Finding Upper Bounds on Complex Function Extrema

Questions
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1

Hello together,

I am new to Mathematica and would like to find an upper bound on the number of extrema for a complex function, defined in terms of the variable h:

g[h, l, d]:=100*sqrt[((cos[d]*sin[h]/arccos[cos[d]*cos[h]*cos[l] + sin[d]*sin[l]] + ...;

The formula here is abbreviated since it’s quite large.

I’ve attempted to find possible extrema in terms of h using:

Solve[{D[g[h],h]==0},h];

However, Solve is not capable to find any solution. What else can I try?

Thank you very much for your help in advance. :slight_smile:

Greetings, Patrick

2

What else can I try?

Well, a good start could be to use a correct syntax! Mathematica is case sensitive!, So, your function definition should rather read like so:

  g[h_, l_, d_] := 100*Sqrt[((Cos[d]*Sin[h]/ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]] + ...))];

And then you can try to call

Solve[{D[g[h,l,d],h]==0},h];

because a function g[h] is not defined.

3

Thank you for your response. I’ve corrected the syntax and attempted to derive the function with respect to h. This function is the general form of the one I presented previously:

f[h_,l_,d_,x_,y_]:=Sqrt[(200*Sqrt[-Cos[d]^2*Cos[h]^2*Cos[l]^2 - 2*Cos[d]*Cos[h]*Cos[l]*Sin[d]*Sin[l] - Sin[d]^2*Sin[l]^2 + 1]*(Cos[d]^2*Cos[h]*Cos[l]*Sin[h] + Cos[d]*Sin[d]*Sin[h]*Sin[l])*x*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^2 + 10000*(Cos[d]^2*Cos[h]^2*Cos[l]^2 + 2*Cos[d]*Cos[h]*Cos[l]*Sin[d]*Sin[l] + Sin[d]^2*Sin[l]^2 - 1)*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^3 + ((Cos[d]^2*Cos[h]^2*Cos[l]^2 + 2*Cos[d]*Cos[h]*Cos[l]*Sin[d]*Sin[l] + Sin[d]^2*Sin[l]^2 - 1)*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^3 - (2*Cos[d]^4*Cos[h]^2*Cos[l]^2*Sin[h]^2 + 4*Cos[d]^3*Cos[h]*Cos[l]*Sin[d]*Sin[h]^2*Sin[l] + 2*Cos[d]^2*Sin[d]^2*Sin[h]^2*Sin[l]^2 - Cos[d]^2*Sin[h]^2)*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]])*x^2 - ((Cos[d]^2*Cos[h]^2*Cos[l]^2 + 2*Cos[d]*Cos[h]*Cos[l]*Sin[d]*Sin[l] + Sin[d]^2*Sin[l]^2)*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^3 - (2*Cos[d]^4*Cos[h]^2*Cos[l]^2*Sin[h]^2 + 4*Cos[d]^3*Cos[h]*Cos[l]*Sin[d]*Sin[h]^2*Sin[l] + 2*Cos[d]^2*Sin[d]^2*Sin[h]^2*Sin[l]^2 - Cos[d]^2*Sin[h]^2)*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]])*y^2 - 2*(100*Sqrt[-Cos[d]^2*Cos[h]^2*Cos[l]^2 - 2*Cos[d]*Cos[h]*Cos[l]*Sin[d]*Sin[l] - Sin[d]^2*Sin[l]^2 + 1]*(Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l])*Sqrt[-(Cos[d]^2*Sin[h]^2 - ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^2)/ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^2]*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^3 + ((Cos[d]^3*Cos[h]^2*Cos[l]^2*Sin[h] + 2*Cos[d]^2*Cos[h]*Cos[l]*Sin[d]*Sin[h]*Sin[l] + Cos[d]*Sin[d]^2*Sin[h]*Sin[l]^2 - Cos[d]*Sin[h])*(-(Cos[d]^2*Sin[h]^2 - ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^2)/ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^2)^(3/2)*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^2 + (Cos[d]^5*Cos[h]^2*Cos[l]^2*Sin[h]^3 + 2*Cos[d]^4*Cos[h]*Cos[l]*Sin[d]*Sin[h]^3*Sin[l] + Cos[d]^3*Sin[d]^2*Sin[h]^3*Sin[l]^2 - Cos[d]^3*Sin[h]^3 - (3*Cos[d]^3*Cos[h]^2*Cos[l]^2*Sin[h] + 6*Cos[d]^2*Cos[h]*Cos[l]*Sin[d]*Sin[h]*Sin[l] + 3*Cos[d]*Sin[d]^2*Sin[h]*Sin[l]^2 - 2*Cos[d]*Sin[h])*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^2)*Sqrt[-(Cos[d]^2*Sin[h]^2 - ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^2)/ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^2])*x)*y)/((Cos[d]^2*Cos[h]^2*Cos[l]^2 + 2*Cos[d]*Cos[h]*Cos[l]*Sin[d]*Sin[l] + Sin[d]^2*Sin[l]^2 - 1)*ArcCos[Cos[d]*Cos[h]*Cos[l] + Sin[d]*Sin[l]]^3)];

solutions = Solve[{D[f[h,l,d,x,y],h]==0, -Pi < l, l < Pi, -Pi/4 < d, d < Pi/4}, h];

However, Solve and Reduce do not produce useful outputs or do not halt. Is there an automated way to detect substitution patterns, or could I use a mathematical trick to estimate the upper bound on the number of extrema for any initial values of l,d,x,y with l∈[−π,π] and d∈[−π/4,π/4]?

4

Since you are looking for extrema, I would perhaps remove the Sqrt that envelops the function. Even then, the problem seems too complicated to be solved symbolically. I would make some conjectures based on numerical data, like this:

Table[Length[Union@NSolve[{D[f[h, l, d, 1, 1], h] == 0,
     -Pi < h < Pi}, h, Reals]],
 {l, -Pi, Pi, Pi/2}, {d, -Pi/4, Pi/4, Pi/8}]