QR algorithm implementation 3x3 fixed point

Question

I want to find the eigenvalues and eigenvectors of a 3x3 matrix (mostly if not always symmetric!!). My numbers are stored in fixed-point format (16.16 to be exact).

Note that I don't mind much about the performance, but simply implementing an algorithm that does the job.

The code below, when you build it and run it (with libfixmath library) produces the right eigenvalues but not the correct eigenvectors.

If I understand correctly the algorithm the eigenvectors are the product of all the calculated Qs.

Does anyone know about what might be wrong? (Even correction about the code (writing style etc) anything you can think of, but ofcourse try to center your mind on the eigenvectors! :) :P

The actual loop is like 3 lines...and what it does is this:

eigenvectors = identity matrix

1) QR decomposition A = Q*R

2) Anew = R*Q (multiply the factors in the reverse order, and iterate)

3) eigenvectors = eigenvectors * Q

Thank you!! Oh and C newbie here....

the code :

Answer 1

I have used for many years and with several benefits LAPACK library ,

To compare results, you could use R (that uses the LINPACK routine DQRDC2 as a default)m even MATLAB.

And then you can use the qr() command in R with the LAPACK=TRUE option to use a LAPACK routine :

> QR <- qr(Mat,LAPACK=TRUE)
> QR

However, you should be aware that the function qr() in this case uses the LAPACK routine DGEQP3. Contrary to the DGEQRF routine you're using, DGEQP3 calculates the QR decomposition of a matrix with column pivoting.

If you get different results it could be that you're not using the same methods (do not know what method uses the code you posted).

You should remember that a QR decomposition is not a unique solution. To know whether your QR decomposition is correct, you can simply check if the Q and R matrices fulfill the requirements. eg in R :

> Q <- qr.Q(QR)
> round( t(Q) %*% Q , 10 )
     [,1] [,2] [,3] [,4]
[1,]    1    0    0    0
[2,]    0    1    0    0
[3,]    0    0    1    0
[4,]    0    0    0    1

> all.equal(Q %*% qr.R(QR),Mat)
[1] TRUE