Carl Love

21905 Reputation

9 years, 57 days
Natick, Massachusetts, United States
My name was formerly Carl Devore.

fsolve for each point...

@phil76600

The reason that it is slow is that an fsolve command is used for every point plotted. In other words, in order to evaluate the implicit function numerically at a value of Vb, I solve equation eq numerically for pH. (Do you understand that?) From the gaps that you're getting, I guess that the fsolve may be having trouble converging for some points. What version of Maple are you using? I am not having that trouble in Maple 17. If it really matters, it may be possible to adjust some options and fill those gaps. Are you getting gaps in the plot of the primary also? I can't figure out a reason that you would only get gaps in the derivative. Both the primary and the derivative are applying fsolve to the same equation.

You can improve the speed by reducing the value of numpoints in the plots. The default value is about 200.

`f'`:= (-D[2]/D[1])(F) @ (V__b-> (f(V__b),V__b)):

Why is 'f' necessary.

Are you asking why the quotes are necessary? If that's what you're asking: I wanted to name the derivative f ', that being one of the normal representations of a derivative and the one that you used. But f ' is not a valid name in Maple because it contains a special character. However, anything at all can be made into a valid name by enclosing it in back quotes (``). Does that answer your question?

And what is that @ in this line ?

The @ is how one expresses functional composition. In other words, if h(x) = f(g(x)), then h = f@g. I did it this way to separate the symbolic computation (the left side) and the numeric computation (the right side). That way the symbolic differentiation is only done one time total, and the fsolve is only done one time for each point. Does that answer your question?

fsolve for each point...

@phil76600

The reason that it is slow is that an fsolve command is used for every point plotted. In other words, in order to evaluate the implicit function numerically at a value of Vb, I solve equation eq numerically for pH. (Do you understand that?) From the gaps that you're getting, I guess that the fsolve may be having trouble converging for some points. What version of Maple are you using? I am not having that trouble in Maple 17. If it really matters, it may be possible to adjust some options and fill those gaps. Are you getting gaps in the plot of the primary also? I can't figure out a reason that you would only get gaps in the derivative. Both the primary and the derivative are applying fsolve to the same equation.

You can improve the speed by reducing the value of numpoints in the plots. The default value is about 200.

`f'`:= (-D[2]/D[1])(F) @ (V__b-> (f(V__b),V__b)):

Why is 'f' necessary.

Are you asking why the quotes are necessary? If that's what you're asking: I wanted to name the derivative f ', that being one of the normal representations of a derivative and the one that you used. But f ' is not a valid name in Maple because it contains a special character. However, anything at all can be made into a valid name by enclosing it in back quotes (``). Does that answer your question?

And what is that @ in this line ?

The @ is how one expresses functional composition. In other words, if h(x) = f(g(x)), then h = f@g. I did it this way to separate the symbolic computation (the left side) and the numeric computation (the right side). That way the symbolic differentiation is only done one time total, and the fsolve is only done one time for each point. Does that answer your question?

phi[0]?...

The second ODE contains a constant phi[0]. Unless you say otherwise, I am going to assume that that should be phi(0), which equals 1 according to your boundary conditions.

Maple IDE...

In the Maplesoft Blog Posts sections of this forum, look at the most recent Post, entitled "Introducing the Maple IDE". That product will let you step through the code line by line in a separate window.

No inequality sign...

Your expression has no inequality sign.

8, 9, 23 wrong...

Your values for n = 8, 9, and 23 are obviousy wrong. And why do some have only 9 digits?

Considering how difficult this is to compute, I think that you should post your Mathematica code so that it can be verified.

How to compute?...

Recursively, we have a(n) = n^a(n-1), a(1)=1. And we're displaying a(n) mod 10^10.

I don't know how to use Maple to compute this for n > 6. Anyone have an idea? How does Mathematica do it? How do you reduce the exponent to a computable value in modular arithmetic? In other words, we want to compute

(A &^ B) mod 10^10

where B itself is too large to compute. Is there some way to reduce B using the modulus? Surely Mathematica cannot compute a(49) directly. Surely that number has more bits in its representation than there are electrons in the Universe.

But there is a pattern to the moduli of a(n) based on the last digit of n.

Marvin Ray Burns wrote:

I've been playing around with power towers and thought my latest might be interesting enough to post.

It's only interesting to me if we discuss algorithms for computing such things.

One execution group...

Do you see that as being substantially different from having a single worksheet with all the code in one execution group?

Yes...

@tuGUTS Yes, you can add initial and boundary conditions. These will, if they work at all, remove some or all of the arbitrary functions and constants from the solution. I also don't know what would be appropriate initial and boundary conditions for this PDE. But see ?pdsolve for numerous examples.

Yes...

@tuGUTS Yes, you can add initial and boundary conditions. These will, if they work at all, remove some or all of the arbitrary functions and constants from the solution. I also don't know what would be appropriate initial and boundary conditions for this PDE. But see ?pdsolve for numerous examples.

Try dsolve...

You equations are effectively a system of ordinary differential equation. Try changing alpha(r,t) to alpha(r) and then using dsolve. (Although I wouldn't be surprised if pdsolve had already tried that.) It might help to make assumptions on c and kappa such as assume(c>0, kappa>0).

I would try these things myself, but I don't feel like typing in your equations. Please post them in plaintext or upload a worksheet.

Extract points? Veq?...

Are you saying that you want to extract the point data from the plot?

What is Veq?

Post equations...

Can you post the equations?

error message, or no result?...

Are you getting an error message from pdsolve, or no result at all?

Simplfied plot command...

A simplified version of your plotting command:

plot(
[zip(`[]`,X1, Y1), zip(`[]`, X2, Y2)],
color= [blue, red],
style= point, symbol= solidcircle, symbolsize= 8,
view= [0 .. 2, 0 .. 1], scaling= constrained
);

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