RE: [Algorithms] matrix inverse

[mark_me <ma...@so...>]

Transposing the 3x3 orthonormal matrix is always a good idea , and should be
used if you know for sure that the matrix is orthonormal ( specially in
articulated objects like skeletons and that stuff ) it's very fast and
stable , simply because there is no operation that makes you lose precision.

For the translation part of the 4x4 matrix , you need to transform it by the
transposed 3x3 matrix first then negate it. if we take a 2X2 rotation with
translation as a simple example :

	|A	-B	0|
	|B  	 A	0|
	|X  	 Y	1|

The  Inverse would be :

	|A	 	B		0|
	|-B  	 	A		0|
	|-AX+BY  	-BX-AY	1|


The same for 3x3 with translation.


-----Original Message-----
From: gda...@li...
[mailto:gda...@li...]On Behalf Of Peter
Warden
Sent: February 6, 2002 1:28 PM
To: 'Algorithms List'
Subject: RE: [Algorithms] matrix inverse


Since we're discussing speeding up matrix inversions:

In cases where you have 'simple' orthogonal 3x3 matrices containing only
rotations and no scaling or skewing, the inverse is just the transpose
of the original matrix. One thing I have found occasionally useful in
the past is extending this to simple 4x4 matrices containing just a
rotation and a translation by transposing the 3x3 rotation part of the
matrix as before.  Then feed the original translation through this
transpose and multiply it by -1 to get the inverted translation. (IIRC?
I don't have the code in front of me)

Of course you should only use this if your profiler is telling you that
doing the inverse properly is a performance problem _and_ you're sure
that the matrices you're dealing with are orthogonal. There may be other
problems with this approach, YMMV, etc, but it did the job for me at the
time.

Pete

-----Original Message-----
From: gda...@li...
[mailto:gda...@li...] On Behalf Of
Idahosa Edokpayi
Sent: Wednesday, February 06, 2002 9:39 AM
To: Algorithms List
Subject: RE: [Algorithms] matrix inverse


I wanted it for an explicit inverse. I actually only use the inverse for
the camera so yes that is what I wanted.

Idahosa Edokpayi

-----Original Message-----
From: gda...@li...
[mailto:gda...@li...]On Behalf Of
Willmott, Andrew
Sent: Tuesday, February 05, 2002 9:46 PM
To: Algorithms List
Subject: RE: [Algorithms] matrix inverse


> Well, I have contraditory results from Intel:
>
> http://developer.intel.com/design/pentiumiii/sml/24504301.pdf
>
> Cramer's rule seems pretty efficient even with a 4x4. When coupled
> with SIMD, it totally rocks!

Note that these results aren't inconsistent: Graham's link is measuring
Cramer's rule against Gaussian elimination for solving Ax = b, whereas
the Intel guys are measuring it against computing the entire inverse
iteratively. It's not surprising elimination does better for the first
case.

FWIW I use cramer's rule for 3x3 and 4x4 matrix inversion and swap to
iterative methods thereafter. As everyone else has said, you should be
using elimination to solve most Ax = b cases, but having to have an
explicit inverse does crop up a fair bit in camera calculations, enough
to care about having a fast explicit inverse.

Cheers,

Andrew

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