Re: [eigen] request for help: 4x4 matrix inverse

[Benoît Jacob <jacob@xxxxxxxxxxxxxxx>]

Well, thinking further about this: I do have a problem, since when P is not 
invertible I don't know how to compute M^{-1} then!

Cheers,

Benoit

On Monday 14 April 2008 19:50:47 Konstantinos Margaritis wrote:
> On Monday 14 April 2008, Benoît Jacob wrote:
> > Hi List,
>
> Hi all, this is my first msg to the list!
>
> > I have started coding the LU module and committed already this:
> > - optimized determinants for sizes <=4
> >   (Thanks to Geoff for pointing me to the 30-mul technique for size
> > 4). - general matrix inversion (good for sizes >= 5)
> > - bruteforce matrix inversion using cofactors for sizes <=3
> > - unrolled version of the general method for size 4
> >
> > Since the latter unrolled version is clumsy and is only used for
> > size 4, since after all it seems that a bruteforce (cofactor)
> > technique would be faster, and since size 4 is a common case that
> > we want to optimize, I am asking:
> >
> > do you have good 4x4 matrix inversion code that you could
> > contribute? Or can you point me to a good "algorithm" ? I put
> > quotation marks because we know the algorithm is just brutefore,
> > the question is how to implement it to minimize the number of ops
> > and perhaps to take advantage of vectorization...
> >
> > Markos: if I remember correctly, you mentioned you had AltiVec code
> > for that. When you mentioned it I was still hoping that the
> > unrolled gaussian elimination would be good but now I realize that
> > it's not, so I'm very much interested in your code if you want to
> > contribute it :) Extra points if it also works with SSE.
>
> Might I suggest choosing the matrix partitioning/matrix blocks/Liebniz
> formula for determinants, etc (don't remember the exact name, I got
> it from Numerical Recipes in C).
>
> Anyway it goes like this:
>            | a1 a2 a3 a4 |
>
> Assume M = | b1 b2 b3 b4 |
>
>            | c1 c2 c3 c4 |
>            | d1 d2 d3 d4 |
>
> We partition the matrix in 4 2x2 matrices:
>
> M= | P Q |
>
>    | R S |
>
> where P = | a1 a2 |, Q = | a3 a4 |, R = | c1 c2 |, S = | c3 c4 |
>
>           | b1 b2 |      | b3 b4 |      | d1 d2 |      | d3 d4 |
>
> (actually the method is generic for any NxM matrix, but it becomes
> quite more complicated in these cases).
> So, M's determinant is calculated thus:
>
> detM = detP * det(S - R*P1*Q)
>
> ( P1 = P^(-1) )
>
> (so if detP = 0, detM = 0 and M has no inverse, but detP is calculated
> in just one step!)
>
> And M^(-1) is calculated thus:
>
> M^(-1) = | P' Q' |
>
>          | R' S' |
>
> where
>
> P' = P1 + (P1*Q) * (S - R*P1*Q)*(R*P1)
> Q' = -(P1*Q)*(S - R*P1*Q)
> R' = -(S - R*P1*Q)*(R*P1)
> S' = (S - R*P1*Q)^(-1)
>
> Notice the repeated expression (S - R*P1*Q). This is much easier to
> vectorize than LU decomposition or Gaussian or the coefficients
> methods. In fact I'm writing exactly this version for AltiVec right
> this moment for another project I'm contributing to (SIMDx86 math
> library, this is just for 3D stuff and is not intented to be as
> generic as Eigen). I'll post the code when I'm finished but it should
> not be hard for someone with good SSE experience to come up with this
> version.
>
> > I am also -- and most importantly -- looking for an optimized
> > non-vectorized path!
> >
> > As you'll see in src/LU/Inverse.h, we have a template parameter
> > bool CheckExistence. If true, we must carefully check that the
> > inverse exists. Ideally, your implementation would do that in an
> > optimized way when CheckExistence==true, and we would not pay any
> > cost when
> > CheckExistence==false.
>
> With this method, CheckExistence should just check the detP and
> det(S - R*P1*Q), which should be quite fast to compute :)
>
> Konstantinos
>
> > The API is:
> >
> > Matrix4d m;
> > m.inverse(); // CheckExistence==true
> > m.quickInverse(); // CheckExistence==false
> >
> > Inverse<Matrix4d> inverse_of_m(m);
> >   // or, if you prefer, Inverse<Matrix4d> inverse_of_m =
> > m.inverse(); if(inverse_of_m.exists())
> > {
> > 	cout << inverse_of_m << endl;
> > }
> >
> > Cheers,
> >
> > Benoit

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