mathematical optimization - Fitting one function to another

I'd like to see how close I can fit an exponential function to a cosine function in the domain of $\;-\frac{\pi}{2}$ to $\frac{\pi}{2}$ in the least squares sense.

FindFit[Cos[x], {A*Exp[l*(Cos[x] - 1)], -π/2 <= x <= π/2}, {A, l},  x]

FindFit::fitd: First argument Cos[x] in FindFit is not a list or a rectangular array.

I could create a table for Cos[x] to pass as the first parameter but I feel like I'm approaching this wrong and should be able to give Mathematica my problem in symbolic form. Perhaps, I should try Minimize but I didn't get any results out of it.

Minimize[{(Cos[x] - A*Exp[l*(Cos[x] - 1)])^2, -π/2 <= x <= π/2}, {A, l, x}]

Minimize[{(-A E^(l (-1 + Cos[x])) + Cos[x])^2, -π/2 <= x <= π/2}, {A, l, x}]

Answer

You can do the integral symbolically, then minimize numerically:

distance = 
 Integrate[(A*Exp[l*(Cos[x] - 1)] - Cos[x])^2, {x, -Pi/2, Pi/2}, 
  Assumptions -> (A | l) \[Element] Reals]

{min, sol} = NMinimize[distance, {A, l}]

Plot[{A*Exp[l*(Cos[x] - 1)], Cos[x]} /. sol // Evaluate, {x, -Pi/2, Pi/2}, 
  Filling -> {1 -> {2}}, PlotStyle -> Black]

enter image description here

Comments

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...

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...

What is and isn't a valid variable specification for Manipulate?

I have an expression whose terms have arguments (representing subscripts), like this: myExpr = A[0] + V[1,T] I would like to put it inside a Manipulate to see its value as I move around the parameters. (The goal is eventually to plot it wrt one of the variables inside.) However, Mathematica complains when I set V[1,T] as a manipulated variable: Manipulate[Evaluate[myExpr], {A[0], 0, 1}, {V[1, T], 0, 1}] (*Manipulate::vsform: Manipulate argument {V[1,T],0,1} does not have the correct form for a variable specification. >> *) As a workaround, if I get rid of the symbol T inside the argument, it works fine: Manipulate[ Evaluate[myExpr /. T -> 15], {A[0], 0, 1}, {V[1, 15], 0, 1}] Why this behavior? Can anyone point me to the documentation that says what counts as a valid variable? And is there a way to get Manpiulate to accept an expression with a symbolic argument as a variable? Investigations I've done so far: I tried using variableQ from this answer , but it says V[1...

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