Thread overview  


April 19 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to Stanislav Blinov  On Wed, Apr 19, 2017 at 07:54:02PM +0000, Stanislav Blinov via Digitalmarsd wrote: > Awesome! Congrats and thanks for sharing. > > On Wednesday, 19 April 2017 at 19:32:14 UTC, H. S. Teoh wrote: > > > Haha, it seems that the only roadblocks were related to the implementation quality of std.numeric.gcd... nothing that a few relativelysimple PRs couldn't fix. So overall, D is still awesome. > > There's another one, which is more about dmd: dmd does not inline gcd, which, when arguments are const, turns gcd into a double function call :D If I weren't such a sucker for the bleeDing edge with dmd (I actually compile even my serious projects with git HEAD dmd, except when performance matters), I'd be standardizing on ldc or gdc, which have far superior optimizers. I consistently find that my CPUintensive projects perform at least 2030% worse on dmd than gdc (and I presume ldc), sometimes even as bad as 4050%, due to dmd's inliner giving up far too easily. I don't know if this has been fixed yet, but the last time I checked, if you wrote this: int func(int x, int y) { if (x<0) return y; return x; } instead of: int func(int x, int y) { if (x<0) return y; else return x; } then the dmd inliner would not inline the function. Because of sensitivities like this, the inliner gives up far too early in the process, thus the optimizer misses out on further optimization opportunities that would have opened up, had the function been inlined. The last time I checked, I also found that dmd was rather weak at loop optimizations (and loops are very important in performance as we all know) compared to gdc. Again, the same domino effect (or rather, the missing thereof) applies: by failing to, for example, hoist a constant expression out of the loop, further optimization opportunities are missed, whereas gdc, after hoisting the expression out, would discover that the loop can be reduced further, perhaps via a strength reduction, and then unrolled, and then inlined inside the caller, then vectorized, etc.. This chain of optimizations were missed because of one missed optimization early in the process. Hence the suboptimal resulting code. T  If blunt statements had a point, they wouldn't be blunt... 
April 19 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to Timon Gehr  On Wed, Apr 19, 2017 at 10:47:04PM +0200, Timon Gehr via Digitalmarsd wrote: > On 19.04.2017 21:32, H. S. Teoh via Digitalmarsd wrote: > > I alluded to this in D.learn some time ago, and finally decided to take the dip and actually write the code. So here it is: exact arithmetic with numbers of the form (a+b√r)/c where a, b, c are integers, c!=0, and r is a (fixed) squarefree integer. > > > > Code: https://github.com/quickfur/qrat > > > > ... > > Nice. :) > > Some suggestions: > >  You might want to support ^^ (it is useful for examples like the one > below). I would, except that I doubt it would perform any better than an actual recursive or iterative algorithm for computing Fibonacci sequences, because I don't know of any simple way to exponentiate a quadratic rational using only integer arithmetic other than repeated multiplication. (For all I know, it may perform even worse, because multiplying n quadratic rationals involves a lot more than just summing n+1 integers as in an iterative implementation of Fibonacci.) Hmm, come to think of it, I *could* expand the numerator using the binomial theorem, treating (a+b√r) as a binomial (a+bx) where x=√r, and folding even powers into the integral part (since x^2 = r, so x^(2k) = r^k). The denominator could be exponentiated using plain ole integer exponentiation. Then it's just a matter of summing coefficients. But it still seems to be about the same amount of work as (or more than) summing n+1 integers in an iterative implementation of Fibonacci. Am I missing something? >  constructor parameter _b should have a default value of 0. Good point, done. >  formatting should special case b==1 like it special cases b==1. Done, good catch! > (also: as you are using Unicode anyway, you could also use · instead > of *. Another cute thing to do is to add a vinculum.) Well, I would, but it gets a bit too fancy for my tastes and may not render well on all displays. But I'll put it on my list of things to consider. Another module I'm thinking about is an extension of QRat that allows you to mix different radicals in the same expression. For example, (√3+√5)/√7 and so forth. I have discovered algorithms that, given n distinct radicals, allow a closedform expression with 2^n coefficients (+1 denominator), closed under field operations. The only trouble in this case is that reciprocating such things will be very slow, as will comparisons, and both have a high chance of overflow (so BigInt is probably a necessity). And 2^n+1 coefficients for n radicals quickly gets expensive spacewise as n increases. Yesterday I also discovered an algorithm for expressing the reciprocal of numbers of the form: (a + b∛r + c∛r^2)/d in the same form. I.e., for rewriting: d/(a + b∛r + c∛r^2) in the first form. Which means it's possible to implement a QRatlike representation for cubic rationals. (The actual computation is rather expensive, as it involves quite a lot of multiplications, squaring, and cubing. But it's *possible*.) I'm still trying to verify the correctness of the formula I obtained, since while checking a concrete example last night I discovered a possible error. If this works out, I might consider 4th roots as well  though I'm expecting that might be near the limit of the usefulness of these representations, since the complexity becomes so great that symbolic manipulation like in Mathematica may turn out to be more feasible after all. But it may be of some theoretical interest whether such representations are possible, even if they are ultimately impractical. A particularly interesting question is whether such representations exist for *all* algebraic numbers (of bounded degree). Currently I have a conjecture that given a rational extension of n radicals of degree k, field closure can be achieved with a representation of k^n+1 coefficients. But it's still too early to say whether algorithms exist for inverting radicals of degree k for large k  I have a creeping suspicion that perhaps somewhere around k=5 or k=6 the unsolvability of the general quintic may raise its ugly head and prevent further progress. T  INTEL = Only half of "intelligence". 
