Eigen LDLT slower than LLT?

Question

I'm using the Cholesky module of Eigen 3 for solving a linear equation system. The Eigen documentation states, that using LDLT instead of LLT would be faster for this purpose, but my benchmarks show a different result.

I using the following code for benchmarking:

#include <iostream>
#include <chrono>
#include <Eigen/Core>
#include <Eigen/Cholesky>
using namespace std;
using namespace std::chrono;
using namespace Eigen;

int main()
{
    MatrixXf cov = MatrixXf::Random(4200, 4200);
    cov = (cov + cov.transpose()) + 1000 * MatrixXf::Identity(4200, 4200);
    VectorXf b = VectorXf::Random(4200), r1, r2;

    r1 = b;
    LLT<MatrixXf> llt;
    auto start = high_resolution_clock::now();
    llt.compute(cov);
    if (llt.info() != Success)
    {
        cout << "Error on LLT!" << endl;
        return 1;
    }
    auto middle = high_resolution_clock::now();
    llt.solveInPlace(r1);
    auto stop = high_resolution_clock::now();
    cout << "LLT decomposition & solving in  " << duration_cast<milliseconds>(middle - start).count()
         << " + " << duration_cast<milliseconds>(stop - middle).count() << " ms." << endl;

    r2 = b;
    LDLT<MatrixXf> ldlt;
    start = high_resolution_clock::now();
    ldlt.compute(cov);
    if (ldlt.info() != Success)
    {
        cout << "Error on LDLT!" << endl;
        return 1;
    }
    middle = high_resolution_clock::now();
    ldlt.solveInPlace(r2);
    stop = high_resolution_clock::now();
    cout << "LDLT decomposition & solving in " << duration_cast<milliseconds>(stop - start).count()
         << " + " << duration_cast<milliseconds>(stop - middle).count() << " ms." << endl;

    cout << "Total result difference: " << (r2 - r1).cwiseAbs().sum() << endl;
    return 0;
}

I've compiled it with g++ -std=c++11 -O2 -o llt.exe llt.cc on Windows and this is what I get:

LLT decomposition & solving in  6515 + 15 ms.
LDLT decomposition & solving in 8562 + 15 ms.
Total result difference: 1.27354e-006

So, why is LDLT slower than LLT? Am I doing something wrong or do I missunderstand the documentation?

Answer 1

This sentence of the documentation is outdated. With a recent version of Eigen, LLT should be much faster than LDLT for quite large matrices because the LLT implementation leverage cache-friendly matrix-matrix operations, while the LDLT implementation involves pivoting and matrix-vector operations only. With the devel branch your example gives me:

LLT decomposition & solving in  380 + 4 ms.
LDLT decomposition & solving in 2746 + 4 ms.