This Legacy goes back 20 years ago.
In applied mathematics, interpolation is the most difficult task.
The Smath Akima is a piece of gold, well exemplified.
In the mean time of eventual refresh ... enjoy/benefit.
Take care ... Jean.

Spline Interpolation TREASURY-InspectDATA Copy.sm (25 KiB) downloaded 82 time(s).
Spline Interpolation TREASURY-MATRIX Copy.sm (82 KiB) downloaded 80 time(s).
Spline Interpolation TREASURY-ainterp(x) Copy.sm (117 KiB) downloaded 76 time(s).
Spline Interpolation TREASURY Copy.sm (1 MiB) downloaded 77 time(s).

Thanks for your hard work. I would like to ask you about continued fraction interpolation. Is it possible to use this method to approximate a function like the one in the picture?

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Is it possible to use this method to approximate a function like the one in the picture?

If you have it working in Smath, please attach the document.
The Thiele model you have red in my attachment can be reduced in J_Fraction
via Maple and manual suite. I have very many of those J_Fraction.
Mathcad 8, 11 converts directly Thiele and rational fraction to
J_Fraction, but not Smath/Maple.
Thanks for your appreciation ... Jean

I have attempted to construct your picture.
It produces a graph. I see no way to reduce further whereas
it is not simply numerical. If you leave it this way, then
it will track the parameters [R0, Lo, k, a] which are
presumed resulting from previous calculations.
Does it help ? ... Jean

Unknown CF.sm (26 KiB) downloaded 63 time(s).

... at this point, your continued fraction if numerical
can be reduced to J_Fraction of substantial visibility.
You will have to carry the manual procedure for each
numerical valued system ... observe the β construction
Please give news ... Jean

Thiele UNKNOWN parametric [4].sm (50 KiB) downloaded 65 time(s).

Wow!!!!! Thanks for you ultra very quickly answer
I thought about using Thieles interpolation too. but I don't understand how adapt this function for my needs, because I am very far in this area of knowledge.
THIELE_INTERPOLATION_ AND_REGRESSION_SS7251.sm (210 KiB) downloaded 59 time(s).
This's example from CBG(It's user this forum) and I wanted used it.

Thank you Jean

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I thought about using Thieles interpolation too. but I don't understand how adapt this function for my needs, because I am very far in this area of knowledge.

Beautiful Annunciation Cathedral ... I love Russian Coupoles.
I can make zillions of this shapes in Smath.
Continued fractions originate from Mesopotamia 1000's years BC.
Probably from Aliens visit card.
They appeared in modern computing machinery in the earlier IBM Main Frame.
Explained in details in C.T. Fikes.
Since then, functions resident in Windows are normalized rational fractions.
Normalized is Num(x)/[1+Denom(x)... in Bible Hart et Al.
In fact, the sole interpolation method that relates directly X, Y
is the continued fraction as applied Thiele from true/clean data.
All other methods rescue themselves from accessory maths.
If you have more applications, please don't hesitate.
Commonly, Cholesky passes monomials through all or some points.
The pieces need not be monomials, as per the attached *.sm.
INTERPOLATION was not intended to fit model functions.
Fitting model function(s) to data is a much longer story.
An immense Compendium of fitted data in my Smath.
Take my friend ... Jean.

Genfit [Algo Minimize(Cholesky)].sm (27 KiB) downloaded 55 time(s).
Thiele UNKNOWN parametric [4].sm (45 KiB) downloaded 52 time(s).
Inst_Water viscosity [cP].sm (55 KiB) downloaded 56 time(s).
Inst_Type Pt100 XFR Thiele.sm (50 KiB) downloaded 53 time(s).
Inst_Type T_J NBS Resume.sm (152 KiB) downloaded 60 time(s).

This Thiele MERITAS ... as applied.



Thiele McCabe Companion MERITAS Copy.sm (76 KiB) downloaded 52 time(s).

Thanks Jean. How many interesting materials have you provided. Study now more than one evening. Yes, our cathedral is not bad at all. What tourist attractions do you have in Canada?
Oh!!! Your country have many interesting and beautiful places.

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What tourist attractions do you have in Canada ?

1. The Canadians themselves and spending a year with our Native Ancestors.
2. Niagara Falls are impressive !
3. Trecking the Rockies, wait wild goats have finished snoozing on the road.
4. Plenty of National/Provincial Parks.
Have a good day ... Jean.