April 20 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to H. S. Teoh  On 19.04.2017 23:39, H. S. Teoh via Digitalmarsd wrote: > On Wed, Apr 19, 2017 at 10:47:04PM +0200, Timon Gehr via Digitalmarsd wrote: >> On 19.04.2017 21:32, H. S. Teoh via Digitalmarsd wrote: >>> I alluded to this in D.learn some time ago, and finally decided to >>> take the dip and actually write the code. So here it is: exact >>> arithmetic with numbers of the form (a+b√r)/c where a, b, c are >>> integers, c!=0, and r is a (fixed) squarefree integer. >>> >>> Code: https://github.com/quickfur/qrat >>> >>> ... >> >> Nice. :) >> >> Some suggestions: >> >>  You might want to support ^^ (it is useful for examples like the one >> below). > > I would, except that I doubt it would perform any better than an actual > recursive or iterative algorithm for computing Fibonacci sequences, > because I don't know of any simple way to exponentiate a quadratic > rational using only integer arithmetic other than repeated > multiplication. (For all I know, it may perform even worse, because > multiplying n quadratic rationals involves a lot more than just summing > n+1 integers as in an iterative implementation of Fibonacci.) > > Hmm, come to think of it, I *could* expand the numerator using the > binomial theorem, treating (a+b√r) as a binomial (a+bx) where x=√r, and > folding even powers into the integral part (since x^2 = r, so x^(2k) = > r^k). The denominator could be exponentiated using plain ole integer > exponentiation. Then it's just a matter of summing coefficients. > > But it still seems to be about the same amount of work as (or more than) > summing n+1 integers in an iterative implementation of Fibonacci. Am I > missing something? > ... Yes, there is in fact a beautifully simple way to do better. :) Assume we want to compute some power of x. With a single multiplication, we obtain x². Multiplying x² by itself, we obtain x⁴. Repeating this a few times, we get: x, x², x⁴, x⁸, x¹⁶, x³², etc. In general, we only need n operations to compute x^(2ⁿ). To compute xʸ, it basically suffices to express y as a sum of powers of two (i.e., we write it in binary). For example, 22 = 16 + 4 + 2, and x²² = x¹⁶·x⁴·x². My last post includes an implementation of this algorithm. ;) In particular, this leads to multiple ways to compute the nth Fibonacci number using O(log n) basic operations. (One way is to use your QRat type, but we can also use the matrix (1 1; 1 0).) > ... > > Another module I'm thinking about is an extension of QRat that allows > you to mix different radicals in the same expression. For example, > (√3+√5)/√7 and so forth. ... > That would certainly be nice. Note that QRat is basically already there for different quadratic radicals, the only reason it does not work already is that we cannot use a QRat as the base field instead of ℚ (because ℚ is hardcoded). This is the relevant concept from algebra: https://en.wikipedia.org/wiki/Splitting_field All your conjectures are true, except the last one. (Galois theory is not an obstacle, because here, we only need to consider splitting fields of particularly simple polynomials that are solvable in radicals.) You can even mix radicals of different degrees. To get the formula for multiplicative inverses, one possible algorithm is: https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Polynomial_extended_Euclidean_algorithm 
April 20 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to Timon Gehr  On 20.04.2017 02:01, Timon Gehr wrote: > > To get the formula for multiplicative inverses, one possible algorithm is: > https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Polynomial_extended_Euclidean_algorithm > > Better reference: https://en.wikipedia.org/wiki/Polynomial_greatest_common_divisor#Arithmetic_of_algebraic_extensions 
April 20 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to Timon Gehr  On 20.04.2017 02:01, Timon Gehr wrote:
>
> My last post includes an implementation of this algorithm. ;)
But in that implementation I used the parameter 'a' instead of the variable 'x' as a result of being tired, which makes it slightly more confusing than necessary even though it is correct. More readable version:
