Skip to main content

symbolic - Improving the performance of a package for working with rational functions



As Mathematica gets slow for large symbolic calculations, the cost of putting terms over a common denominator (Together), in particular, gets too high. It occurred to me that, if one has a small number of variables, it should be much faster if we represent our expressions by arrays of their coefficients.


I have implemented a small package that does this with an improvement of a factor of 10 in both speed and memory (in putting Together a sum of rational terms, more or less independent of the size of the sum), but it seems to me that I should be able to do better. I post my package here with the hope that someone might point out glaring performance bottlenecks.


First, the public initializations, with their usability comments.


AA::usage = "DATA STRUCTURE AA[num,den] that stores polynomials as their tensor coefficients of a basis, multiplication is the tensor product"
PolytoAA::usage = "PolytoAA[poly_,vars_List] returns an AA[num_SparseArray,1]"
AAtoPoly::usage = "AAtoPoly[aa_AA,vars_] gives back a rational function of the polynomials num/den"

The actual code begins here, PolProd takes care of multiplying polynomials


SetAttributes[PolProd, Orderless]
PolProd[i__Integer] := Times @ i;

PolProd[sa__SparseArray] := Outer[Times, sa];
  (*multiplying 2 polynomials is the tensor product in their vector space*)
PolProd[i__Integer, sa__SparseArray] := Times[i] Outer[Times, sa];

(*It would be cleaner and more object oriented to call two instances of polprod 
  on i and sa but it would force more pattern matching so it would be less efficient*)

This code performs the arithmetic operations on the AA[num,den] data structure


AA /: i_Integer*AA[num_, den_] := AA[i*num, den]
 (*absorb all integer into AA, this means that the sum rule has to do less 

   pattern matching and is more efficient, overall more efficient?*)
AA /: AA[a_, a_] := 1
AA /: AA[0, _] := 0
AA /: AA[num1_, den1_]*AA[num2_, den2_] := 
  AA @@ (PolProd[#1, #2]& @@@ {{num1, num2}, {den1, den2}})
AA /: AA[num1_, den_] + AA[num2_, den_] := 
  AA[num1 + num2, den]
  (*can't check for a SparseArray cos it might be an Integer, 
    by dimensional analysis we should get two arrays here though*)
AA /: AA[num1_, den1_] + AA[num2_, den2_] := 

  AA[PolProd[num1, den2] + PolProd[num2, den1], PolProd[den1, den2]]
  (*together rule, this a recursive implementation that doesn't treat the 
    sum of many AA all at once, this is intentional cause it minimises the 
    number of array outer products that are calculated and I believe it's 
    the more efficient but maybe not*)
AA /: aa_AA^n_Integer := 
  If[n < 0, Reverse @ #, #]& @ (PolProd[Sequence @@ ConstantArray[#, Abs @ n]]& /@ aa)
  (*this takes care of both efficiency of recursively pattern matching the 
    product rule on large n and dividing by AA*)
(*NOTICE THE WEIRD SYNTAX/BUG If[n<0, Reverse @ #, #]& @ THERE ARE TWO INTANCES OF @*)


I also post the the functions that go between AA and the polynomials for the sake of clarity and completeness


AAtoPoly[aa_AA,vars_] := #1/#2& @@ (aa /. l_SparseArray :> 
  Dot[l, Sequence @@ ConstantArray[vars, Depth[l] - 1 (*gives rank of the tensor*)]])
PolytoAA[expr_^n_, vars_List] := PolytoAA[expr, vars]^n
PolytoAA[expr : (_Times | _Plus), vars_List] := PolytoAA[#, vars]& /@ expr
PolytoAA[expr_,vars_List] /; PolynomialQ[expr, vars] := 
  AA[Last[CoefficientArrays[expr, vars]](*num*), 1 (*den*)]

BENCHMARKS



I define some random, small polynomials


vars = a /@ Range[10]
pols = Table[
   a[RandomInteger[{1, 10}]] + a[RandomInteger[{1, 10}]], {20}] // 
  DeleteDuplicates
% // Length


