Hello,

im new to eigen and i try to solve this equation system:

1 -3 2 | 8
-1 3 -4 | -16

The expected solution after some pivots is:

1 -3 0 | 0
0 0 1 | 4
....
1 -3 0 | 0
0 -1 0 | 0
0 0 1 | 4
...
x1=0+3t
x2=0+1t
x3=4

So i've written this little program and it works fine:

MatrixXd A(2,3);
VectorXd b(2);
A << 1,-3,2, -1,3,-4;
b << 8,-16;
VectorXd x(3);
x = A.fullPivHouseholderQr().solve(b);
cout << "The solution is:\n" << x << endl;

...
The solution is:
0
-0
4
And this is the expected right hand side!

But how can i get the left-hand-side to see the second column?

1 -3 0
0 -1 0
0 0 1

Thanks,
Ralf

If you want an expression of the full space of solution, you can use FullPivLU::kernel(): http://eigen.tuxfamily.org/dox/classEig ... 13b6e7200a. The space of solution is then:

v0+lu.kernel()*x

where v0 is one solution (e.g., obtained using FullPivLU::solve) and x is any vector of size lu.kernel().cols().

Thank you! That helped.

Is it possible to see the whole left hand side solution like

1 -3 0
0 -1 0
0 0 1

?

Ralf

This is the matrix U, which you can get as in the topmost example of http://eigen.tuxfamily.org/dox/classEig ... PivLU.html . However, Eigen uses a more sophisticated algorithm which also allows for column swaps, as indicated by "full pivoting", which means you will not get the same left-hand side matrix. Instead I got

-4 3 -1
0 -1.5 0.5

We can reconstruct what Eigen does. Starting from your original matrix, Eigen swaps the rows and also first and third column, giving

-4 3 -1
2 -3 1

and then it adds 0.5 times the first row to the second row, which gives the solution as above (the first matrix in this post).