Skip to main content

plotting - Testing the accuracy of numerically computed derivatives


I am calculating approximate derivatives by using NDSolve`FiniteDifferenceDerivative, so this works:


 Subscript[Der, i_][yyy_] := 
   Module[{xx}, 
     xx = Length[yyy];
     NDSolve`FiniteDifferenceDerivative[

       Derivative[i], 
       N[yyy], 
       DifferenceOrder -> 2] @ "DifferentiationMatrix"
         // Normal // Developer`ToPackedArray // SparseArray];

xi = 1.;
xf = -1;
yy = 100; 
xgrid = Table[xi + i (xf - xi/yy), {i, 0, yy}];


(Der1 = Subscript[Der, 1][xgrid]) // MatrixForm;

numerical = Der1.Exp[-xgrid^2];
exact = -2*xgrid*Exp[-xgrid^2];

diff = numerical - exact; 
diffError = yy^2*diff

ListLinePlot[yy^2 Abs[diff]]


I want to show my solution is accurate by demonstrating that the difference between the numerical solution and the exact solution goes to zero as $\mathtt{yy}^{-2}$. For this I want to plot $\mathtt{yy}^2 |\mathrm{numerical} - \mathrm{exact}|$ for different values of $\mathtt{yy}$ but am not sure how to do this.


The code gives reasonable values for the differences, though I am not sure how to plot them for different $\mathtt{yy}$ values.


I obtained the follow plot from the code shown above.


accuracytest



Answer



xi = -1.;
xf = 1;
xgrid[yy_] := Range[xi, xf, (xf - xi)/yy]

Subscript[Der, i_][yyy_] := Module[{xx}, xx = Length[yyy];

 NDSolve`FiniteDifferenceDerivative[Derivative[i], N[yyy], 
    DifferenceOrder -> 2]@"DifferentiationMatrix" // Normal // 
 Developer`ToPackedArray // SparseArray];

Der1[yy_] := Subscript[Der, 1][xgrid[yy]]
numerical[yy_] := Der1[yy].Exp[-xgrid[yy]^2]
exact[yy_] := -2*xgrid[yy]*Exp[-xgrid[yy]^2]
diff[yy_] := numerical[yy] - exact[yy]

yyvals = {100, 300, 1000};


ListLinePlot[
 Table[Transpose[{xgrid[yy], yy^2 Abs[diff[yy]]}], {yy, yyvals}], 
 PlotRange -> All, PlotLegends -> StringTemplate["yy = ``"] /@ yyvals]

enter image description here


Max[diff[100]] / Max[diff[1000]] = 99.9756

This means the error ~ 1/yy^2. For better see this scaling low one can use logarithmic scale:


ListLinePlot[

 Table[Transpose[{xgrid[yy], Abs[diff[yy]]}], {yy, yyvals}], 
 PlotRange -> All, PlotLegends -> StringTemplate["yy = ``"] /@ yyvals,
 ScalingFunctions -> "Log", Frame -> True]

enter image description here


NonlinearModelFit[Table[{yy, Max[diff[yy]]}, {yy, 100, 10000, 500}], 
 a + b/x^2, {a, b}, x]


Blockquote




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