is highly optimized, supports arbitrary-sized matrices, and integrates seamlessly with other C++ libraries.

Eigen is a high-performance C++ template library for linear algebra, including matrices, vectors, numerical solvers, and related algorithms. It is widely used in scientific computing, machine learning, robotics, and computer graphics.

Installation

linux: sudo apt install libeigen3-dev

mac: brew install eigen

windows: Download from https://gitlab.com/libeigen/eigen and include headers in your project

Usage

Eigen allows defining fixed-size and dynamic-size matrices and vectors, performing arithmetic, solving linear systems, computing eigenvalues, performing decompositions (LU, QR, SVD), and supporting advanced operations like tensor computations.

Defining matrices and vectors

#include <Eigen/Dense>
#include <iostream>
int main() {
    Eigen::Matrix3d mat;
    mat << 1, 2, 3,
           4, 5, 6,
           7, 8, 9;
    Eigen::Vector3d vec(1, 2, 3);
    std::cout << mat << std::endl;
    std::cout << vec << std::endl;
    return 0;
}

Creates a 3x3 matrix and a 3-dimensional vector, initializing them with values and printing them.

Matrix arithmetic

Eigen::Matrix2d A;
A << 1, 2, 3, 4;
Eigen::Matrix2d B;
B << 5, 6, 7, 8;
Eigen::Matrix2d C = A + B;
std::cout << C << std::endl;

Performs element-wise addition of two 2x2 matrices.

Solving linear systems

Eigen::Matrix2d A;
A << 3, 1, 1, 2;
Eigen::Vector2d b(9, 8);
Eigen::Vector2d x = A.colPivHouseholderQr().solve(b);
std::cout << x << std::endl;

Solves a linear system Ax = b using QR decomposition.

Eigenvalues and eigenvectors

Eigen::Matrix2d A;
A << 1, 2, 2, 3;
Eigen::EigenSolver<Eigen::Matrix2d> solver(A);
std::cout << 'Eigenvalues: ' << solver.eigenvalues() << std::endl;
std::cout << 'Eigenvectors: ' << solver.eigenvectors() << std::endl;

Computes eigenvalues and eigenvectors of a 2x2 matrix.

Matrix decompositions

Eigen::Matrix3d A;
A << 1, 2, 3, 0, 1, 4, 5, 6, 0;
Eigen::FullPivLU<Eigen::Matrix3d> lu(A);
std::cout << 'Rank: ' << lu.rank() << std::endl;

Performs LU decomposition and computes the rank of a matrix.

Dynamic-size matrices

Eigen::MatrixXd mat(4,4);
mat.setRandom();
std::cout << mat << std::endl;

Defines a dynamic-size 4x4 matrix and fills it with random values.

Error Handling

Assertion failed: Occurs when matrix dimensions are incompatible. Check that operations are dimensionally correct.

Eigen decomposition fails: Ensure matrices are square when required and check that numerical stability conditions are met.

Best Practices

Use fixed-size matrices for small, performance-critical computations.

Use `.noalias()` when performing chained operations to avoid unnecessary temporaries.

Leverage built-in decompositions (LU, QR, SVD) instead of implementing your own.

Use Eigen’s expression templates for efficient vectorized operations.

Include only necessary headers (Dense, Sparse, LU, etc.) to reduce compilation times.

Official Docs Github