Feigenbaum constant calculation
You are encouraged to solve this task according to the task description, using any language you may know.
- Task
Calculate the Feigenbaum constant.
- See
-
- Details in the Wikipedia article: Feigenbaum constant.
11l
<lang 11l>V max_it = 13 V max_it_j = 10 V a1 = 1.0 V a2 = 0.0 V d1 = 3.2 V a = 0.0
print(‘ i d’) L(i) 2..max_it
a = a1 + (a1 - a2) / d1
L(j) 1..max_it_j
V x = 0.0
V y = 0.0
L(k) 1..(1 << i)
y = 1.0 - 2.0 * y * x
x = a - x * x
a = a - x / y
V d = (a1 - a2) / (a - a1)
print(‘#2 #.8’.format(i, d))
d1 = d
a2 = a1
a1 = a</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Ada
<lang Ada> with Ada.Text_IO; use Ada.Text_IO; with Ada.Integer_Text_IO; use Ada.Integer_Text_IO;
procedure Main is
procedure feigenbaum is
subtype i_range is Integer range 2 .. 13;
subtype j_range is Integer range 1 .. 10;
-- the number of digits in type Real is reduced to 15 to produce the
-- results reported by C, C++, C# and Ring. Increasing the number of
-- digits in type Real produces the results reported by D.
type Real is digits 15;
package Real_Io is new Float_IO (Real);
use Real_Io;
a, x, y, d : Real;
a1 : Real := 1.0;
a2 : Real := 0.0;
d1 : Real := 3.2;
begin
Put_Line (" i d");
for i in i_range loop
a := a1 + (a1 - a2) / d1;
for j in j_range loop
x := 0.0;
y := 0.0;
for k in 1 .. 2**i loop
y := 1.0 - 2.0 * x * y;
x := a - x * x;
end loop;
a := a - x / y;
end loop;
d := (a1 - a2) / (a - a1);
Put (Item => i, Width => 2);
Put (Item => d, Fore => 5, Aft => 8, Exp => 0);
New_Line;
d1 := d;
a2 := a1;
a1 := a;
end loop;
end feigenbaum;
begin
feigenbaum;
end Main; </lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
ALGOL 68
<lang algol68># Calculate the Feigenbaum constant #
print( ( "Feigenbaum constant calculation:", newline ) ); INT max it = 13; INT max it j = 10; REAL a1 := 1.0; REAL a2 := 0.0; REAL d1 := 3.2; print( ( "i ", "d", newline ) ); FOR i FROM 2 TO max it DO
REAL a := a1 + (a1 - a2) / d1;
FOR j TO max it j DO
REAL x := 0;
REAL y := 0;
FOR k TO 2 ^ i DO
y := 1 - 2 * y * x;
x := a - x * x
OD;
a := a - x / y
OD;
REAL d = (a1 - a2) / (a - a1);
IF i < 10 THEN
print( ( whole( i, 0 ), " ", fixed( d, -10, 8 ), newline ) )
ELSE
print( ( whole( i, 0 ), " ", fixed( d, -10, 8 ), newline ) )
FI;
d1 := d;
a2 := a1;
a1 := a
OD</lang>
- Output:
Feigenbaum constant calculation: i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
AWK
<lang AWK>
- syntax: GAWK -f FEIGENBAUM_CONSTANT_CALCULATION.AWK
BEGIN {
a1 = 1
a2 = 0
d1 = 3.2
max_i = 13
max_j = 10
print(" i d")
for (i=2; i<=max_i; i++) {
a = a1 + (a1 - a2) / d1
for (j=1; j<=max_j; j++) {
x = y = 0
for (k=1; k<=2^i; k++) {
y = 1 - 2 * y * x
x = a - x * x
}
a -= x / y
}
d = (a1 - a2) / (a - a1)
printf("%2d %.8f\n",i,d)
d1 = d
a2 = a1
a1 = a
}
exit(0)
} </lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
C
<lang c>#include <stdio.h>
void feigenbaum() {
int i, j, k, max_it = 13, max_it_j = 10;
double a, x, y, d, a1 = 1.0, a2 = 0.0, d1 = 3.2;
