NMinimize a function of NMaximize

*To*: mathgroup at smc.vnet.net*Subject*: [mg79947] NMinimize a function of NMaximize*From*: sapsi <saptarshi.guha at gmail.com>*Date*: Thu, 9 Aug 2007 05:24:10 -0400 (EDT)

I have a function g[x_]:=1 (Pi - 2 x*Sqrt[1 - x^2] - 2 ArcSin[x]) d[x_, o_, n_] := Abs[g[x] - Normal[Series[g[y], {y, o, n}]]] /. y -> x d[] is the absolute difference of g[] and its Taylor series approximation dm[p_] := NMaximize[{d[x, p, 3], 0 <= x <= 1}, x][[1]] dm[] is the maximum absolute difference between the Taylor series approximation(3rd order) around p and g[]. I wish to find that value of 'p' that minimizes this maximum difference - i tried plotting and can see where the minima occurs but would like the exact value. So, it thought this would work NMinimize[{NMaximize[{ d[x, p, 3], 0 <= x <= 1}, x], 0 <= p <= 1}, p]. Instead i get errors (briefly) NMaximize::nnum: The function value -Abs[-2.41057-(2 (<<19>>-p)^2 \ p)/Sqrt[1-p^2]+2 p Sqrt[1-p^2]+(2 (<<1>>)^3)/(3 Sqrt[1-<<1>>] \ (-1+p^2))-(4 (0.652468-p) (-1+p^2))/Sqrt[1-p^2]+2 ArcSin[p]] is not a \ number at {x} = {0.652468}. >> NMaximize::nnum: "The function value \ -Abs[-2.41057-(2\(<<19>>-p)^2\p)/Sqrt[1-p^2]+2\ p\ Sqrt[1-p^2]+(2\(<<\ 1>>)^3)/(3\Sqrt[1-<<1>>]\(-1+p^2))-(4\(0.652468-p)\(-1+p^2))/Sqrt[1-p^ \ 2]+2\ ArcSin[p]] is not a number at {x} = {0.6524678079740285`}." NMaximize::nnum: The function value -Abs[-2.41057-(2 (<<19>>-p)^2 \ p)/Sqrt[1-p^2]+2 p Sqrt[1-p^2]+(2 (<<1>>)^3)/(3 Sqrt[1-<<1>>] \ (-1+p^2))-(4 (0.652468-p) (-1+p^2))/Sqrt[1-p^2]+2 ArcSin[p]] is not a \ number at {x} = {0.652468}. >> NMinimize::nnum: Can anyone provide any pointers on how to find the minimum of dm[]? Thank you for your time Saptarshi

**Follow-Ups**:**Re: NMinimize a function of NMaximize***From:*Daniel Lichtblau <danl@wolfram.com>