From: Rik Cabanier <cabanier@gmail.com>

Date: Thu, 21 Mar 2013 11:26:26 -0700

Message-ID: <CAGN7qDDvJnR=siRvE+iPaxw29pe_sSSFePs6zu51Lg8S_aV8-A@mail.gmail.com>

To: Benoit Jacob <jacob.benoit.1@gmail.com>

Cc: public-fx@w3.org

Date: Thu, 21 Mar 2013 11:26:26 -0700

Message-ID: <CAGN7qDDvJnR=siRvE+iPaxw29pe_sSSFePs6zu51Lg8S_aV8-A@mail.gmail.com>

To: Benoit Jacob <jacob.benoit.1@gmail.com>

Cc: public-fx@w3.org

On Thu, Mar 21, 2013 at 7:38 AM, Benoit Jacob <jacob.benoit.1@gmail.com>wrote: > > > 2013/3/21 Benoit Jacob <jacob.benoit.1@gmail.com> > >> >> >> 2013/3/21 Rik Cabanier <cabanier@gmail.com> >> >>> >>> >>> On Wed, Mar 20, 2013 at 8:38 PM, Benoit Jacob <jacob.benoit.1@gmail.com>wrote: >>> >>>> >>>> >>>> 2013/3/20 Rik Cabanier <cabanier@gmail.com> >>>> >>>>> >>>>> >>>>> On Wed, Mar 20, 2013 at 6:29 PM, Benoit Jacob < >>>>> jacob.benoit.1@gmail.com> wrote: >>>>> >>>>>> >>>>>> >>>>>> 2013/3/20 Rik Cabanier <cabanier@gmail.com> >>>>>> >>>>>>> >>>>>>> >>>>>>> On Tue, Mar 19, 2013 at 6:38 PM, Benoit Jacob < >>>>>>> jacob.benoit.1@gmail.com> wrote: >>>>>>> >>>>>>>> Hi, >>>>>>>> >>>>>>>> Seeing that a matrix API was being discussed ( >>>>>>>> https://dvcs.w3.org/hg/FXTF/raw-file/default/matrix/index.html), I >>>>>>>> thought I'd take a look and chime in. >>>>>>>> >>>>>>>> Here are the first few things that scare me, glancing at the >>>>>>>> current draft: >>>>>>>> >>>>>>>> 1. Some functions like inverse() throw on singular matrices. The >>>>>>>> problem is it's impossible to define very firmly what singular means, in >>>>>>>> floating-point arithmetic --- except perhaps by prescribing a mandatory >>>>>>>> order in which the arithmetic operations inside of inverse() are performed, >>>>>>>> which would be insane --- so saying that inverse() throws on singular >>>>>>>> matrices means in effect that there are realistic matrices on which it is >>>>>>>> implementation-defined whether inverse() throws or not. The consensus in >>>>>>>> all the serious matrix libraries that I've seen, for closed-form matrix >>>>>>>> inversion, is to just blindly perform the computation --- in the worst case >>>>>>>> you'll get Inf or NaN values in the result, which you will have anyway on >>>>>>>> some input matrices unless you mandate unduly expensive checks. More >>>>>>>> generally, I would never throw on singular-ness in any function, and in the >>>>>>>> case of 4x4 matrices and closed-form computations I wouldn't attempt to >>>>>>>> report on singular-ness otherwise than by inf/nan values. >>>>>>>> >>>>>>> >>>>>>> Are you suggesting that the user should check all the values to make >>>>>>> sure that the matrix inversion didn't fail? That seems very expensive. >>>>>>> Can you point us to an algorithm for inversion that you think the >>>>>>> spec should include? >>>>>>> >>>>>> >>>>>> No, I'm saying that we should _not_ check anything, or if we do (say >>>>>> in a separate "checked inverse" method), it should be in a graceful way >>>>>> e.g. an output boolean parameter, not throwing an exception which by >>>>>> default halts the program. That's what I meant by "The consensus[...] is to >>>>>> just blindly perform the computation --- in the worst case you'll get Inf >>>>>> or NaN values in the result" above. >>>>>> >>>>> >>>>> That's certainly tempting! I can see how that's easier (and less >>>>> disrupting). >>>>> >>>>> I checked 2 of our internal libraries, as well as Cairo, Skia and >>>>> Flash and they all refuse to calculate the matrix if it's not invertible. >>>>> So, not all matrix libraries do as you suggest. >>>>> SVGMatrix is also documented to throw. >>>>> These are all matrix classes designed for graphics so I *think* it >>>>> makes sense to follow what they did. >>>>> >>>> >>>> I had dedicated matrix libraries in mind, rather than the casual >>>> incidental matrix computation in something like Cairo. >>>> >>> >>> So, are you saying the matrix class should not throw? >>> >> >> Yes. Matrices are just a generalization of numbers (which are 1x1 >> matrices). Numbers don't throw --- not even 1/0 or 0/0. Matrices should >> not, either. >> >> >>> >>> >>>> >>>> >>>>> >>>>> >>>>>> >>>>>> >>>>>> The algorithm for inversion is going to be plain closed-form matrix >>>>>> inversion (i.e. by computing the cofactors) as in >>>>>> http://en.wikipedia.org/wiki/Invertible_matrix#Inversion_of_3.C3.973_matricesbut instead of mandating implementers to use particular formulas, a good >>>>>> matrix API would rather carefully avoid being too sensitive to the exact >>>>>> floating-point values (e.g. by not throwing on singular matrices) because >>>>>> even mandating formulas won't ensure that the results are the same >>>>>> everywhere up to the last bit of precision. >>>>>> >>>>> >>>>> I don't know how important this is as it seems like this would only >>>>> apply to edge cases. >>>>> Are there any w3c specs