printf(" i d\n");
for (i = 2; i <= max_it; ++i) {
a = a1 + (a1 - a2) / d1;
for (j = 1; j <= max_it_j; ++j) {
x = 0.0;
y = 0.0;
for (k = 1; k <= 1 << i; ++k) {
y = 1.0 - 2.0 * y * x;
x = a - x * x;
}
a -= x / y;
}
d = (a1 - a2) / (a - a1);
printf("%2d %.8f\n", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
}
int main() {
feigenbaum(); return 0;
}</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
C#
<lang csharp>using System;
namespace FeigenbaumConstant {
class Program {
static void Main(string[] args) {
var maxIt = 13;
var maxItJ = 10;
var a1 = 1.0;
var a2 = 0.0;
var d1 = 3.2;
Console.WriteLine(" i d");
for (int i = 2; i <= maxIt; i++) {
var a = a1 + (a1 - a2) / d1;
for (int j = 1; j <= maxItJ; j++) {
var x = 0.0;
var y = 0.0;
for (int k = 1; k <= 1<<i; k++) {
y = 1.0 - 2.0 * y * x;
x = a - x * x;
}
a -= x / y;
}
var d = (a1 - a2) / (a - a1);
Console.WriteLine("{0,2:d} {1:f8}", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
}
}
}</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
C++
<lang cpp>#include <iostream>
int main() {
const int max_it = 13; const int max_it_j = 10; double a1 = 1.0, a2 = 0.0, d1 = 3.2;
std::cout << " i d\n";
for (int i = 2; i <= max_it; ++i) {
double a = a1 + (a1 - a2) / d1;
for (int j = 1; j <= max_it_j; ++j) {
double x = 0.0;
double y = 0.0;
for (int k = 1; k <= 1 << i; ++k) {
y = 1.0 - 2.0*y*x;
x = a - x * x;
}
a -= x / y;
}
double d = (a1 - a2) / (a - a1);
printf("%2d %.8f\n", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
return 0;
}</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
D
<lang d>import std.stdio;
void main() {
int max_it = 13; int max_it_j = 10; double a1 = 1.0; double a2 = 0.0; double d1 = 3.2; double a;
writeln(" i d");
for (int i=2; i<=max_it; i++) {
a = a1 + (a1 - a2) / d1;
for (int j=1; j<=max_it_j; j++) {
double x = 0.0;
double y = 0.0;
for (int k=1; k <= 1<<i; k++) {
y = 1.0 - 2.0 * y * x;
x = a - x * x;
}
a -= x / y;
}
double d = (a1 - a2) / (a - a1);
writefln("%2d %.8f", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
}</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920028 12 4.66920099 13 4.66920555
F#
<lang fsharp>open System
[<EntryPoint>] let main _ =
let maxIt = 13
let maxItJ = 10
let mutable a1 = 1.0
let mutable a2 = 0.0
let mutable d1 = 3.2
Console.WriteLine(" i d")
for i in 2 .. maxIt do
let mutable a = a1 + (a1 - a2) / d1
for j in 1 .. maxItJ do
let mutable x = 0.0
let mutable y = 0.0
for _ in 1 .. (1 <<< i) do
y <- 1.0 - 2.0 * y * x
x <- a - x * x
a <- a - x / y
let d = (a1 - a2) / (a - a1)
Console.WriteLine("{0,2:d} {1:f8}", i, d)
d1 <- d
a2 <- a1
a1 <- a
0 // return an integer exit code</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Factor
<lang factor>USING: formatting io locals math math.ranges sequences ;
[let
1 :> a1!
0 :> a2!
3.2 :> d!
" i d" print
2 13 [a,b] [| exp |
a1 a2 - d /f a1 + :> a!
10 [
0 :> x!
0 :> y!
exp 2^ [
1 2 x y * * - y!
a x sq - x!
] times
a x y /f - a!
] times
a1 a2 - a a1 - /f d!
a1 a2! a a1!