auto pow(T,S)(T a,S n){
T r=T(ℕ(1),ℕ(0));
for(auto x=a;n;n>>=1,x*=x)
if(n&1) r*=x;
return r;
}

April 19 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to Timon Gehr  On Thu, Apr 20, 2017 at 02:01:20AM +0200, Timon Gehr via Digitalmarsd wrote: [...] > Yes, there is in fact a beautifully simple way to do better. :) > > Assume we want to compute some power of x. With a single > multiplication, we obtain x². Multiplying x² by itself, we obtain x⁴. > Repeating this a few times, we get: > > x, x², x⁴, x⁸, x¹⁶, x³², etc. > > In general, we only need n operations to compute x^(2ⁿ). > > To compute xʸ, it basically suffices to express y as a sum of powers of two (i.e., we write it in binary). > > For example, 22 = 16 + 4 + 2, and x²² = x¹⁶·x⁴·x². > > My last post includes an implementation of this algorithm. ;) Ahh, so *that's* what it's all about. I figured that's what I was missing. :D Thanks, I'll include this in QRat soon. > In particular, this leads to multiple ways to compute the nth > Fibonacci number using O(log n) basic operations. (One way is to use > your QRat type, but we can also use the matrix (1 1; 1 0).) True, though I'm still jealous that with transcendental functions like with floatingpoint, one could ostensibly compute that in O(1). > > Another module I'm thinking about is an extension of QRat that allows you to mix different radicals in the same expression. For example, (√3+√5)/√7 and so forth. ... > > > > That would certainly be nice. Note that QRat is basically already there for different quadratic radicals, the only reason it does not work already is that we cannot use a QRat as the base field instead of ℚ (because ℚ is hardcoded). Oh? I didn't try it myself, but if QRat itself passes isArithmeticType, I'd venture to say QRat!(n, QRat!m) ought to work... There are some hidden assumptions about properties of the rationals, though, but I surmise none that couldn't be replaced by prerequisites about the relative linear dependence of the mixed radicals over Q. What I had in mind, though, was a more direct approach that perhaps may reduce the total number of operations, since if the code is aware that multiple radicals are involved, it could potentially factor out some commonalities to minimize recomputations. Also, the current implementation of QRat fixes the radical at compiletime; I wanted to see if I could dynamically handle arbitrary radicals at runtime. It would have to be restricted by only allowing operations between two QRats of the same extension, of course, but if the code could handle arbitrary extensions dynamically, then that restriction could be lifted and we could (potentially, anyway) support arbitrary combinations of expressions involving radicals. That would be far more useful than QRat, for some of the things I'd like to use it for. > This is the relevant concept from algebra: https://en.wikipedia.org/wiki/Splitting_field > > > All your conjectures are true, except the last one. (Galois theory is not an obstacle, because here, we only need to consider splitting fields of particularly simple polynomials that are solvable in radicals.) That's nice to know. But I suppose Galois theory *would* become an obstacle if I wanted to implement, for example, Q(x) for an arbitrary algebraic x? > You can even mix radicals of different degrees. Yes, I've thought about that before. So it should be possible to represent Q(r1,r2,...rn) using deg(r1)*deg(r2)*...*deg(rn)+1 coefficients? > To get the formula for multiplicative inverses, one possible algorithm is: https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Polynomial_extended_Euclidean_algorithm [...] Thanks, will look into this at some point. :) T  Some ideas are so stupid that only intellectuals could believe them.  George Orwell 