that define how precise a calculation should >>>>> happen? I can see how this would be very important for mathematical >>>>> programs but less so for graphics. >>>>> >>>> >>>> That's my point: the last bits of mantissa are irrelevant to graphics, >>>> so we should just make sure that our API doesn't accidentally make them >>>> unduly important (e.g. by throwing on singular matrix or trying to avoid >>>> methods like is2D() that can return true or false depending on them). Once >>>> we've ensured that, we really don't need no worry about different browsers >>>> giving slightly different values. >>>> >>> >>> I'm a bit confused. >>> It seems that if the last bits are unimportant and the determinant >>> returns something that is in the range of those small bits, 'invert' should >>> fail and throw. >>> If not, you might generate an inverted matrix with very large values and >>> things will go very wrong... >>> >> >> If the determinant is only a few times machine epsilon then the result is >> indeed unreliable, i.e. it will sometimes be very imprecise. >> > > I need to qualify this a bit: > > If the determinant is only a few times machine epsilon TIMES a quantity > that depends on the order of magnitude of the matrix coefficients then the > result is unreliable. > > Let me explain this: > > Let's use floats. Machine epsilon is roughly 1e-7 and the range of > absolute values is roughly from 1e-30 to 1e+30. The following matrix: > > 1e-10 0 > 0 1e-10 > > is perfectly fine (not even close to approaching the boundaries of the > range of float values) and has determinant 1e-20 which is far below machine > epsilon. Yet it is perfectly invertible, and its inverse is > > 1e10 0 > 0 1e10 > > So just comparing the determinant to machine epsilon doesn't work in full > generality. It might work if the matrix coefficient were on the order of > magnitude of 1 but that's not even the case of graphics matrices, were > coefficients can typically be of the order of magnitude of the size of the > page in CSS pixels, I suppose, which would be about 1e3. > > Another issue is that it should be clear that any determinant check is at > most a necessary condition, not a sufficient condition for the inverse to > be reliable. For example, take this matrix: > > 1 0 1e4 > 1 1 1e4 > 1 1e4 1 > > If we compute its determinant by developing it along the first column, as > it is most usually computed, we get > > 1 - 1e8 + 1e8 + 1e4 > > which is mathematically equal to 1e4 + 1, but a with floats, 1 - 1e8 == - > 1e8 so we get 1e4 instead. > > That's a relative imprecision of 1e-4 only, but that's already over 1e3 > time machine epsilon, and that was only a very naive example, and the > determinant of 1e4 would have passed any check guarding against very small > (or very large) determinants. > > Just to make it clear that simple checks like checking against small > determinants only buy you so much. > I don't question your knowledge of matrices. However, this does not seem to be a problem in the field. > > >> But if on that basis one decides to do something like >> >> function inverse() { >> if (determinant() < THRESHOLD) { >> // return an error >> } >> // compute and return the inverse >> } >> >> Then one creates a huge discontinuity at the threshold point, which will >> disrupt a majority of application much more than this risk of imprecision. >> >> It is legitimate for an application to say that they really want this >> check to be done. In this case, if you want (to maximize efficiency) to do >> this without computing the determinant twice, you can add a >> checkedInverse() method returning not only the inverse but also a boolean >> indicating whether it is to be trusted. >> >> But: >> - please don't throw because that defaults to killing the application, >> which is not something that should happen as a result of floating point >> imprecision. >> - be very wary of hardcoding an arbitrary fixed THRESHOLD. There's no >> one-size-fits-all there. That's what we learnt the hard way in Eigen and we >> eventually decided to let the only inverse-with-check function we have, >> take a 'threshold' argument (though it has a vaguely reasonable default >> value). >> http://eigen.tuxfamily.org/dox/classEigen_1_1MatrixBase.html#abcd1eb30af2979b147e1f65477caa209 >> >> Benoit >> >> >> >>> >>> >>>> That's a good thing, because it would be completely unrealistic to >>>> expect exact values. Keep in mind that in floating point, a*(b*c) and >>>> (a*b)*c give different results i.e. associativity doesn't even hold. Same >>>> for distributivity: a*b+a*c != a*(b+c). So for example just a single >>>> browser having a SSE and a non-SSE codepath would almost certainly get >>>> different results from them. >>>> >>>> Benoit >>>> >>> >>> >> >Received on Thursday, 21 March 2013 18:26:55 UTC

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