exp d "%2d %.8f\n" printf
] each
]</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Fortran
<lang fortran> program feigenbaum
implicit none
integer i, j, k
real ( KIND = 16 ) x, y, a, b, a1, a2, d1
print '(a4,a13)', 'i', 'd'
a1 = 1.0;
a2 = 0.0;
d1 = 3.2;
do i=2,20
a = a1 + (a1 - a2) / d1;
do j=1,10
x = 0
y = 0
do k=1,2**i
y = 1 - 2 * y * x;
x = a - x**2;
end do
a = a - x / y;
end do
d1 = (a1 - a2) / (a - a1);
a2 = a1;
a1 = a;
print '(i4,f13.10)', i, d1
end do
end</lang>
- Output:
i d 2 3.2185114220 3 4.3856775986 4 4.6009492765 5 4.6551304954 6 4.6661119478 7 4.6685485814 8 4.6690606606 9 4.6691715554 10 4.6691951560 11 4.6692002291 12 4.6692013133 13 4.6692015458 14 4.6692015955 15 4.6692016062 16 4.6692016085 17 4.6692016090 18 4.6692016091 19 4.6692016091 20 4.6692016091
FreeBASIC
<lang freebasic>' version 25-0-2019 ' compile with: fbc -s console
Dim As UInteger i, j, k, maxit = 13, maxitj = 13 Dim As Double x, y, a, a1 = 1, a2, d, d1 = 3.2
Print "Feigenbaum constant calculation:" Print Print " i d" Print "==================="
For i = 2 To maxIt
a = a1 + (a1 - a2) / d1
For j = 1 To maxItJ
x = 0 : y = 0
For k = 1 To 2 ^ i
y = 1 - 2 * y * x
x = a - x * x
Next
a = a - x / y
Next
d = (a1 - a2) / (a - a1)
Print Using "### ##.#########"; i; d
d1 = d
a2 = a1
a1 = a
Next
' empty keyboard buffer While Inkey <> "" : Wend Print : Print "hit any key to end program" Sleep End</lang>
- Output:
Feigenbaum constant calculation: i d =================== 2 3.218511422 3 4.385677599 4 4.600949277 5 4.655130495 6 4.666111948 7 4.668548581 8 4.669060660 9 4.669171555 10 4.669195148 11 4.669200285 12 4.669201301 13 4.669198656
Fōrmulæ
In this page you can see the solution of this task.
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text (more info). Moreover, there can be multiple visual representations of the same program. Even though it is possible to have textual representation —i.e. XML, JSON— they are intended for transportation effects more than visualization and edition.
The option to show Fōrmulæ programs and their results is showing images. Unfortunately images cannot be uploaded in Rosetta Code.
Go
<lang go>package main
import "fmt"
func feigenbaum() {
maxIt, maxItJ := 13, 10
a1, a2, d1 := 1.0, 0.0, 3.2
fmt.Println(" i d")
for i := 2; i <= maxIt; i++ {
a := a1 + (a1-a2)/d1
for j := 1; j <= maxItJ; j++ {
x, y := 0.0, 0.0
for k := 1; k <= 1<<uint(i); k++ {
y = 1.0 - 2.0*y*x
x = a - x*x
}
a -= x / y
}
d := (a1 - a2) / (a - a1)
fmt.Printf("%2d %.8f\n", i, d)
d1, a2, a1 = d, a1, a
}
}
func main() {
feigenbaum()
}</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Groovy
<lang groovy>class Feigenbaum {
static void main(String[] args) {
int max_it = 13
int max_it_j = 10
double a1 = 1.0
double a2 = 0.0
double d1 = 3.2
double a
println(" i d")
for (int i = 2; i <= max_it; i++) {
a = a1 + (a1 - a2) / d1
for (int j = 0; j < max_it_j; j++) {
double x = 0.0
double y = 0.0
for (int k = 0; k < 1 << i; k++) {
y = 1.0 - 2.0 * y * x
x = a - x * x
}
a -= x / y
}
double d = (a1 - a2) / (a - a1)
printf("%2d %.8f\n", i, d)
d1 = d
a2 = a1
a1 = a
}
}
}</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Haskell
<lang haskell>import Data.List (mapAccumL)
feigenbaumApprox :: Int -> [Double] feigenbaumApprox mx = snd $ mitch mx 10
where
mitch :: Int -> Int -> ((Double, Double, Double), [Double])
mitch mx mxj =
mapAccumL
(\(a1, a2, d1) i ->
let a =
iterate
(\a ->
let (x, y) =
iterate
(\(x, y) -> (a - (x * x), 1.0 - ((2.0 * x) * y)))
(0.0, 0.0) !!
