Tutorial page 6 - Linear algebra and decompositions

This tutorial explains how to solve linear systems, compute various decompositions such as LU, QR, SVD, eigendecompositions... for more advanced topics, don't miss our special page on this topic.

Table of contents

Basic linear solving

The problem: You have a system of equations, that you have written as a single matrix equation

\[ Ax \: = \: b \]

Where A and b are matrices (b could be a vector, as a special case). You want to find a solution x.

The solution: You can choose between various decompositions, depending on what your matrix A looks like, and depending on whether you favor speed or accuracy. However, let's start with an example that works in all cases, and is a good compromise:

Example:Output: 
#include <iostream>
#include <Eigen/Dense>

using namespace std;
using namespace Eigen;

int main()
{
   Matrix3f A;
   Vector3f b;
   A << 1,2,3,  4,5,6,  7,8,10;
   b << 3, 3, 4;
   cout << "Here is the matrix A:\n" << A << endl;
   cout << "Here is the vector b:\n" << b << endl;
   Vector3f x = A.colPivHouseholderQr().solve(b);
   cout << "The solution is:\n" << x << endl;
}

   




In this example, the colPivHouseholderQr() method returns an object of class ColPivHouseholderQR. Since here the matrix is of type Matrix3f, this line could have been replaced by: 


ColPivHouseholderQR<Matrix3f> dec(A);
Vector3f x = dec.solve(b);

Here, ColPivHouseholderQR is a QR decomposition with column pivoting. It's a good compromise for this tutorial, as it works for all matrices while being quite fast. Here is a table of some other decompositions that you can choose from, depending on your matrix and the trade-off you want to make:





Decomposition Method Requirements on the matrix Speed Accuracy  



PartialPivLU partialPivLu() Invertible ++ +  



FullPivLU fullPivLu() None - +++  



HouseholderQR householderQr() None ++ +  



ColPivHouseholderQR colPivHouseholderQr() None + ++  



FullPivHouseholderQR fullPivHouseholderQr() None - +++  



LLT llt() Positive definite +++ +  



LDLT ldlt() Positive or negative semidefinite +++ ++  




All of these decompositions offer a solve() method that works as in the above example.


For example, if your matrix is positive definite, the above table says that a very good choice is then the LDLT decomposition. Here's an example, also demonstrating that using a general matrix (not a vector) as right hand side is possible.





Example:Output: 




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

using namespace std;
using namespace Eigen;

int main()
{
   Matrix2f A, b;
   A << 2, -1, -1, 3;
   b << 1, 2, 3, 1;
   cout << "Here is the matrix A:\n" << A << endl;
   cout << "Here is the right hand side b:\n" << b << endl;
   Matrix2f x = A.ldlt().solve(b);
   cout << "The solution is:\n" << x << endl;
}

  
   




For a much more complete table comparing all decompositions supported by Eigen (notice that Eigen supports many other decompositions), see our special page on this topic.



Checking if a solution really exists


Only you know what error margin you want to allow for a solution to be considered valid. So Eigen lets you do this computation for yourself, if you want to, as in this example:





Example:Output: 




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

using namespace std;
using namespace Eigen;

int main()
{
   MatrixXd A = MatrixXd::Random(100,100);
   MatrixXd b = MatrixXd::Random(100,50);
   MatrixXd x = A.fullPivLu().solve(b);
   double relative_error = (A*x - b).norm() / b.norm(); // norm() is L2 norm
   cout << "The relative error is:\n" << relative_error << endl;
}

  
   





Computing eigenvalues and eigenvectors


You need an eigendecomposition here, see available such decompositions on this page. Make sure to check if your matrix is self-adjoint, as is often the case in these problems. Here's an example using SelfAdjointEigenSolver, it could easily be adapted to general matrices using EigenSolver or ComplexEigenSolver.





Example:Output: 




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

using namespace std;
using namespace Eigen;

int main()
{
   Matrix2f A;
   A << 1, 2, 2, 3;
   cout << "Here is the matrix A:\n" << A << endl;
   SelfAdjointEigenSolver<Matrix2f> eigensolver(A);
   cout << "The eigenvalues of A are:\n" << eigensolver.eigenvalues() << endl;
   cout << "Here's a matrix whose columns are eigenvectors of A "
        << "corresponding to these eigenvalues:\n"
        << eigensolver.eigenvectors() << endl;
}

  
   





Computing inverse and determinant


First of all, make sure that you really want this. While inverse and determinant are fundamental mathematical concepts, in numerical linear algebra they are not as popular as in pure mathematics. Inverse computations are often advantageously replaced by solve() operations, and the determinant is often not a good way of checking if a matrix is invertible.


However, for very small matrices, the above is not true, and inverse and determinant can be very useful.


While certain decompositions, such as PartialPivLU and FullPivLU, offer inverse() and determinant() methods, you can also call inverse() and determinant() directly on a matrix. If your matrix is of a very small fixed size (at most 4x4) this allows Eigen to avoid performing a LU decomposition, and instead use formulas that are more efficient on such small matrices.


