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

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*To*: eigen@xxxxxxxxxxxxxxxxxxx*Subject*: Re: [eigen] request for help: 4x4 matrix inverse*From*: Benoît Jacob <jacob@xxxxxxxxxxxxxxx>*Date*: Tue, 15 Apr 2008 14:18:48 +0200*Cc*: Konstantinos Margaritis <markos@xxxxxxxx>

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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**References**:**[eigen] request for help: 4x4 matrix inverse***From:*Benoît Jacob

**Re: [eigen] request for help: 4x4 matrix inverse***From:*Konstantinos Margaritis

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