IAPWS-97 is scalar. From a for loop, create/range your table.
wrt your PLC [Programmable Logic Controller]
ramp a linear SP over 120 elements. Next demo.
Please, confirm it works. Jean

Sattigungsdampfdruck-Stutzpunktberechnung.sm (39kb) downloaded 4 time(s).

This method assumes that the error in linear interpolation is related to the curvature. The problem is reduced from 120 parameters to just a few.
You still have to adjust them by hand, according to what your understanding of optimal distribution is. I tried to minimize the maximum absolute error.

Note that for 120 points, you probably need to increase the sampling points of the plot to not undersample the error.


Optimal sampling points_Kr.sm (29kb) downloaded 11 time(s).

Edited by user 21 January 2024 10:56:33(UTC)  | Reason: Not specified

At this point, as far as it might mean,
at each PLC scan, it will take the next 't,bar(t)'

Worksheet64 bar(degC).sm (28kb) downloaded 3 time(s).

Thank you so much for your replies!

Especially the answer of Martin Kraska looks promising.

As far as I understand it is a major problem that the curve I used in the first row consists of two formulas. So it would be complicated to determine how many sampling points should be assigned to the low and high temperature range. Hence the idea of Jean Giraud is good to use a formula that is valid for the entire range in question. I integrated this IAPWS-97 formula in Martin Kraskas example.

I am not convinced that this question is too much about "taste". With a given assumption there should be a limited number of solutions. This is an optimisation problem. And I assume least squares would be the method to go. So I can either create a solution that minimizes absolute error (guess the unit would be Pa*K) and one that mininizes percentual (relative) error.

In Martins example I marked those parameters that are input as green. There remain four parameters that need calculation. One of them is determined by T.max but I do not see the formula for calculating h.0 out of T.max so any help appreciated.

The example below does not work because Smath cannot calculate the second derivative. I don't know why, this should be possible.

@Jean: H have not understood part of what you are trying to say. What are wrt and XFR? Then I think you have not understood what I am actually trying. I am looking for X in Martins example.

Again, thank you so much for the ideas provided!

Optimal sampling points_v2.sm (27kb) downloaded 4 time(s).

Don't bother, he doesn't have any intention to help you.
He is the master troll of this forum, the great nuisance.
Jean's only interest is promoting his worksheets.
Whether they are related with the question or not.
I really don't know the motivation behind it.
Is he getting paid per download of his samples?
I can't really tell, because he never posts anything useful.

You mean back to square one to your original demand; 2 rows of 120 cols
that you can plug in the PLC, all in ultimate accuracy.

Worksheet64 bar(degC).sm (28kb) downloaded 3 time(s).

1. AFAIK,there is no known technical accuracy.
2. Split the ranges of your choice mesh at will, stack.

Inst_IAPWS region 1 DECADES.sm (54kb) downloaded 1 time(s).

This would work only if the value of the function is proportional to it's curvature, e.g. exp(), sin().

In order to check the approach, make a plot of the error just like further above in my previous post.
Essentially, this is a minimax problem, where you want to minimize the maximum error. It should be sufficient to sample the error at the sampling points and in the center of the intervals.

Perhaps an evolutionary algorithm would be worth a try. The variables are the step sizes h_i. After each recombination/mutation step, you have to calibrate the h_i to enforce the boundary condition of the total x range.

Here is a version with evaluation of the error. This helps with setting the centering parameter.

There are two versions giving the same result.


Optimal sampling points_Kr.sm (45kb) downloaded 2 time(s).

Edited by user 28 January 2024 02:16:02(UTC)  | Reason: Not specified

Nice to see successful applications of Maxima

The two implementations of E(X) give the same result. I added the if-based one because it might be easier to understand. Not everyone is familiar with convolutions.

I wonder why you put a lot of effort in minimizing the error and then being satisfied with some RMS instead of a rigid bracketing. I am not sure at all that I understand your concept of error.

As to the optimization: Do you have to solve multiple such problems or just this one? In the latter case just be pragmatic and play around with the parameters until you are satisfied. There is next to nothing an optimizer could gain in the given example.

As I said, for a general solution I'd skip the curvature based approach and use brute force evolutionary algorithm.

In an engineering application it's generally more about relative error instead of absolute error. I modified your example in order to show relative error and surprisingly the needed exponent p becomes negative! Means I need more points in the low temperature range than in the high temperature range. Blue is absolute error, red is relative error:





I don't have many problems like this. I would like to do another one for the inverse function and one for the derivative. But I agree that for this purpose the room for optimisation is small. The red curve is amplified by 1000 so I have a peak error of 3.7 ‰ at the end and average errors less than 0.5 ‰ in the middle range. In order to further improve I would have to reduce the discretization error but that would require different hardware.

This was an interesting exercise I enjoyed much, thanks for contributing!

Optimal sampling points_Kr-bj2.sm (61kb) downloaded 2 time(s).