(2 ^ i)
in a - (x / y))
(a1 + (a1 - a2) / d1) !!
mxj
d = (a1 - a2) / (a - a1)
in ((a, a1, d), d))
(1.0, 0.0, 3.2)
[2 .. (1 + mx)]
-- TEST ------------------------------------------------------------------ main :: IO () main =
(putStrLn . unlines) $ zipWith (\i s -> justifyRight 2 ' ' (show i) ++ '\t' : s) [1 ..] (show <$> feigenbaumApprox 13) where justifyRight n c s = drop (length s) (replicate n c ++ s)</lang>
- Output:
1 3.2185114220380866 2 4.3856775985683365 3 4.600949276538056 4 4.6551304953919646 5 4.666111947822846 6 4.668548581451485 7 4.66906066077106 8 4.669171554514976 9 4.669195154039278 10 4.669200256503637 11 4.669200975097843 12 4.669205372040318 13 4.669207514010413
J
Translated from the beautiful Fōrmulæ version. Rather than a verb, the conjunction pre-assigns m and n .
Feigenbaum =: conjunction define NB. use: n Feigenbaum m
irange=: <. + i.@:>:@:|@:- NB. inclusive range
a=. 0 1
delta=. , 3.2
for_i. 3 irange n do.
tmp=. ({: + ({:delta) *inv ({: - _2&{)) a
for. i. m do.
'b bp'=. 0
for. i. 2 ^ <: i do.
'b bp'=. (tmp - *: b) , 1 _2 p. b * bp
end.
tmp=. tmp - b % bp
end.
a=. a , tmp
delta=. delta , %/@:(-/"1) (- 2 3 ,: 1 2) { a
end.
2 14j6 14j6 ": (#\ i. # delta) ,. (}. a) ,. delta
)
8 Feigenbaum 13
1 1.000000 3.200000
2 1.310703 3.218511
3 1.381547 4.385678
4 1.396945 4.600949
5 1.400253 4.655130
6 1.400962 4.666112
7 1.401114 4.668549
8 1.401146 4.669061
9 1.401153 4.669172
10 1.401155 4.669195
11 1.401155 4.669200
12 1.401155 4.669201
Java
<lang java>public class Feigenbaum {
public static void main(String[] args) {
int max_it = 13;
int max_it_j = 10;
double a1 = 1.0;
double a2 = 0.0;
double d1 = 3.2;
double a;
System.out.println(" i d");
for (int i = 2; i <= max_it; i++) {
a = a1 + (a1 - a2) / d1;
for (int j = 0; j < max_it_j; j++) {
double x = 0.0;
double y = 0.0;
for (int k = 0; k < 1 << i; k++) {
y = 1.0 - 2.0 * y * x;
x = a - x * x;
}
a -= x / y;
}
double d = (a1 - a2) / (a - a1);
System.out.printf("%2d %.8f\n", i, d);
d1 = d;
a2 = a1;
a1 = a;
}
}
}</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
jq
<lang jq>def feigenbaum_delta(imax; jmax):
def lpad: tostring | (" " * (4 - length)) + .;
def pp(i;x): "\(i|lpad) \(x)";
"Feigenbaum's delta constant incremental calculation:",
pp("i"; "δ"),
pp(1; "3.20"),
( foreach range(2; 1+imax) as $i (
{a1: 1.0, a2: 0.0, d1: 3.2};
.a = .a1 + (.a1 - .a2) / .d1
| reduce range(1; 1+jmax) as $j (.;
.x = 0 | .y = 0
| reduce range(1; 1+pow(2;$i)) as $k (.;
.y = (1 - 2 * .x * .y)
| .x = .a - (.x * .x) )
| .a -= (.x / .y) )
| .d = (.a1 - .a2) / (.a - .a1)
| .emit = pp($i; .d)
| .d1 = .d | .a2 = .a1 | .a1 = .a;
.emit ) ) ;
feigenbaum_delta(13; 10) </lang>
- Output:
<lang sh> Feigenbaum's delta constant incremental calculation:
i δ 1 3.20 2 3.2185114220380866 3 4.3856775985683365 4 4.600949276538056 5 4.6551304953919646 6 4.666111947822846 7 4.668548581451485 8 4.66906066077106 9 4.669171554514976 10 4.669195154039278 11 4.669200256503637 12 4.669200975097843 13 4.669205372040318
</lang>
Julia
<lang julia># http://en.wikipedia.org/wiki/Feigenbaum_constant
function feigenbaum_delta(imax=23, jmax=20)
a1, a2, d1 = BigFloat(1.0), BigFloat(0.0), BigFloat(3.2)
println("Feigenbaum's delta constant incremental calculation:\ni δ\n1 3.20")