April 20 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to H. S. Teoh  On 20.04.2017 03:00, H. S. Teoh via Digitalmarsd wrote: > On Thu, Apr 20, 2017 at 02:01:20AM +0200, Timon Gehr via Digitalmarsd wrote: > [...] >> Yes, there is in fact a beautifully simple way to do better. :) >> ... > > Ahh, so *that's* what it's all about. I figured that's what I was > missing. :D Thanks, I'll include this in QRat soon. > > >> In particular, this leads to multiple ways to compute the nth >> Fibonacci number using O(log n) basic operations. (One way is to use >> your QRat type, but we can also use the matrix (1 1; 1 0).) > > True, though I'm still jealous that with transcendental functions like > with floatingpoint, one could ostensibly compute that in O(1). > ... BTW, you are right that with std.bigint, computation using a linear number of additions is actually faster for my example (100000th Fibonacci number). The asymptotic running time of the version with pow on QRats is lower though, so there ought to be a crossover point. (It is Θ(n^2) vs. O(n^log₂(3)·log(n)). std.bigint does not implement anything that is asymptotically faster than Karatsuba.) For computations over field extensions of (small) finite fields, pow is a lot faster though. > >>> Another module I'm thinking about is an extension of QRat that >>> allows you to mix different radicals in the same expression. For >>> example, (√3+√5)/√7 and so forth. ... >>> >> >> That would certainly be nice. Note that QRat is basically already >> there for different quadratic radicals, the only reason it does not >> work already is that we cannot use a QRat as the base field instead of >> ℚ (because ℚ is hardcoded). > > Oh? I didn't try it myself, but if QRat itself passes isArithmeticType, > I'd venture to say QRat!(n, QRat!m) ought to work... There are some > hidden assumptions about properties of the rationals, though, but I > surmise none that couldn't be replaced by prerequisites about the > relative linear dependence of the mixed radicals over Q. > ... The issue is that gcd does not work on QRats. If QRat had two coefficients from an arbitrary (possibly ordered) field instead of encoding rationals explicitly, I think it would work. > What I had in mind, though, was a more direct approach that perhaps may > reduce the total number of operations, since if the code is aware that > multiple radicals are involved, it could potentially factor out some > commonalities to minimize recomputations. > ... This is probably the case. > Also, the current implementation of QRat fixes the radical at > compiletime; I wanted to see if I could dynamically handle arbitrary > radicals at runtime. It would have to be restricted by only allowing > operations between two QRats of the same extension, of course, but if > the code could handle arbitrary extensions dynamically, then that > restriction could be lifted and we could (potentially, anyway) support > arbitrary combinations of expressions involving radicals. That would be > far more useful than QRat, for some of the things I'd like to use it > for. > ... What applications do you have in mind? Computational geometry? > >> This is the relevant concept from algebra: >> https://en.wikipedia.org/wiki/Splitting_field >> >> >> All your conjectures are true, except the last one. (Galois theory is >> not an obstacle, because here, we only need to consider splitting >> fields of particularly simple polynomials that are solvable in >> radicals.) > > That's nice to know. But I suppose Galois theory *would* become an > obstacle if I wanted to implement, for example, Q(x) for an arbitrary > algebraic x? > ... All that the result about the quintic really says is that you will not, in general, be able to express x using field operations on radicals. It is still possible to compute the roots to arbitrary precision. Computing the field operations in ℚ(x) will actually still be quite straightforward but you'd have to think about what to do with toString and opCmp. (Or more generally, you'd have to think about how to pick one of the roots of a given polynomial.) > >> You can even mix radicals of different degrees. > > Yes, I've thought about that before. So it should be possible to > represent Q(r1,r2,...rn) using deg(r1)*deg(r2)*...*deg(rn)+1 > coefficients? > ... Yes, at most, except you don't need "+1". (For each radical ri, you will at most need to pick a power between 0 to deg(ri)1 to index into the coefficients.) 
