Skip to main content

numerical integration - How to use NIntegrate when there are symbolic constant coefficients


I would like to numerically integrate an equation such as the one below in which there are symbolic constant coefficients. I used a very simple code but it doesn't work in general, that tried to deal with constant coefficients with patterns. Is there any general approach to NIntegrate where symbolic constant coefficients exist?


11.94` a[1, 1]^2 Cos[x]^2 Cos[θ]^2 + 
   21.31` c[1, 1]^2 Cos[x]^2 Cos[θ]^2 + 
   0.14702` a[1, 1] b[1, 1] Cos[x] Cos[θ]^2 Sin[x] - (
   1.395` b[1, 1]^2 Cos[x] Cos[θ]^2 Sin[x])/(1 + x/2)^3 + 
   0.4669` b[1, 1]^2 Cos[θ]^2 Sin[x]^2 + (
   1.395` b[1, 1]^2 Cos[θ]^2 Sin[x]^2)/(1 + x/
     2)^4 /. 

{b[a1_, a2_] b[a3_, a4_] g_ :> b[a1, a2] b[a3, a4] NIntegrate[g, {θ, 0, 2}, {x, 0, 1}], 
 a[a1_, a2_] a[a1_, a2_] g_ :> a[a1, a2] a[a1, a2] NIntegrate[g, {θ, 0, 2}, {x, 0, 1}, 
                               Method -> {Automatic, "SymbolicProcessing" -> 0}],
 a[a1_, a2_] a[a3_, a4_] g_ :> a[a1, a2] a[a3, a4] NIntegrate[g, {θ, 0, 2}, {x, 0, 1}, 
                               Method -> {Automatic, "SymbolicProcessing" -> 0}],
 b[a1_, a2_] b[a1_, a2_] g_ :> b[a1, a2] b[a1, a2] NIntegrate[g, {θ, 0, 2}, {x, 0, 1}, 
                               Method -> {Automatic, "SymbolicProcessing" -> 0}],
 c[a1_, a2_] c[a1_, a2_] g_ :> c[a1, a2] c[a1, a2] NIntegrate[g, {θ, 0, 2}, {x, 0, 1}, 
                               Method -> {Automatic, "SymbolicProcessing" -> 0}],
 a[a1_, a2_] b[a3_, a4_] g_ :> a[a1, a2] b[a3, a4] NIntegrate[g, {θ, 0, 2}, {x, 0, 1}, 

                               Method -> {Automatic, "SymbolicProcessing" -> 0}],
 a[a1_, a2_] c[a3_, a4_] g_ :> a[a1, a2] c[a3, a4] NIntegrate[g, {θ, 0, 2}, {x, 0, 1}, 
                               Method -> {Automatic, "SymbolicProcessing" -> 0}],
 b[a1_, a2_] c[a3_, a4_] g_ :> a[a1, a2] a[a3, a4] NIntegrate[g, {θ, 0, 2}, {x, 0, 1}, 
                               Method -> {Automatic, "SymbolicProcessing" -> 0}],
 c[a1_, a2_] c[a3_, a4_] g_ :> c[a1, a2] c[a3, a4] NIntegrate[g, {θ, 0, 2}, {x, 0, 1}, 
                               Method -> {Automatic, "SymbolicProcessing" -> 0}]
   } // Timing

Answer



You have to be able to reduce the integrand to a sum of terms in which the symbolic constants are factors of the term. Then you can separate the terms and their factors. Gather the factors that are constants, and numerically integrate the (product of the) function-factors of each term.