for i in 2:imax
a = a1 + (a1 - a2) / d1
for j in 1:jmax
x, y = 0, 0
for k in 1:2^i
y = 1 - 2 * x * y
x = a - x * x
end
a -= x / y
end
d = (a1 - a2) / (a - a1)
println(rpad(i, 4), lpad(d, 4))
d1, a2 = d, a1
a1 = a
end
end
feigenbaum_delta()
</lang>
- Output:
Feigenbaum's delta constant incremental calculation: i δ 1 3.20 2 3.218511422038087912270504530742813256028820377971082199141994437483271226037533 3 4.385677598568339085744948568775522346103216356576497808699630752612705940390646 4 4.600949276538075357811694698623834985023552496633543372295593454454329771521727 5 4.655130495391980136486254995856898819475460497385226078363311588165123307017281 6 4.66611194782857138833121369671177648071905897173694216397236891198998639455025 7 4.668548581446840948044543680148146265543287896654348757317309551400403337843036 8 4.66906066064826823913259982263027263779968209542149740052288679867743088942764 9 4.669171555379511388886004609897567088240676573170789783804375113804695091803033 10 4.669195156030017174021108801191492093392147908605756405516325961597435372704323 11 4.669200229086856497938353781004067217408888048906823830162962242800074595934665 12 4.669201313294204171164754941185571183728248888986548913352217226469150028661929 13 4.669201545780906707506058109930429736431564330452605295006142805341042630340361 14 4.669201595537493910292470639289646040074547412490596040512777985387237785978782 15 4.669201606198152157723831097078594524421336516011873717994000712976201143278191 16 4.669201608480804423294067945898622842792868381815074127672747764898152898198069 17 4.669201608969744700482485321938373343907385540992447405883605282416375303280911 18 4.669201609074452566227981520370886753946099646679618270214759101315481224820708 19 4.669201609096878794705135037864783677622666525741836726064298799595215295927305 20 4.66920160910168168118696016084580172992808889324407617097679098039831535247408 21 4.669201609102710327837210208629111857781724142614997392167298168695631199065625 22 4.669201609102930630539778141205517641783439121041016813735799961205502985593042 23 4.66920160910297781286849594159066394676896043144121209732784416240857379387701
Kotlin
<lang scala>// Version 1.2.40
fun feigenbaum() {
val maxIt = 13
val maxItJ = 10
var a1 = 1.0
var a2 = 0.0
var d1 = 3.2
println(" i d")
for (i in 2..maxIt) {
var a = a1 + (a1 - a2) / d1
for (j in 1..maxItJ) {
var x = 0.0
var y = 0.0
for (k in 1..(1 shl i)) {
y = 1.0 - 2.0 * y * x
x = a - x * x
}
a -= x / y
}
val d = (a1 - a2) / (a - a1)
println("%2d %.8f".format(i,d))
d1 = d
a2 = a1
a1 = a
}
}
fun main(args: Array<String>) {
feigenbaum()
}</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Lambdatalk
Following the Python code in a recursive mode. <lang scheme> {feigenbaum 11} // on my computer stackoverflow for values greater than 11 -> [3.2185114220380866,4.3856775985683365,4.600949276538056,4.6551304953919646,4.666111947822846, 4.668548581451485,4.66906066077106,4.669171554514976,4.669195154039278,4.669200256503637]
with:
{def feigenbaum
{lambda {:maxi}
{f3 :maxi 10 1 0 3.2 0 {A.new} 2}}}
{def f3
{lambda {:maxi :maxj :a1 :a2 :d1 :a3 :s :i}
{if {< :i {+ :maxi 1}}
then {let { {:maxi :maxi} {:maxj :maxj} {:a1 :a1} {:a2 :a2}