April 20 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to Timon Gehr  On Thu, Apr 20, 2017 at 02:51:12PM +0200, Timon Gehr via Digitalmarsd wrote: > On 20.04.2017 03:00, H. S. Teoh via Digitalmarsd wrote: > > On Thu, Apr 20, 2017 at 02:01:20AM +0200, Timon Gehr via Digitalmarsd wrote: [...] > > > Yes, there is in fact a beautifully simple way to do better. :) ... > > > > Ahh, so *that's* what it's all about. I figured that's what I was missing. :D Thanks, I'll include this in QRat soon. Update: QRat now supports ^^. :) Integral exponents only, of course. I also implemented negative exponents, because QRat supports division and the same algorithm can be easily reused for that purpose. Interestingly enough, std.math has an overload of pow() that pretty much does exactly the same thing, except that its sig constraints require a builtin floatingpoint type. I'm halftempted to submit a Phobos PR to relax the sig constraints so that we could actually use it for QRat without essentially duplicating the code. [...] > BTW, you are right that with std.bigint, computation using a linear > number of additions is actually faster for my example (100000th > Fibonacci number). The asymptotic running time of the version with > pow on QRats is lower though, so there ought to be a crossover point. > (It is Θ(n^2) vs. O(n^log₂(3)·log(n)). std.bigint does not implement > anything that is asymptotically faster than Karatsuba.) Yeah, probably there is a crossover point. But it might be quite large. I suppose one could make a graph of the running times for increasing n, and either find the crossover point that way or extrapolate using the known curve shapes. Having said that, I haven't scrutinized the performance characteristics of QRat too carefully just yet  there is probably room for optimization. [...] > > > That would certainly be nice. Note that QRat is basically already there for different quadratic radicals, the only reason it does not work already is that we cannot use a QRat as the base field instead of ℚ (because ℚ is hardcoded). > > > > Oh? I didn't try it myself, but if QRat itself passes isArithmeticType, I'd venture to say QRat!(n, QRat!m) ought to work... There are some hidden assumptions about properties of the rationals, though, but I surmise none that couldn't be replaced by prerequisites about the relative linear dependence of the mixed radicals over Q. ... > > The issue is that gcd does not work on QRats. If QRat had two coefficients from an arbitrary (possibly ordered) field instead of encoding rationals explicitly, I think it would work. You're right, without gcd it won't work. The current implementation is a bit overzealous on using gcd (cf. the division algorithm), mainly because I'm concerned with integer overflow on native types. Probably some of these uses can be dispensed with, where BigInt is involved. But normalize() still needs gcd, otherwise sgn() and opCmp() may produce wrong results. One thought is that if there is a QInt base type (i.e., implementing numbers of the form a+b√r, without the denominator), then we could implement a gcd algorithm for it, and we'd be able to instantiate QRat!(r, QInt). > > What I had in mind, though, was a more direct approach that perhaps may reduce the total number of operations, since if the code is aware that multiple radicals are involved, it could potentially factor out some commonalities to minimize recomputations. ... > > This is probably the case. Upon reviewing the algorithms I've come up with in the past, it appears that QRat!(r, QInt) may in fact produce essentially the same code. > > Also, the current implementation of QRat fixes the radical at compiletime; I wanted to see if I could dynamically handle arbitrary radicals at runtime. It would have to be restricted by only allowing operations between two QRats of the same extension, of course, but if the code could handle arbitrary extensions dynamically, then that restriction could be lifted and we could (potentially, anyway) support arbitrary combinations of expressions involving radicals. That would be far more useful than QRat, for some of the things I'd like to use it for. ... > > What applications do you have in mind? Computational geometry? Yes. In particular, manipulating the coordinates of certain kinds of polytopes. I currently do have code that can do this with floatingpoint, but I'd like to be able to deal with exact coordinates rather than floatingpoint approximations. [...] > > > All your conjectures are true, except the last one. (Galois theory is not an obstacle, because here, we only need to consider splitting fields of particularly simple polynomials that are solvable in radicals.) > > > > That's nice to know. But I suppose Galois theory *would* become an obstacle if I wanted to implement, for example, Q(x) for an arbitrary algebraic x? ... > > All that the result about the quintic really says is that you will not, in general, be able to express x using field operations on radicals. It is still possible to compute the roots to arbitrary precision. Oh I know that; I'm not really concerned with computing roots to arbitrary precision here though, but more with implementing precise arithmetic on expressions involving said roots. > Computing the field operations in ℚ(x) will actually still be quite straightforward but you'd have to think about what to do with toString and opCmp. (Or more generally, you'd have to think about how to pick one of the roots of a given polynomial.) Hmm, good point. I suppose I haven't really thought through the consequences of what might happen if I implemented a reciprocation algorithm for an algebraic number k where the defining polynomial for k may have multiple roots. At some point, assumptions about which root is being used need to come into play, I suppose. How to encode this in the API and internally in the code is an interesting question. > > > You can even mix radicals of different degrees. > > > > Yes, I've thought about that before. So it should be possible to > > represent Q(r1,r2,...rn) using deg(r1)*deg(r2)*...