Three simple examples of recursion usage taken from this document: http://twt.mpei.ac.ru/ochkov/Recursion.pdf (created by V.F. Ochkov).

recursionExamples.sm (16,36 KiB) scaricato 1523 volta(e).



Best regards.

Regarding 3-rd example: solution can be tested here: http://www.softschools.com/games/logic_games/tower_of_hanoi/

Regarding 2-nd example: as I mentioned, I took this example from the document, created by V.Ochkov. But I did a significant change against to the source - I've used vectorize function. Please see screenshot from referenced PDF:

What I don't understand, is why Mathcad gives correct result after call Fibonacci function with vector passed as an argument? How it determines that user wants this function to be called for every single vector element? Miracle!

Best regards.

Hello Andrey,

Thank you for your recent elevated activity on SMath

Wrote

What I don't understand, is why Mathcad gives correct result after call Fibonacci function with vector passed as an argument? How it determines that user wants this function to be called for every single vector element? Miracle!

It is not a miracle. If you use an interval variable in Mathcad (n in this example is an interval variable - not a vector) then if you use a function and pass that interval variable to it, you will have the result of every element in that interval variable (Fibonacci(n)= ). There is a restriction here though. You can call that function, but you can not assign its result to a variable.

Regards,
Radovan

P.S. V.Ochkov is one of the person who knows quite a lot about Mathcad and Mathcad Prime

What you are saying Radovan is true, but NOT exactly: explain the nuance.
In Mathcad n:=1..5 is a "scalar vector". It will tabulate w/o rows
identification. It will not transpose as it does not exist as numbers,
just a "vanishing iterator" working when invoked.
Don't know how it is implemented in the Mathcad version Andrey is talking
about. Presumably it has Fibonacci built-in "Fibonacci(n)". It may have
two built-in algo: one giving Fibonacci(12)=144 ,,, and another one or the
same built-in around "IfIsArray". So, if given Fibonacci(12)=144
and for n:=1..12 it will run over each values of n=0,1,2,3....
thus tabulating Fibonacci(for each n)=1,1,2,3,5,8,13 ...

Fibonacci has a non-recursive form, something like pi from Simon Plouffe.

Jean



Pi [Simon Plouffe].sm (7,9 KiB) scaricato 1278 volta(e).

Hello Andrey,

If you ever meet Valery F. Ochov, his "colleagues" Konstantin, Viktor...
They helped me a lot in Mathsoft Collab and wish to thank them for Eternity.
My small brain is telling me the Mathcad "Vectorise Operator" is a creation
from the main inventors, mostly Allan Razdow. Generally, we can visualise
how it works under the hood like vectorising icfft(fft) ....

The code is deeper than reasonning. Vectorise over an RGB image will explode
the 3 R,G,B components side by side.

Radovan,

Wrote

It is not a miracle. If you use an interval variable in Mathcad (n in this example is an interval variable - not a vector) then if you use a function and pass that interval variable to it, you will have the result of every element in that interval variable (Fibonacci(n)= ). There is a restriction here though. You can call that function, but you can not assign its result to a variable.

Thank you! Now I see... it is just another type of data.




Jean,
Thank you for clarifications and example!

Best regards.

Wrote

Thank you! Now I see... it is just another type of data.

Yes Andrey: just a scalar vector, abusely used in Mathcad ,
mostly for quick plot and gone wth the wind, like butter in the pan.

Jean

Two things I want to note:

1. Over time, recursion has changed in SMath: in the past, it was possible to use recursion directly; currently it seems to require a programming function like line(), otherwise it reports "function not defined" error.
   
   
2. Recursion isn't safe; improper termination condition that leads to infinite recursion will crash SMath. Try f(x):line(f(x),1,1) followed by f(2): (just remember to save your work first).
   

The latter indicates stack overflow, and absence of recursion depth checking. I hope that SMath will eventually implement a means of stack overflow protection, so that in such cases it will simply report "cannot evaluate: nesting too deep" instead of crash.

Also, it would be great if Andrey considered a kind of tail recursion optimization in the future.

Hi. A way to handle recursions.

recursion.sm (85,22 KiB) scaricato 1459 volta(e).
recursion.pdf (330,55 KiB) scaricato 977 volta(e).

Best regards.
Alvaro.

Recursion comes in many ways ... some attached.
1. Pure maths [Fibonacci, Factorial, Powers ...]
2. Curiosity [Hofstadter sequence]
3. Engineering application [Process Control Instrumentation]
4. Generate vector of functions [ChebyShev ...]
Jean.

PolyOrtho ChebyShev.sm (20,65 KiB) scaricato 1429 volta(e).
Maths Sequence Hofstadter.sm (33,26 KiB) scaricato 1489 volta(e).
Maths Fibonnaci.sm (71,28 KiB) scaricato 1471 volta(e).

Wrote

3. Engineering application [Process Control Instrumentation]

Few more productive advanced applications.

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