{:a3 {f2 {+ :a1 {/ {- :a1 :a2} :d1}} :i :maxj 1} }
{:s :s} {:i :i}
} {f3 :maxi :maxj :a3 :a1 {/ {- :a1 :a2} {- :a3 :a1}} :a3
{A.addlast! {/ {- :a1 :a2} {- :a3 :a1}} :s} {+ :i 1}} }
else :s}}}
{def f2
{lambda {:a :i :maxj :j}
{if {< :j {+ :maxj 1}}
then {f2 {f1 :a :i 0 0 1} :i :maxj {+ :j 1}}
else :a}}}
{def f1
{lambda {:a :i :y :x :k}
{if {< :k {+ {pow 2 :i} 1}}
then {f1 :a :i {- 1 {* 2 :y :x}} {- :a {* :x :x}} {+ :k 1}}
else {- :a {/ :x :y}} }}}
</lang>
Lua
<lang lua>function leftShift(n,p)
local r = n
while p>0 do
r = r * 2
p = p - 1
end
return r
end
-- main
local MAX_IT = 13 local MAX_IT_J = 10 local a1 = 1.0 local a2 = 0.0 local d1 = 3.2
print(" i d") for i=2,MAX_IT do
local a = a1 + (a1 - a2) / d1
for j=1,MAX_IT_J do
local x = 0.0
local y = 0.0
for k=1,leftShift(1,i) do
y = 1.0 - 2.0 * y * x
x = a - x * x
end
a = a - x / y
end
d = (a1 - a2) / (a - a1)
print(string.format("%2d %.8f", i, d))
d1 = d
a2 = a1
a1 = a
end</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Modula-2
<lang modula2>MODULE Feigenbaum; FROM FormatString IMPORT FormatString; FROM LongStr IMPORT RealToStr; FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
VAR
buf : ARRAY[0..63] OF CHAR; i,j,k,max_it,max_it_j : INTEGER; a,x,y,d,a1,a2,d1 : LONGREAL;
BEGIN
max_it := 13; max_it_j := 10;
a1 := 1.0; a2 := 0.0; d1 := 3.2;
WriteString(" i d");
WriteLn;
FOR i:=2 TO max_it DO
a := a1 + (a1 - a2) / d1;
FOR j:=1 TO max_it_j DO
x := 0.0;
y := 0.0;
FOR k:=1 TO INT(1 SHL i) DO
y := 1.0 - 2.0 * y * x;
x := a - x * x
END;
a := a - x / y
END;
d := (a1 - a2) / (a - a1);
FormatString("%2i ", buf, i);
WriteString(buf);
RealToStr(d, buf);
WriteString(buf);
WriteLn;
d1 := d;
a2 := a1;
a1 := a
END;
ReadChar
END Feigenbaum.</lang>
Nim
<lang Nim>import strformat
iterator feigenbaum(): tuple[n: int; δ: float] =
## Yield successive approximations of Feigenbaum constant.
const MaxI = 13 MaxJ = 10 var a1 = 1.0 a2 = 0.0 δ = 3.2
for i in 2..MaxI:
var a = a1 + (a1 - a2) / δ
for j in 1..MaxJ:
var x, y = 0.0
for _ in 1..(1 shl i):
y = 1 - 2 * y * x
x = a - x * x
a -= x / y
δ = (a1 - a2) / (a - a1) a2 = a1 a1 = a yield (i, δ)
echo " i δ" for n, δ in feigenbaum():
echo fmt"{n:2d} {δ:.8f}"</lang>
- Output:
i δ 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Perl
<lang perl>use strict; use warnings; use Math::AnyNum 'sqr';
my $a1 = 1.0; my $a2 = 0.0; my $d1 = 3.2;
print " i δ\n";
for my $i (2..13) {
my $a = $a1 + ($a1 - $a2)/$d1;
for (1..10) {
my $x = 0;
my $y = 0;
for (1 .. 2**$i) {
$y = 1 - 2 * $y * $x;
$x = $a - sqr($x);
}
$a -= $x/$y;
}
$d1 = ($a1 - $a2) / ($a - $a1); ($a2, $a1) = ($a1, $a); printf "%2d %17.14f\n", $i, $d1;
}</lang>
- Output:
2 3.21851142203809 3 4.38567759856834 4 4.60094927653808 5 4.65513049539198 6 4.66611194782857 7 4.66854858144684 8 4.66906066064827 9 4.66917155537951 10 4.66919515603002 11 4.66920022908686 12 4.66920131329420 13 4.66920154578091
Phix
<lang Phix>constant maxIt = 13,
maxItJ = 10
atom a1 = 1.0,
a2 = 0.0,
d1 = 3.2
puts(1," i d\n") for i=2 to maxIt do
atom a = a1 + (a1 - a2) / d1
for j=1 to maxItJ do
atom x = 0, y = 0
for k=1 to power(2,i) do
y = 1 - 2*y*x
x = a - x*x
end for
a = a - x/y
end for
atom d = (a1-a2)/(a-a1)
printf(1,"%2d %.8f\n",{i,d})
d1 = d
a2 = a1
a1 = a
end for</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Python