*deg(rn)+1 > > coefficients? > > ... > > Yes, at most, except you don't need "+1". (For each radical ri, you > will at most need to pick a power between 0 to deg(ri)1 to index into > the coefficients.) [...] The +1 is for the denominator, assuming integer coefficients. Since having 2^n rational coefficients is equivalent to having 2^n integer coefficients (which are half the size of rational coefficients, computerrepresentationwise) + 1 denominator. Though it's arguable whether this really saves that much once you get out of the realm of native machine integer types into BigInt. It's debatable whether that much is really saved in terms of space and CPU time if you have two or more rational coefficients with denominators that are relatively prime to each other, so that merging them all into a single denominator may just cause all coefficients to explode in size whereas separate rational coefficients could in fact be more compact. But, at least in theory, if you're dealing with relatively small members in the field, you could fit everything in native int types and having 2^n + 1 integer coefficients may perform better than having 2^n rational coefficients. I suspect the applicability of this is rather narrow, however, because once you get past a small handful of radicals, the coefficients (esp. intermediate coefficients computed during reciprocation) will easily overflow native machine int types, thus necessitating BigInt coefficients pretty quickly. The last time I attempted an implementation with 34 separate radicals many years ago, I found that even small starting coefficients (i.e., 12 digits) quickly exploded in internal algorithms due to repeated multiplication, so that after just a small number of operations I was already running into integer overflows. This was back when I was still doing it in C/C++... I did attempt a reimplementation using libgmp, but never finished. T  "Real programmers can write assembly code in any language. :)"  Larry Wall 
April 20 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to H. S. Teoh  On 20.04.2017 20:29, H. S. Teoh via Digitalmarsd wrote: > On Thu, Apr 20, 2017 at 02:51:12PM +0200, Timon Gehr via Digitalmarsd wrote: >> On 20.04.2017 03:00, H. S. Teoh via Digitalmarsd wrote: >>> On Thu, Apr 20, 2017 at 02:01:20AM +0200, Timon Gehr via Digitalmarsd wrote: >>> [...] >>>> Yes, there is in fact a beautifully simple way to do better. :) >>>> ... >>> >>> Ahh, so *that's* what it's all about. I figured that's what I was >>> missing. :D Thanks, I'll include this in QRat soon. > > Update: QRat now supports ^^. :) Integral exponents only, of course. I > also implemented negative exponents, because QRat supports division and > the same algorithm can be easily reused for that purpose. > ... Nice! :) > ... > > [...] >> BTW, you are right that with std.bigint, computation using a linear >> number of additions is actually faster for my example (100000th >> Fibonacci number). The asymptotic running time of the version with >> pow on QRats is lower though, so there ought to be a crossover point. >> (It is Θ(n^2) vs. O(n^log₂(3)·log(n)). std.bigint does not implement >> anything that is asymptotically faster than Karatsuba.) > > Yeah, probably there is a crossover point. But it might be quite large. > I suppose one could make a graph of the running times for increasing n, > and either find the crossover point that way or extrapolate using the > known curve shapes. > > Having said that, I haven't scrutinized the performance characteristics > of QRat too carefully just yet  there is probably room for > optimization. > Gcd is the problem. The following code which implements a strategy based on matrix multiplication instead of QRat multiplication is significantly faster than naive linear computation: BigInt fib(long n){ BigInt[2] a=[BigInt(0),BigInt(1)],b=[BigInt(1),BigInt(2)],c=[BigInt(1),BigInt(1)]; for(;n;n>>=1){ foreach(i;1n&1..2){ auto d=a[i]*a[1]; a[i]=a[i]*b[1]+c[i]*a[1]; b[i]=b[i]*b[1]d; c[i]=c[i]*c[1]d; } } return a[0]; } If I change the gcd computation in QRat (line 233) from auto g = gcd(abs(a), abs(b), c); to auto g = gcd(abs(a), c, abs(b)); I get performance that is a lot closer to the matrix version and also beats the linear computation. (This is because if one of the operands is 1, gcd is cheap to compute.) > ... >> >> Yes, at most, except you don't need "+1". (For each radical ri, you >> will at most need to pick a power between 0 to deg(ri)1 to index into >> the coefficients.) > [...] > > The +1 is for the denominator, assuming integer coefficients. Since > having 2^n rational coefficients is equivalent to having 2^n integer > coefficients (which are half the size of rational coefficients, > computerrepresentationwise) + 1 denominator. > ... Ah, I see. Personally, I'm more in the one denominator per coefficient camp. :) I think having a designated ℚ type is cleaner, and it might even be more performant. 
April 20 Re: Exact arithmetic with quadratic irrationals  

 
Posted in reply to Timon Gehr  On 20.04.2017 21:11, Timon Gehr wrote:
>> Update: QRat now supports ^^. :) Integral exponents only, of course. I
>> also implemented negative exponents, because QRat supports division and
>> the same algorithm can be easily reused for that purpose.
>> ...
>
> Nice! :)
It does not work with BigIntbased QRats (T(1) does not work, as 1 does not implicitly convert to BigInt.)

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