Skip to main content

calculus and analysis - Interpolating an Antiderivative



I'd like to be able to make InterpolatingFunctions for antiderivatives of functions that can't be integrated symbolically. However, the following code returns several error messages:


FunctionInterpolation[Integrate[Sqrt[1 + x^3], {x, 0, t}], {t, 0, 10}]

Here are the messages:



Thread::tdlen: "Objects of unequal length in {-1.25,-0.416667,0.416667,1.25}^{} cannot be combined. "


Thread::tdlen: "Objects of unequal length in {0.223144 +3.14159\ I,-0.875469+3.14159\ I,-0.875469,0.223144}\ {}\n cannot be combined. "


Thread::tdlen: "Objects of unequal length in {-1.25,-0.416667,0.416667,1.25}\ {} cannot be combined."


General::stop: "Further output of Thread::tlden will be suppressed during this calculation."


FunctionInterpolation::nreal: Near t = 1.25`, the function did not evaluate to a real number.



FunctionInterpolation::nreal: Near t = 1.3277777777777777`, the function did not evaluate to a real number.



What's going on? Is there a simple way to make this work? Changing Integrate to NIntegrate doesn't help, though the error messages are different:



NIntegrate::nlim: x = t is not a valid limit of integration.


NIntegrate::nlim: x = t is not a valid limit of integration.


NIntegrate::nlim: x = t is not a valid limit of integration.


General::stop: "Further output of NIntegrate::nlim will be suppressed during this calculation. "




Answer




Use NDSolve


antiD = NDSolveValue[{f'[x] == Sqrt[1 + x^3], f[0] == 0}, f, {x, 0, 10}]

Example usage:


Plot[antiD[x], {x, 0, 10}]

Mathematica graphics



Alternatively...


This works because this function can be antidifferentiated (by Mathematica).



antiD = FunctionInterpolation[
 Evaluate @ Integrate[Sqrt[1 + x^3], {x, 0, t}, Assumptions -> 0 < t < 10],
 {t, 0, 10}]

or...


integral[t_?NumericQ] := NIntegrate[Sqrt[1 + x^3], {x, 0, t}];
FunctionInterpolation[integral[t], {t, 0, 10}]

FunctionInterpolation evaluated its argument on a symbolic t. The pattern test ?NumericQ prevents evaluation of NIntegrate until an actual number is substituted for t. See also What are the most common pitfalls awaiting new users?. Note also that this way does many evaluations of NIntegrate, whereas the NDSolve method does just one integration.


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

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

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