<lang python>max_it = 13 max_it_j = 10 a1 = 1.0 a2 = 0.0 d1 = 3.2 a = 0.0
print " i d" for i in range(2, max_it + 1):
a = a1 + (a1 - a2) / d1
for j in range(1, max_it_j + 1):
x = 0.0
y = 0.0
for k in range(1, (1 << i) + 1):
y = 1.0 - 2.0 * y * x
x = a - x * x
a = a - x / y
d = (a1 - a2) / (a - a1)
print("{0:2d} {1:.8f}".format(i, d))
d1 = d
a2 = a1
a1 = a</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Racket
<lang racket>#lang racket (define (feigenbaum #:max-it (max-it 13) #:max-it-j (max-it-j 10))
(displayln " i d" (current-error-port))
(define-values (_a _a1 d)
(for/fold ((a 1) (a1 0) (d 3.2))
((i (in-range 2 (add1 max-it))))
(let* ((a′ (for/fold ((a (+ a (/ (- a a1) d))))
((j (in-range max-it-j)))
(let-values (([x y] (for/fold ((x 0) (y 0))
((k (expt 2 i)))
(values (- a (* x x))
(- 1 (* 2 y x))))))
(- a (/ x y)))))
(d′ (/ (- a a1) (- a′ a))))
(eprintf "~a ~a\n" (~a i #:width 2) (real->decimal-string d′ 8))
(values a′ a d′))))
d)
(module+ main
(feigenbaum))</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537 4.669205372040318
Raku
(formerly Perl 6)
<lang perl6>my $a1 = 1; my $a2 = 0; my $d = 3.2;
say ' i d';
for 2 .. 13 -> $exp {
my $a = $a1 + ($a1 - $a2) / $d;
do {
my $x = 0;
my $y = 0;
for ^2 ** $exp {
$y = 1 - 2 * $y * $x;
$x = $a - $x²;
}
$a -= $x / $y;
} xx 10;
$d = ($a1 - $a2) / ($a - $a1);
($a2, $a1) = ($a1, $a);
printf "%2d %.8f\n", $exp, $d;
}</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
REXX
<lang rexx>/*REXX pgm calculates the (Mitchell) Feigenbaum bifurcation velocity, #digs can be given*/ parse arg digs maxi maxj . /*obtain optional argument from the CL.*/ if digs== | digs=="," then digs= 30 /*Not specified? Then use the default.*/ if maxi== | maxi=="," then maxi= 20 /* " " " " " " */ if maxJ== | maxJ=="," then maxJ= 10 /* " " " " " " */
- = 4.669201609102990671853203820466201617258185577475768632745651343004134330211314737138,
|| 68974402394801381716 /*◄──Feigenbaum's constant, true value.*/
numeric digits digs /*use the specified # of decimal digits*/
a1= 1 a2= 0 d1= 3.2
say 'Using ' maxJ " iterations for maxJ, with " digs ' decimal digits:' say say copies(' ', 9) center("correct", 11) copies(' ', digs+1) say center('i', 9, "─") center('digits' , 11, "─") center('d', digs+1, "─")
do i=2 for maxi-1
a= a1 + (a1 - a2) / d1
do maxJ
x= 0; y= 0
do 2**i; y= 1 - 2 * x * y
x= a - x*x
end /*2**i*/
a= a - x / y
end /*maxj*/
d= (a1 - a2) / (a - a1) /*compute the delta (D) of the function*/
t= max(0, compare(d, #) - 2) /*# true digs so far, ignore dec. point*/
say center(i, 9) center(t, 11) d /*display values for I & D ──►terminal*/
parse value d a1 a with d1 a2 a1 /*assign 3 variables with 3 new values.*/
end /*i*/
/*stick a fork in it, we're all done. */
say left(, 9 + 1 + 11 + 1 + t )"↑" /*show position of greatest accuracy. */ say ' true value= ' # / 1 /*true value of Feigenbaum's constant. */</lang>
- output when using the default inputs:
Using 10 iterations for maxJ, with 30 decimal digits:
correct
────i──── ──digits─── ───────────────d───────────────
2 0 3.21851142203808791227050453077
3 1 4.3856775985683390857449485682
4 2 4.60094927653807535781169469969
5 2 4.65513049539198013648625498649
6 3 4.66611194782857138833121364654