{a[1], a[2], a[3], a[4], a[5], a[6], a[7], a[8], a[9], a[10]}


{a[1] + a[7], a[4] + a[6], a[2] + a[4], a[5] + a[9], 2 a[8], a[5] + a[6], a[2] + a[9], a[4] + a[7], a[4] + a[9], a[7] + a[9], a[3] + a[7], a[2] + a[6], a[1] + a[9], a[2] + a[3], a[7] + a[10], 2 a[2], a[3] + a[9]}



17



First I put them Together using my code


aa = PolytoAA[#, vars] & /@ pols;
togaa = Sum[aa[[i]]/aa[[i + 1]], {i, Length@pols - 1}] // 
  AbsoluteTiming
% // ByteCount


{0.8960513,AA[SparseArray[<274432>,{10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}],SparseArray[<16384>,{10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10}]]}



18614128



and now without my implementation


togpols = (Sum[pols[[i]]/pols[[i + 1]], {i, Length@pols - 1}] // 
    Together) // AbsoluteTiming//
% // ByteCount


{8.2054693,...}


34567480




I check that both give the same result


togpols2 = AAtoPoly[togaa // Last, vars](*//Simplify*);
Last[togpols] - togpols2 // Together


0





Comments

Post a Comment

Popular posts from this blog

plotting - How to draw lines between specified dots on ListPlot?

I would like to create a plot where I have unconnected dots and some connected. So far, I have figured out how to draw the dots. My code is the following: ListPlot[{{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4,13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full] I have thought using ListLinePlot command, but I don't know how to specify to the command to draw only selected lines between the dots. Do have any suggestions/hints on how to do that? Thank you. Answer One possibility would be to use Epilog with Line : ListPlot[ {{1, 1}, {2, 2}, {3, 3}, {4, 4}, {1, 4}, {2, 5}, {3, 6}, {4, 7}, {1, 7}, {2, 8}, {3, 9}, {4, 10}, {1, 10}, {2, 11}, {3, 12}, {4, 13}, {2.5, 7}}, Ticks -> {{1, 2, 3, 4}, None}, AxesStyle -> Thin, TicksStyle -> Directive[Black, Bold, 12], Mesh -> Full, Epilog -> { Line[ ...
Post a Comment
Read more

equation solving - Invert and fit implicitly defined curve

I need to fit an implicitly defined curve. I thought I could get some data out of Solve , and then using FindFit . Therefore, I would like to find the relation the parametric curve defined by $F(x,y)=0$: Solve[-(1/2) + 1/2 (0.41202 BesselK[0, 0.1 Sqrt[x^2 + y^2]] + (0.101483 x BesselK[1, 0.1 Sqrt[x^2 + y^2]])/Sqrt[x^2 + y^2]) == 0, y] But I can't get an output: Solve was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Solve require exact input, providing Solve with an exact version of the system may help. >> Edit: In particular, I would like to fit the data coming from the curve with the expression of another curve, and not with a function $f(x)$. In particular, since this clearly looks like a cardioid , I would like it to fit to something like it. What other strategies could I try?
Post a Comment
Read more

dynamic - How can I make a clickable ArrayPlot that returns input?

I would like to create a dynamic ArrayPlot so that the rectangles, when clicked, provide the input. Can I use ArrayPlot for this? Or is there something else I should have to use? Answer ArrayPlot is much more than just a simple array like Grid : it represents a ranged 2D dataset, and its visualization can be finetuned by options like DataReversed and DataRange . These features make it quite complicated to reproduce the same layout and order with Grid . Here I offer AnnotatedArrayPlot which comes in handy when your dataset is more than just a flat 2D array. The dynamic interface allows highlighting individual cells and possibly interacting with them. AnnotatedArrayPlot works the same way as ArrayPlot and accepts the same options plus Enabled , HighlightCoordinates , HighlightStyle and HighlightElementFunction . data = {{Missing["HasSomeMoreData"], GrayLevel[ 1], {RGBColor[0, 1, 1], RGBColor[0, 0, 1], GrayLevel[1]}, RGBColor[0, 1, 0]}, {GrayLevel[0], GrayLevel...
Post a Comment
Read more