expr = 11.94` a[1, 1]^2 Cos[x]^2 Cos[θ]^2 + 
   21.31` c[1, 1]^2 Cos[x]^2 Cos[θ]^2 + 
   0.14702` a[1, 1] b[1, 1] Cos[x] Cos[θ]^2 Sin[x] -
   (1.395` b[1, 1]^2 Cos[x] Cos[θ]^2 Sin[x]) / (1 + x/2)^3 + 
   0.4669` b[1, 1]^2 Cos[θ]^2 Sin[x]^2 +
   (1.395` b[1, 1]^2 Cos[θ]^2 Sin[x]^2) / (1 + x/2)^4;
terms = List @@@ List @@ expr
(*
   {{11.94`, a[1,1]^2, Cos[x]^2, Cos[θ]^2},
    {21.31`, c[1,1]^2, Cos[x]^2, Cos[θ]^2},

    {0.14702`, a[1,1], b[1,1], Cos[x], Cos[θ]^2, Sin[x]},
    {-1.395`, 1/(1+x/2)^3, b[1,1]^2, Cos[x], Cos[θ]^2, Sin[x]},
    {0.4669`, b[1,1]^2, Cos[θ]^2, Sin[x]^2},
    {1.395`, 1/(1+x/2)^4, b[1,1]^2, Cos[θ]^2, Sin[x]^2}}
*)

integrals = GatherBy[#, MemberQ[#, x | θ, Infinity] &] & /@ terms;
(Times @@@ integrals[[All, 1]]) *
  (NIntegrate[Times @@ #,
      {θ, 0, 2}, {x, 0, 1}, 

      Method -> {Automatic, "SymbolicProcessing" -> 0}] & /@ 
    integrals[[All, 2]]) // Total
(*
   7.04119 a[1, 1]^2 + 0.0422025 a[1, 1] b[1, 1] +
    0.00809433 b[1, 1]^2 + 12.5668 c[1, 1]^2
*)

Comments

Post a Comment

Popular posts from this blog

front end - keyboard shortcut to invoke Insert new matrix

I frequently need to type in some matrices, and the menu command Insert > Table/Matrix > New... allows matrices with lines drawn between columns and rows, which is very helpful. I would like to make a keyboard shortcut for it, but cannot find the relevant frontend token command (4209405) for it. Since the FullForm[] and InputForm[] of matrices with lines drawn between rows and columns is the same as those without lines, it's hard to do this via 3rd party system-wide text expanders (e.g. autohotkey or atext on mac). How does one assign a keyboard shortcut for the menu item Insert > Table/Matrix > New... , preferably using only mathematica? Thanks! Answer In the MenuSetup.tr (for linux located in the $InstallationDirectory/SystemFiles/FrontEnd/TextResources/X/ directory), I changed the line MenuItem["&New...", "CreateGridBoxDialog"] to read MenuItem["&New...", "CreateGridBoxDialog", MenuKey["m", Modifiers-...
Post a Comment
Read more

How to thread a list

I have data in format data = {{a1, a2}, {b1, b2}, {c1, c2}, {d1, d2}} Tableform: I want to thread it to : tdata = {{{a1, b1}, {a2, b2}}, {{a1, c1}, {a2, c2}}, {{a1, d1}, {a2, d2}}} Tableform: And I would like to do better then pseudofunction[n_] := Transpose[{data2[[1]], data2[[n]]}]; SetAttributes[pseudofunction, Listable]; Range[2, 4] // pseudofunction Here is my benchmark data, where data3 is normal sample of real data. data3 = Drop[ExcelWorkBook[[Column1 ;; Column4]], None, 1]; data2 = {a #, b #, c #, d #} & /@ Range[1, 10^5]; data = RandomReal[{0, 1}, {10^6, 4}]; Here is my benchmark code kptnw[list_] := Transpose[{Table[First@#, {Length@# - 1}], Rest@#}, {3, 1, 2}] &@list kptnw2[list_] := Transpose[{ConstantArray[First@#, Length@# - 1], Rest@#}, {3, 1, 2}] &@list OleksandrR[list_] := Flatten[Outer[List, List@First[list], Rest[list], 1], {{2}, {1, 4}}] paradox2[list_] := Partition[Riffle[list[[1]], #], 2] & /@ Drop[list, 1] RM[list_] := FoldList[Transpose[{First@li...
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