<content> <p>rec Examples</p> </content>

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Recursions in SMath

SMath seems to have a limit of 66 for call recursive functions

n f n n 1 + : n f n try 21 line :

0 f 66

Also, it could be very slow

n F 0 n 0 ≡ 1 n 1 ≡ n 1 - F n 2 - F + eval cases 11 line :

to 0 time :

20 F 6765

0 time to - 2.905 s *

This small piece of code can call a function for make a list of the recursion values

2 f\032E\ a\032E\ \032E\N rec v\0324\ a\032E\ stack : \0324\k v\0324\ cols 1 + (\032E\N 1 + (range v\0324\ v\0324\ v\0324\ \0324\k f\032E\ augment eval : for v\0324\ 31 line :

<content> <p>rec Examples</p> </content>

Examples

Fibonacci numbers

a n r a n 1 - el a n 2 - el + :

https://mathworld.wolfram.com/FibonacciNumber.html

a n r 0112 mat 15 rec 01123581321345589144233377610116 mat

ϕ 15 sqrt + 2 / :

n F ϕ n^n π * cos ϕ n -^* - 5 sqrt / vectorize 11 line :

015 range F transpose 01123581321345589144233377610116 mat

Lichtenberg sequence

a n r 2 a n 2 - el * a n 1 - el + 1 + :

https://oeis.org/A000975

a n r 0112 mat 15 rec 0125102142851703416821365273054611092221845116 mat

n L 2 n 1 +^2 - 3 / Ceil vectorize 11 line :

015 range L transpose 0125102142851703416821365273054611092221845116 mat

Triangular numbers

a n r n 2^a n 1 - el - :

https://en.wikipedia.org/wiki/Triangular_number

a n r 115 rec 13610152128364555667891105120136116 mat

Also:

a n r a n 2 - el 2 n * + 1 - :

a n r 1312 mat 15 rec 13610152128364555667891105120136116 mat

Somos-5 sequence

a n r a n 1 - el a n 4 - el * a n 2 - el a n 3 - el * + a n 5 - el / :

https://mathworld.wolfram.com/SomosSequence.html

a n r 1111115 mat 15 rec 111112351137832741217616122833165713116 mat

Nonlinear map of the plane

xy n r 0.7 xy 1 n 1 - el * xy 2 n 1 - el + 0.7989995 - xy 1 n 1 - el (2^+ 21 mat eval :

where

x n el xy 1 n el ≡ y n el xy 2 n el ≡ 21 sys

P a n r 0.142857 0.33 21 mat 500 rec :

P transpose

Also r could be

a n r x y 12 mat a 1 row a 2 row 12 mat : 0.7 x n 1 - el * y n 1 - el + 0.7989995 - x n 1 - el (2^+ 21 mat 21 line :

Solving difference or differential equations.

ic 0121 mat :

vf y x 1 x 2^- (* 21 mat :

for using RK methods:

t u D vf x u 1 el ≡ y u 2 el ≡ 21 sys at :

Discretizing for Euler Method

ho Clear 1

vfn x n 1 - el y n 1 - el 21 mat ho vf x x n 1 - el ≡ y y n 1 - el ≡ 21 sys at * + : x 1 - n + el ho y 1 - n + el * + y 1 - n + el x 1 - n + el 1 x 1 - n + el 2^- (* ho * + 21 mat

Discretizing for Symplectic Euler Method

vfn' vf x x n 1 - el ≡ y y n 1 - el ≡ 21 sys at :

vfn' 1 el x n 1 - el ho vfn' 1 el * + :

vfn' 2 el y n 1 - el ho vf 2 el x vfn' 1 el ≡ y y n 1 - el ≡ 21 sys at * + :

vfn' ho y 1 - n + el * x 1 - n + el + ho ho y 1 - n + el * x 1 - n + el + (* 1 ho y 1 - n + el * x 1 - n + el + (2^- (* y 1 - n + el + 21 mat

Rewriting for

x n el xy 1 n el ≡

and

y n el xy 2 n el ≡

xy n vfn ho xy 2 n 1 - el * xy 1 n 1 - el + xy 2 n 1 - el xy 1 n 1 - el 1 xy 1 n 1 - el (2^- (* ho * + 21 mat :

xy n vfn' ho xy 2 n 1 - el * xy 1 n 1 - el + ho ho xy 2 n 1 - el * xy 1 n 1 - el + (* 1 ho xy 2 n 1 - el * xy 1 n 1 - el + (2^- (* xy 2 n 1 - el + 21 mat :

ho 0.06 : N 100 : TXY ic 0 N ho * N t u D rkfixed : Plot xy n vfn ic N rec transpose xy n vfn' ic N rec transpose TXY 2 col TXY 3 col augment 31 sys : 41 line

Plot

blue,
red,
green.

Linear ODE with constant coefficients

u'' 2 u' * + 3 u * + 0 ≡

vf y 2 y * - 3 x * - 21 mat :

t u D vf x u 1 el ≡ y u 2 el ≡ 21 sys at :

ic 1021 mat :

ho Clear 1

Euler

xy n vfn ho xy 2 n 1 - el * xy 1 n 1 - el + xy 2 n 1 - el ho xy 2 n 1 - el 3 xy 1 n 1 - el * + (* - 21 mat :

Same process than the previous example

Symplectic Euler

xy n vfn' ho xy 2 n 1 - el * xy 1 n 1 - el + ho 2 xy 2 n 1 - el * 3 ho xy 2 n 1 - el * xy 1 n 1 - el + (* + (* - xy 2 n 1 - el + 21 mat :

ho 0.08 : N 100 : TXY ic 0 N ho * N t u D rkfixed : Plot xy n vfn ic N rec transpose xy n vfn' ic N rec transpose TXY 2 col TXY 3 col augment 31 sys : 41 line

Plot

blue,
red,
green.

Iterate a matrix recurrence

M n r M n 1 - el 0.1 10 0.1 22 mat * M n 1 - el * 11 mat 11 line :

M n r 01 - 1022 mat 11 mat 3 rec 01 - 1022 mat 0.1 - 01 0.1 - 22 mat 0.099 - 0.01 0.98 0.099 - 22 mat 0.0951 - 0.0096 0.941 0.0951 - 22 mat 14 mat

Cliff random number generator

x n Cliff 100 x n 1 - el log10 * 1 mod abs eval :

seed 0.3 :

x n Cliff seed 6 rec 0.3 0.2879 0.0797 0.8672 0.1885 0.4593 0.7912 17 mat

N 1000 : r 1 N 1 + (range : 12 mat ρ x n Cliff seed N rec transpose : Plot r ρ augment : 31 line

Plot

Compare with SMath's random

ρ' r el 109 -^109^random * eval :

ρ Mean 0.4856

ρ' Mean 0.4862

ρ Kurtosis 1.7938

ρ' Kurtosis 1.7933

ρ Skewness 0.0654

ρ' Skewness 0.0697

Alvaro

ToDO: rec for 2 index vars.