Here is an example: 





Example:Output: 




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

using namespace std;
using namespace Eigen;

int main()
{
   Matrix3f A;
   A << 1, 2, 1,
        2, 1, 0,
        -1, 1, 2;
   cout << "Here is the matrix A:\n" << A << endl;
   cout << "The determinant of A is " << A.determinant() << endl;
   cout << "The inverse of A is:\n" << A.inverse() << endl;
}

  
   





Least squares solving


The best way to do least squares solving is with a SVD decomposition. Eigen provides one as the JacobiSVD class, and its solve() is doing least-squares solving.


Here is an example: 





Example:Output: 




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

using namespace std;
using namespace Eigen;

int main()
{
   MatrixXf A = MatrixXf::Random(3, 2);
   cout << "Here is the matrix A:\n" << A << endl;
   VectorXf b = VectorXf::Random(3);
   cout << "Here is the right hand side b:\n" << b << endl;
   cout << "The least-squares solution is:\n"
        << A.jacobiSvd(ComputeThinU | ComputeThinV).solve(b) << endl;
}

  
   




Another way, potentially faster but less reliable, is to use a LDLT decomposition of the normal matrix. In any case, just read any reference text on least squares, and it will be very easy for you to implement any linear least squares computation on top of Eigen.



Separating the computation from the construction


In the above examples, the decomposition was computed at the same time that the decomposition object was constructed. There are however situations where you might want to separate these two things, for example if you don't know, at the time of the construction, the matrix that you will want to decompose; or if you want to reuse an existing decomposition object.


What makes this possible is that: 



For example:





Example:Output: 




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

using namespace std;
using namespace Eigen;

int main()
{
   Matrix2f A, b;
   LLT<Matrix2f> llt;
   A << 2, -1, -1, 3;
   b << 1, 2, 3, 1;
   cout << "Here is the matrix A:\n" << A << endl;
   cout << "Here is the right hand side b:\n" << b << endl;
   cout << "Computing LLT decomposition..." << endl;
   llt.compute(A);
   cout << "The solution is:\n" << llt.solve(b) << endl;
   A(1,1)++;
   cout << "The matrix A is now:\n" << A << endl;
   cout << "Computing LLT decomposition..." << endl;
   llt.compute(A);
   cout << "The solution is now:\n" << llt.solve(b) << endl;
}

  
   




Finally, you can tell the decomposition constructor to preallocate storage for decomposing matrices of a given size, so that when you subsequently decompose such matrices, no dynamic memory allocation is performed (of course, if you are using fixed-size matrices, no dynamic memory allocation happens at all). This is done by just passing the size to the decomposition constructor, as in this example: 


HouseholderQR<MatrixXf> qr(50,50);
MatrixXf A = MatrixXf::Random(50,50);
qr.compute(A); // no dynamic memory allocation


Rank-revealing decompositions


Certain decompositions are rank-revealing, i.e. are able to compute the rank of a matrix. These are typically also the decompositions that behave best in the face of a non-full-rank matrix (which in the square case means a singular matrix). On this table you can see for all our decompositions whether they are rank-revealing or not.


Rank-revealing decompositions offer at least a rank() method. They can also offer convenience methods such as isInvertible(), and some are also providing methods to compute the kernel (null-space) and image (column-space) of the matrix, as is the case with FullPivLU:





Example:Output: 




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

using namespace std;
using namespace Eigen;

int main()
{
   Matrix3f A;
   A << 1, 2, 5,
        2, 1, 4,
        3, 0, 3;
   cout << "Here is the matrix A:\n" << A << endl;
   FullPivLU<Matrix3f> lu_decomp(A);
   cout << "The rank of A is " << lu_decomp.rank() << endl;
   cout << "Here is a matrix whose columns form a basis of the null-space of A:\n"
        << lu_decomp.kernel() << endl;
   cout << "Here is a matrix whose columns form a basis of the column-space of A:\n"
        << lu_decomp.image(A) << endl; // yes, have to pass the original A
}

  
   




Of course, any rank computation depends on the choice of an arbitrary threshold, since practically no floating-point matrix is exactly rank-deficient. Eigen picks a sensible default threshold, which depends on the decomposition but is typically the diagonal size times machine epsilon. While this is the best default we could pick, only you know what is the right threshold for your application. You can set this by calling setThreshold() on your decomposition object before calling rank() or any other method that needs to use such a threshold. The decomposition itself, i.e. the compute() method, is independent of the threshold. You don't need to recompute the decomposition after you've changed the threshold.





Example:Output: 




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

using namespace std;
using namespace Eigen;

int main()
{
   Matrix2d A;
   A << 2, 1,
        2, 0.9999999999;
   FullPivLU<Matrix2d> lu(A);
   cout << "By default, the rank of A is found to be " << lu.rank() << endl;
   lu.setThreshold(1e-5);
   cout << "With threshold 1e-5, the rank of A is found to be " << lu.rank() << endl;
}

  
   











libicr

Author(s): Robert Krug

autogenerated on Mon Jan 6 2014 11:34:07