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Mathematics Equation Solving Symbolic Computations

Finding Upper Bounds on Complex Function Extrema

Patrick DS.

Posted 2 months ago

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]:=100sqrt[((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. :-) Greetings, Patrick

POSTED BY: Patrick DS.

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Patrick DS.

Posted 2 months ago

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[(200Sqrt[-Cos[d]^2Cos[h]^2Cos[l]^2 - 2Cos[d]Cos[h]Cos[l]Sin[d]Sin[l] - Sin[d]^2Sin[l]^2 + 1](Cos[d]^2Cos[h]Cos[l]Sin[h] + Cos[d]Sin[d]Sin[h]Sin[l])xArcCos[Cos[d]Cos[h]Cos[l] + Sin[d]Sin[l]]^2 + 10000(Cos[d]^2Cos[h]^2Cos[l]^2 + 2Cos[d]Cos[h]Cos[l]Sin[d]Sin[l] + Sin[d]^2Sin[l]^2 - 1)ArcCos[Cos[d]Cos[h]Cos[l] + Sin[d]Sin[l]]^3 + ((Cos[d]^2Cos[h]^2Cos[l]^2 + 2Cos[d]Cos[h]Cos[l]Sin[d]Sin[l] + Sin[d]^2Sin[l]^2 - 1)ArcCos[Cos[d]Cos[h]Cos[l] + Sin[d]Sin[l]]^3 - (2Cos[d]^4Cos[h]^2Cos[l]^2Sin[h]^2 + 4Cos[d]^3Cos[h]Cos[l]Sin[d]Sin[h]^2Sin[l] + 2Cos[d]^2Sin[d]^2Sin[h]^2Sin[l]^2 - Cos[d]^2Sin[h]^2)ArcCos[Cos[d]Cos[h]Cos[l] + Sin[d]Sin[l]])x^2 - ((Cos[d]^2Cos[h]^2Cos[l]^2 + 2Cos[d]Cos[h]Cos[l]Sin[d]Sin[l] + Sin[d]^2Sin[l]^2)ArcCos[Cos[d]Cos[h]Cos[l] + Sin[d]Sin[l]]^3 - (2Cos[d]^4Cos[h]^2Cos[l]^2Sin[h]^2 + 4Cos[d]^3Cos[h]Cos[l]Sin[d]Sin[h]^2Sin[l] + 2Cos[d]^2Sin[d]^2Sin[h]^2Sin[l]^2 - Cos[d]^2Sin[h]^2)ArcCos[Cos[d]Cos[h]Cos[l] + Sin[d]Sin[l]])y^2 - 2(100Sqrt[-Cos[d]^2Cos[h]^2Cos[l]^2 - 2Cos[d]Cos[h]Cos[l]Sin[d]Sin[l] - Sin[d]^2Sin[l]^2 + 1](Cos[d]Cos[h]Cos[l] + Sin[d]Sin[l])Sqrt[-(Cos[d]^2Sin[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]^3Cos[h]^2Cos[l]^2Sin[h] + 2Cos[d]^2Cos[h]Cos[l]Sin[d]Sin[h]Sin[l] + Cos[d]Sin[d]^2Sin[h]Sin[l]^2 - Cos[d]Sin[h])(-(Cos[d]^2Sin[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]^5Cos[h]^2Cos[l]^2Sin[h]^3 + 2Cos[d]^4Cos[h]Cos[l]Sin[d]Sin[h]^3Sin[l] + Cos[d]^3Sin[d]^2Sin[h]^3Sin[l]^2 - Cos[d]^3Sin[h]^3 - (3Cos[d]^3Cos[h]^2Cos[l]^2Sin[h] + 6Cos[d]^2Cos[h]Cos[l]Sin[d]Sin[h]Sin[l] + 3Cos[d]Sin[d]^2Sin[h]Sin[l]^2 - 2Cos[d]Sin[h])ArcCos[Cos[d]Cos[h]Cos[l] + Sin[d]Sin[l]]^2)Sqrt[-(Cos[d]^2Sin[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]^2Cos[h]^2Cos[l]^2 + 2Cos[d]Cos[h]Cos[l]Sin[d]Sin[l] + Sin[d]^2Sin[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]?

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]?

POSTED BY: Patrick DS.

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 1 month ago

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}]

POSTED BY: Gianluca Gorni

Henrik Schachner

Henrik Schachner, Radiation Therapy Center, Weilheim, Germany

Posted 2 months ago

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_] := 100Sqrt[((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.

POSTED BY: Henrik Schachner

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