How to get the maximum value of an implicit function?

I have the following implicit function:

I-Ipv+Io.*exp(((V+Rs.*I)/(Vt.*a))-1)-(V+Rs.*I)/Rp = 0

where I and V are unknowns and everything else is known. From this equation, I am trying to acheive the maximum value of P where P = V*I. To acheive this, I wrote the following codes:

fun = @(V,I) I-Ipv+Io.*exp(((V+Rs.*I)/(Vt.*a))-1)-(V+Rs.*I)/Rp;
f = fimplicit(fun,[0,40,0,9])
voltage = f.XData;
current = f.YData;
power = voltage.*current
maximum_power = max(power)

I believe this method does not allow me to change the number of points in X, therefore the calculated maximum power is not as accurate.

I would appreciate any suggestion regarding calculating a maximum power.

Thank you.

Answer 1

[John D'Errico]

If you have the otimization toolbox, then just formulate it as a nonlinear optimization. fmincon will be the correct tool. That is, minimize -V*I, subject to a nonlinear equality constraint, as you have written. So two variables, one equality constraint.

[Eungkyun Kim]

Thanks. Do you mind elaborating on how to use fmincon to find the maximum value of V*I?

[Eungkyun Kim]

I think I got it actually. Thanks.

[Matt J]

@Eungkyun Kim

If John's advice worked for you, please Accept-click his answer.

Answer 2

[Matt J]

I believe this method does not allow me to change the number of points in X,

It does, e.g.,

f = fimplicit(fun,[0,40,0,9],'MeshDensity',1000)

If it were me, though, I would probably use this method to initialize fmincon, as recommended by John.

[Eungkyun Kim]

Thanks! I didn't know I could do that.