Solving system Ax=b in linear least squares fashio

Question

I would like to solve the linear system Ax = b in a linear least squares fashion, thereby obtaining x. Matrices A, x and b contain elements that are complex numbers.

Matrix A has dimensions of n by n, and A is a square matrix that is also lower triangular. Vectors b and x have lengths of n. There are as many unknowns as there are equations in this system, but since b is a vector filled with actual measured "data", I suspect that it would be better to do this in a linear least squares fashion.

I am looking for an algorithm that will efficiently solve this system in a LLS fashion, using perhaps a sparse matrix data structure for lower-triangular matrix A.

Perhaps there is a C/C++ library with such an algorithm already available? (I suspect that it is best to use a library due to optimized code.) Looking around in the Eigen matrix library, it appears that SVD decomposition can be used to solve a system of equations in a LLS fashion (link to Eigen documentation). However, how do I work with complex numbers in Eigen?

It appears that the Eigen library works with the SVD, and then uses this for LLS solving.

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Here is a code snippet demonstrating what I would like to do:

#include <iostream>
#include <Eigen/Dense>
#include <complex>

using namespace Eigen;

int main()

{

    // I would like to assign complex numbers
    // to A and b

    /*
    MatrixXcd A(4, 4);
    A(0,0) = std::complex(3,5);     // Compiler error occurs here
    A(1,0) = std::complex(4,4);
    A(1,1) = std::complex(5,3);
    A(2,0) = std::complex(2,2);
    A(2,1) = std::complex(3,3);
    A(2,2) = std::complex(4,4);
    A(3,0) = std::complex(5,3);
    A(3,1) = std::complex(2,4);
    A(3,2) = std::complex(4,3);
    A(3,3) = std::complex(2,4);
    */

    // The following code is taken from:
    // http://eigen.tuxfamily.org/dox/TutorialLinearAlgebra.html#TutorialLinAlgLeastsquares

    // This is what I want to do, but with complex numbers
    // and with A as lower triangular

    MatrixXf A = MatrixXf::Random(3, 3);
    std::cout << "Here is the matrix A:\n" << A << std::endl;
    VectorXf b = VectorXf::Random(3);
    std::cout << "Here is the right hand side b:\n" << b << std::endl;
    std::cout << "The least-squares solution is:\n"
    << A.jacobiSvd(ComputeThinU | ComputeThinV).solve(b) << std::endl;
}// end

Here is the compiler error:

 error: missing template arguments before '(' token

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UPDATE

Here is an updated program showing how to deal with the LLS solving using Eigen. This code does indeed compile correctly.

#include <iostream>

#include <Eigen/Dense>

#include <complex>


using namespace Eigen;


int main()

{

    MatrixXcd A(4, 4);
    A(0,0) = std::complex<double>(3,5);
    A(1,0) = std::complex<double>(4,4);
    A(1,1) = std::complex<double>(5,3);
    A(2,0) = std::complex<double>(2,2);
    A(2,1) = std::complex<double>(3,3);
    A(2,2) = std::complex<double>(4,4);
    A(3,0) = std::complex<double>(5,3);
    A(3,1) = std::complex<double>(2,4);
    A(3,2) = std::complex<double>(4,3);
    A(3,3) = std::complex<double>(2,4);

    VectorXcd b(4);
    b(0) = std::complex<double>(3,5);
    b(1) = std::complex<double>(2,0);
    b(2) = std::complex<double>(8,2);
    b(3) = std::complex<double>(4,8);

        std::cout << "Here is the A matrix:" << std::endl;
    std::cout << A << std::endl;

        std::cout << "Here is the b vector:" << std::endl;
        std::cout << b << std::endl;

    std::cout << "The least-squares solution is:\n"

        << A.jacobiSvd(ComputeThinU | ComputeThinV).solve(b) << std::endl;


}// end

Answer 1

Since std::complex is a template class, and you init with std::complex(1,1); the compiler doesn't know what type it is.

Use std::complex<double>(1, 1); instead.