Skip to main content

performance tuning - How does Internal`PolynomialFunctionQ work?


I found that Internal`PolynomialFunctionQ performs much better than PolynomialQ.


Here is a huge random polynomial in 12 variables with around 120k terms:



myPoly = Product[(RandomInteger[{-2, 2}] + RandomInteger[{-2, 2}] a + 
  RandomInteger[{-6, 6}] b + RandomInteger[{-6, 6}] c + 
  RandomInteger[{-6, 6}] d + RandomInteger[{-6, 6}] e + 
  RandomInteger[{-2, 2}] f + RandomInteger[{-1, 1}] g + 
  RandomInteger[{-6, 6}] h + RandomInteger[{-6, 6}] i + 
  RandomInteger[{-6, 6}] j + RandomInteger[{-6, 6}] k + 
  RandomInteger[{-1, 1}] l), {go, 1, 8}] // Expand;

Let's make it not a polynomial in a by replacing the 1000th term with Sin[a]:


myPoly = ReplacePart[myPoly, 1000 -> Sin[a]];


So, now let's see how Internal`PolynomialFunctionQ and PolynomialQ perform:


AbsoluteTiming[Internal`PolynomialFunctionQ[myPoly, a]]
AbsoluteTiming[Internal`PolynomialFunctionQ[myPoly, b]]
AbsoluteTiming[Internal`PolynomialFunctionQ[myPoly, {a, b, c, d, e, f, g, h, i, j, k, l}]]
AbsoluteTiming[Internal`PolynomialFunctionQ[myPoly, {b, c, d, e, f, g, h, i, j, k, l}]]
(*  {0.033989, False}  *)
(*  {0.032627, True}   *)
(*  {0.056368, False}  *)
(*  {0.074603, True}   *)


and


AbsoluteTiming[PolynomialQ[myPoly, a]]
AbsoluteTiming[PolynomialQ[myPoly, b]]
AbsoluteTiming[PolynomialQ[myPoly, {a, b, c, d, e, f, g, h, i, j, k, l}]]
AbsoluteTiming[PolynomialQ[myPoly, {b, c, d, e, f, g, h, i, j, k, l}]]
(*  {3.98786, False}  *)
(*  {4.00939, True}   *)
(*  {3.222, False}  *)
(*  {3.32597, True}   *)


It seems like Internal`PolynomialFunctionQ performs 100 times better than PolynomialQ when checking for polynomialness in one variable, and about 50 times better for multiple variables.


Is anyone aware of this? Is Internal`PolynomialFunctionQ a low-level version of PolynomialQ?



Answer



First note they are not equivalent:


PolynomialQ[x + x[1], x]
Internal`PolynomialFunctionQ[x + x[1], x]
(*
  True
  False

*)

Second note that PolynomialQ does a lot of checking:


Trace[
 PolynomialQ[x^2 Sin[y], x],
 TraceInternal -> True]

Mathematica graphics


If you care to, you can verify that every factor of every term seems to be checked:


With[{e = (a + 2 b + 3 c + 4 y)^2 // Expand},

 Trace[
  PolynomialQ[e, x],
  TraceInternal -> True]
 ]

What exactly this checking consists of seems to be inaccessible. I've seen Integrate`FakeIntervalElement before, but I don't know what it's for or why it is used here.


On the OP's example, 600,000 expressions are checked, I suppose:


Count[myPoly, a | b | c | d | e | f | g | h | i | j | k | l, Infinity]
(*  602194  *)


Probably Internal`PolynomialFunctionQ is meant for a narrower range of use and goes straight to work on determining whether the variables appear only in nonnegative powers, etc.


Comments

Post a Comment

Popular posts from this blog

functions - Get leading series expansion term?

Given a function f[x] , I would like to have a function leadingSeries that returns just the leading term in the series around x=0 . For example: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x)] x and leadingSeries[(1/x + 2 + (1 - 1/x^3)/4)/(4 + x)] -(1/(16 x^3)) Is there such a function in Mathematica? Or maybe one can implement it efficiently? EDIT I finally went with the following implementation, based on Carl Woll 's answer: lds[ex_,x_]:=( (ex/.x->(x+O[x]^2))/.SeriesData[U_,Z_,L_List,Mi_,Ma_,De_]:>SeriesData[U,Z,{L[[1]]},Mi,Mi+1,De]//Quiet//Normal) The advantage is, that this one also properly works with functions whose leading term is a constant: lds[Exp[x],x] 1 Answer Update 1 Updated to eliminate SeriesData and to not return additional terms Perhaps you could use: leadingSeries[expr_, x_] := Normal[expr /. x->(x+O[x]^2) /. a_List :> Take[a, 1]] Then for your examples: leadingSeries[(1/x + 2)/(4 + 1/x^2 + x), x] leadingSeries[Exp[x], x] leadingSeries[(1/x + 2 + (1 - 1/x...
Post a Comment
Read more

mathematical optimization - Minimizing using indices, error: Part::pkspec1: The expression cannot be used as a part specification

I want to use Minimize where the variables to minimize are indices pointing into an array. Here a MWE that hopefully shows what my problem is. vars = u@# & /@ Range[3]; cons = Flatten@ { Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; Minimize[{Total@((vec1[[#]] - vec2[[u[#]]])^2 & /@ Range[1, 3]), cons}, vars, Integers] The error I get: Part::pkspec1: The expression u[1] cannot be used as a part specification. >> Answer Ok, it seems that one can get around Mathematica trying to evaluate vec2[[u[1]]] too early by using the function Indexed[vec2,u[1]] . The working MWE would then look like the following: vars = u@# & /@ Range[3]; cons = Flatten@{ Table[(u[j] != #) & /@ vars[[j + 1 ;; -1]], {j, 1, 3 - 1}], 1 vec1 = {1, 2, 3}; vec2 = {1, 2, 3}; NMinimize[ {Total@((vec1[[#]] - Indexed[vec2, u[#]])^2 & /@ R...
Post a Comment
Read more

How to remap graph properties?

Graph objects support both custom properties, which do not have special meanings, and standard properties, which may be used by some functions. When importing from formats such as GraphML, we usually get a result with custom properties. What is the simplest way to remap one property to another, e.g. to remap a custom property to a standard one so it can be used with various functions? Example: Let's get Zachary's karate club network with edge weights and vertex names from here: http://nexus.igraph.org/api/dataset_info?id=1&format=html g = Import[ "http://nexus.igraph.org/api/dataset?id=1&format=GraphML", {"ZIP", "karate.GraphML"}] I can remap "name" to VertexLabels and "weights" to EdgeWeight like this: sp[prop_][g_] := SetProperty[g, prop] g2 = g // sp[EdgeWeight -> (PropertyValue[{g, #}, "weight"] & /@ EdgeList[g])] // sp[VertexLabels -> (# -> PropertyValue[{g, #}, "name"]...
Post a Comment
Read more