How to Graph and Use the Results of Numerical Integration as a Function

Mathmagic, I'm very new to this and don't know if my initial response to you was posted. If so, I apologize for being redundant. Thank you again for your response. I don't see how to achieve either the accuracy or precision I need by using the freehand tool as you suggest. Perhaps I don't understand the tool. Could you explain in more detail, please?

I think my problem must have been solved. It is analogous to the acceleration, speed, and distance problem, i.e., one integrates the acceleration to get speed and uses speed and time to get distance. In my case the acceleration is not constant (there is jerk) and the acceleration must be integrated numerically to get the speed as a function of time. Only with that (numerical) function can one calculate distance as a function of time. In my case, the result of the numerical integration is an expansion parameter, which is analogous to speed as a function of time.

Pete Willson

the method for numerical integral is runge-kutta I think. I can not speak about this question because I do not know the accuracy of the method in GG.

nintegral command give a colection of points that are an aprox solution of ODE

you can see it in my file typing l1 = First(Element({NSolveODE({f', g', h'}, 0, {1, 2, -2}, 3)}, 1), Length(Element({NSolveODE({f', g', h'}, 0, {1, 2, -2}, 3)}, 1)))

freehand creates a function with this list

Loco,

Here you go. The constants are currently hard coded in the function, but they will become parameters.

Loco, I lost version control. Here is the correct file. Sorry!

okay, there are different possible approaches in GGb

you could determine some data points and define linear or some other functions over some subsets and use them for further computation kinda like (FEM)

or you could use some function fit over the data points that you want to have as function

or you use the approach of mathmagic which was unknown to me (for sure undocumented functionality) that approach uses the given function to numerically integrate extract the data points and use them for a linear interpolated free hand function?

g(x) = If(x > 0, 1 / (-0.00008 + 0.00008 / x² + 0.3 / x + 0.7x²)^0.5, 0)

f(x) = Function(Join({{0, 40}, Zip(y(P), P, First(Element({NSolveODE({g}, 0, {0}, 40)}, 1), Length(Element({NSolveODE({g}, 0, {0}, 40)}, 1))))}))

DataFunction() might be better than Freehand function, eg https://www.geogebra.org/m/nrxmfnmf

Well did not knew that either

g(x) = If(x > 0, 1 / (-0.00008 + 0.00008 / x^2 + 0.3 / x + 0.7 x^2)^0.5, 0)

DataFunction(Join({{0},Sequence(t, t, 0.1, 20, 0.1)}), Join({{0},Sequence(Integral(g, 0, t), t, 0.1, 20, 0.1)}))

note: datafunction is not updated automatically. it needs to be updated with a script

Thanks very much to each of the three of you. I have enough to give it a try--I was stumped! I can also compare results with an excel calculation from an independent party. Unfortunately it will be a good four hours before I am able to do so, but I will report the results.

Michael, you asked for an example of subsequent calculation. This function is the so-called scale factor in cosmology that comes from the Friedman solution to Einstein's (gravitational) field equation. I will use it subsequently in a model to calculate and plot the expansion of the universe as function of time following the "big bang". I'll also use it to calculate the (geodesic) paths that light travels as space (the universe) expands. I'm trying to animate some of this to help others visualize the expanding universe. The dynamic geometry is not easy to visualize. This is neither new science nor aimed at any commercial endeavor--I'm a (retired) volunteer who likes to teach.

PERFECT! Thank each of you--it works perfectly. I had spent soooo much time trying figure out how to do this. The most important check of accuracy/precision is spot-on to at least four significant figures.

Perhaps there are two options for dynamic updates. The only things that change are the individual coefficients in the denominator of the the function (and their sum, which is also in the denominator). I thought perhaps a simple If statement that checked the sum of the coefficients to trigger a recalculation may be simpler than a script. I visited Michael's Piece-Wise Curve on the website, but am unable to figure out how to see the underlying script.

Thank you again. I hope that I, too, will be able to help somebody in the future.

There must be a way to mark this as "solved", but I don't find the button.

Pete Willson