Four examples of Continuous Fractions



Continuous_Fractions_SS7251.sm (79 KiB) downloaded 63 time(s).

Continuous_Fractions_SS6179.sm (76 KiB) downloaded 62 time(s).

Best Regards

Carlos

Wrote

Four examples of Continued Fractions

Thanks Carlos ... works well.
More from Maple ... Jean

Symbolic Convert Parfrac Primary FNCTS.sm (27 KiB) downloaded 62 time(s).

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Thanks Carlos ... works well.
More from Maple ... Jean

Hola Jean
Thank you very much for your examples, they have helped me to correct
a fault that was presented in the ACOS (X) function, where the value
of "Phi () / 2" was not included.

I have merged all the algorithms in a single function for ease of use.

All the examples you provided work perfectly with the algorithm for
the conversion of function to a continuous function.

SMATH STUDIO VERSION 7251, DOES NOT ACCEPT THE ACOS (X), ASIN (x) AND
ATAN (X) FUNCTIONS FOR MAPLE, GIVING THE "EMPTY STACK" ERROR.



Continuous_Fraction_algorith_SS7251.sm (76 KiB) downloaded 56 time(s).

Continuous_Fraction_algorith_SS6179.sm (77 KiB) downloaded 61 time(s).

Best Regards

Carlos

Wrote

SMATH STUDIO VERSION 7251, DOES NOT ACCEPT THE ACOS (X), ASIN (x) AND
ATAN (X) FUNCTIONS FOR MAPLE, GIVING THE "EMPTY STACK" ERROR.

Maybe Viacheslav can doctor the sick Maple.
Few more in there Carlos ... Jean

Symbolic Convert Parfrac Primary FNCTS.sm (101 KiB) downloaded 60 time(s).

In his book, C.T. Fike mentions few system having implemented
the continued fraction for some functions [IBM 360 ...], 1965.
From there, the next step was to economize the CF ...
from which the J_Fraction originates.
By same token, minimax rational fractions to replace the CF.
Those style rational fraction aren't very numerically stable.
Then the Padé normalized rational fractions origin, more stable.
Built-in Microsoft functions are Padé [AFAIK], normalized 15 D.
More decimals can be obtained from Chebyshev, rated 25 D,
only available in Main Frames.

Wrote

I have merged all the algorithms in a single function for ease of use.

All the examples you provided work perfectly with the algorithm for
the conversion of function to a continuous function.

Superb, Carlos ... saved, thanks.

Continued fractions are pure fascination.
They date back long time ago from Mesopotamians, brought back by the Templars.
As a matter of fact, the first Crusade never came back, and bypassed Jerusalem.
Wallis [Newton's teacher] developed a method to compute top/down.
It is certainly coded this way in modern computing machinery [advanced CAS].
Damned hard work to compute in Excel and more not CAS.

If you were with Mathcad/Mathsoft before Smath, there is "confrac"
work sheet from Jesper Gundermann [Danish].
He also provided the BigNumber engine to display > 15 decimals
that imitate Mathcad symbolic.
In his book, F.B. Hildebrand you will find an interesting chapter.
It became the source of approximating data sets that delighted you so much,
and elevates common Engineering works

Hola ! carlos ... Jean

Carlos,
If you have extra willingness, here is a delight for you !
Cheers ... Jean

Untitled Altitude [m_kPa].sm (26 KiB) downloaded 58 time(s).

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... in my old style

Under the hood, there is a "Big Number" red issue,
solved manually, not exposed in the document.
Otherwise: 1/1 happy ... Publishing style.

Inst_ISA [kPa] solve NASA altitude [20200215] Copy.sm (105 KiB) downloaded 57 time(s).