Conjugate Gradient in Eigen for non positive definite matrices

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In the description of the CG method in the Eigen library one can find the statement:

This class allows to solve for A.x = b linear problems using an iterative conjugate gradient algorithm. The matrix A must be selfadjoint.

However in the literature the conjugate gradient method is typically presented for real symmetric positive-definite matrices. Examples shows that Eigen CG actually work for non positive definite matrices that matlab pcg cannot handle.

For example running the code:

#include <cstdio>
#include <iostream>
#include <vector>
#include "Eigen/Dense"
#include "Eigen/IterativeLinearSolvers" 
#include "Eigen/Eigenvalues"

int main()
{
    srand(static_cast<unsigned int>(time(0)));
    const int N = 10;
    Eigen::Matrix<double,Eigen::Dynamic,Eigen::Dynamic,Eigen::RowMajor>  S(N,N);
    const Eigen::Matrix<double,Eigen::Dynamic,2> sources = Eigen::MatrixXd::Random(N,2);

    for(size_t iEx = 0; iEx < 4; iEx++ )
    {
        std::cout<<"EX "<<iEx<<":\n";
        if(iEx == 0)
            for(int i = 0; i < N; i++)
                for(int j = i; j < N; j++)
                    S(i,j) = S(j,i) = 1./std::sqrt((sources.row(i) - sources.row(j)).squaredNorm() +1.);
        if(iEx == 1)
            for(int i = 0; i < N; i++)
                for(int j = i; j < N; j++)
                    S(i,j) = S(j,i) = (sources.row(i) - sources.row(j)).norm();
        if(iEx == 2)
            for(int i = 0; i < N; i++)
                for(int j = i; j < N; j++)
                    S(i,j) = S(j,i) = sources.row(i).dot(sources.row(j));

        if(iEx == 3)
            S = Eigen::MatrixXd::Random(N,N).selfadjointView<Eigen::Lower>();

        Eigen::Matrix<double,Eigen::Dynamic,Eigen::Dynamic,Eigen::RowMajor> Sadj = S.selfadjointView<Eigen::Lower>();
        std::cout<<"\tIS SELFADJOINT: "<<((Sadj.array() == S.array()).all()?"YES\n":"NO\n");
        Eigen::EigenSolver< Eigen::Matrix<double,Eigen::Dynamic,Eigen::Dynamic,Eigen::RowMajor> > eigensolver(S);
        std::cout<<"\tNUMBER OF NEGATIVE EIGEN VALUES: "<<(eigensolver.eigenvalues().real().array() < 0.).count()<<" OUT OF "<<N<<"\n";

        const Eigen::Matrix<double,Eigen::Dynamic,1> xExact = Eigen::VectorXd::Ones(N);
        const Eigen::Matrix<double,Eigen::Dynamic,1> b = S * xExact;

        Eigen::ConjugateGradient< Eigen::MatrixXd, Eigen::Lower|Eigen::Upper, Eigen::IdentityPreconditioner> cg(S);
        cg.setMaxIterations(3000);
        cg.setTolerance(1e-10);

        const Eigen::Matrix<double,Eigen::Dynamic,1> xSol = cg.solve(b);
        std::cout<<"\tITERATIONS       : " << cg.iterations() << "\n";
        std::cout<<"\tESTIMATED ERROR  : " << cg.error()      << "\n";

        std::cout<<"\tNORM 2 ERROR     : "<<(xExact-xSol).norm()<<"\n";
        std::cout<<"\tNORM 2 AVG ERROR : "<<(xExact-xSol).norm()/static_cast<double>(N)<<"\n";
        std::cout<<"\tNORM INF ERROR   : "<<(xExact-xSol).lpNorm<Eigen::Infinity>()<<"\n";
        std::cout<<std::flush;
    }
}

Gives the output:

EX 0:
        IS SELFADJOINT: YES
        NUMBER OF NEGATIVE EIGEN VALUES: 0 OUT OF 10
        ITERATIONS       : 11
        ESTIMATED ERROR  : 1.01319e-11
        NORM 2 ERROR     : 2.49293e-10
        NORM 2 AVG ERROR : 2.49293e-11
        NORM INF ERROR   : 1.20759e-10
EX 1:
        IS SELFADJOINT: YES
        NUMBER OF NEGATIVE EIGEN VALUES: 9 OUT OF 10
        ITERATIONS       : 10
        ESTIMATED ERROR  : 2.43788e-12
        NORM 2 ERROR     : 1.77969e-11
        NORM 2 AVG ERROR : 1.77969e-12
        NORM INF ERROR   : 8.2061e-12
EX 2:
        IS SELFADJOINT: YES
        NUMBER OF NEGATIVE EIGEN VALUES: 4 OUT OF 10
        ITERATIONS       : 1
        ESTIMATED ERROR  : 1.72812e-16
        NORM 2 ERROR     : 2.97281
        NORM 2 AVG ERROR : 0.297281
        NORM INF ERROR   : 1.45547
EX 3:
        IS SELFADJOINT: YES
        NUMBER OF NEGATIVE EIGEN VALUES: 5 OUT OF 10
        ITERATIONS       : 9
        ESTIMATED ERROR  : 7.73713e-14
        NORM 2 ERROR     : 8.55003e-14
        NORM 2 AVG ERROR : 8.55003e-15
        NORM INF ERROR   : 5.29576e-14

Example 0 is a positive definite matrix. Examples 1 2 3 are symmetric NON positive definite matrices. Examples 1 and 3 are solved correctly whereas example 2 fails.

The implementation looks similar to classical CG implementations.

QUESTIONS: Is there any trick in Eigen that allows handling non-positive definite matrices? Is example 2 not respecting some requirements in order to be solved by Eigen with CG?

eigen linear-algebra matrix
1 Answers

The CG can be used for solving systems with matrices that aren't positive definite and symmetric, in the following way: the CG algorithm must be applied to the system [A]T[A]x=[A]Tb where [A]T stands for the transposed matrix. In that case [A]T[A] is symmetric and positive definite unless [A] is singular. A drawback is that [A]T[A] has the original matrix condition ratio squared, so if cond([A]) exceeds appox. 10e7, the CG iterations are unlikely to converge at all, and/or the resulting vector x may not have any significant digits. If your matrix is reasonably "good" in numerical sense, say cond([A]) doesn't exceed about 10e3 or 10e4, you may expect the iterations to converge, and the solution will have several significant digits. The following publication contains source codes implementing such algorithm: https://www.amazon.com/Solution-Systems-Algebraic-Equations-Matrices/dp/0646990454

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