7 3 4.66854858144684094804454708811
8 4 4.66906066064826823913257549468
9 4 4.6691715553795113888859465442
10 4 4.66919515603001717402161720542
11 6 4.66920022908685649793393149233
12 7 4.66920131329420417113719511412
13 7 4.66920154578090670783369507315
14 7 4.66920159553749390966169074155
15 9 4.66920160619815215840788706632
16 9 4.66920160848080435144581223484
17 9 4.66920160896974538458267849027
18 10 4.66920160907444981238909862845
19 10 4.66920160909687888294310165196
20 12 4.66920160910169069039564432665
↑
true value= 4.66920160910299067185320382047
Ring
<lang ring># Project : Feigenbaum constant calculation
decimals(8) see "Feigenbaum constant calculation:" + nl maxIt = 13 maxItJ = 10 a1 = 1.0 a2 = 0.0 d1 = 3.2 see "i " + "d" + nl for i = 2 to maxIt
a = a1 + (a1 - a2) / d1
for j = 1 to maxItJ
x = 0
y = 0
for k = 1 to pow(2,i)
y = 1 - 2 * y * x
x = a - x * x
next
a = a - x / y
next
d = (a1 - a2) / (a - a1)
if i < 10
see "" + i + " " + d + nl
else
see "" + i + " " + d + nl
ok
d1 = d
a2 = a1
a1 = a
next</lang> Output:
Feigenbaum constant calculation: i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Ruby
<lang ruby>def main
maxIt = 13
maxItJ = 10
a1 = 1.0
a2 = 0.0
d1 = 3.2
puts " i d"
for i in 2 .. maxIt
a = a1 + (a1 - a2) / d1
for j in 1 .. maxItJ
x = 0.0
y = 0.0
for k in 1 .. 1 << i
y = 1.0 - 2.0 * y * x
x = a - x * x
end
a = a - x / y
end
d = (a1 - a2) / (a - a1)
print "%2d %.8f\n" % [i, d]
d1 = d
a2 = a1
a1 = a
end
end
main()</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Scala
Imperative, ugly
<lang Scala>object Feigenbaum1 extends App {
val (max_it, max_it_j) = (13, 10) var (a1, a2, d1, a) = (1.0, 0.0, 3.2, 0.0)
println(" i d")
var i: Int = 2
while (i <= max_it) {
a = a1 + (a1 - a2) / d1
for (_ <- 0 until max_it_j) {
var (x, y) = (0.0, 0.0)
for (_ <- 0 until 1 << i) {
y = 1.0 - 2.0 * y * x
x = a - x * x
}
a -= x / y
}
val d: Double = (a1 - a2) / (a - a1)
printf("%2d %.8f\n", i, d)
d1 = d
a2 = a1
a1 = a
i += 1
}
}</lang>
Functional Style, Tail recursive
- Output:
Best seen running in your browser either by ScalaFiddle (ES aka JavaScript, non JVM) or Scastie (remote JVM).
<lang Scala>object Feigenbaum2 extends App {
private val (max_it, max_it_j) = (13, 10)
private def result = {
@scala.annotation.tailrec
def outer(i: Int, d1: Double, a2: Double, a1: Double, acc: Seq[Double]): Seq[Double] = {
@scala.annotation.tailrec
def center(j: Int, a: Double): Double = {
@scala.annotation.tailrec
def inner(k: Int, end: Int, x: Double, y: Double): (Double, Double) =
if (k < end) inner(k + 1, end, a - x * x, 1.0 - 2.0 * y * x) else (x, y)
val (x, y) = inner(0, 1 << i, 0.0, 0.0)
if (j < max_it_j) {
center(j + 1, a - (x / y))
} else a
}
if (i <= max_it) {
val a = center(0, a1 + (a1 - a2) / d1)
val d: Double = (a1 - a2) / (a - a1)
outer(i + 1, d, a1, a, acc :+ d)
} else acc
}
outer(2, 3.2, 0, 1.0, Seq[Double]()).zipWithIndex }
println(" i ≈ δ")
result.foreach { case (δ, i) => println(f"${i + 2}%2d $δ%.8f") }
}</lang>
Sidef
<lang ruby>var a1 = 1 var a2 = 0 var δ = 3.2.float
say " i\tδ"
for i in (2..15) {
var a0 = ((a1 - a2)/δ + a1)
10.times {
var (x, y) = (0, 0)
2**i -> times {
y = (1 - 2*x*y)
x = (a0 - x²)
}
a0 -= x/y
}
δ = ((a1 - a2) / (a0 - a1))
(a2, a1) = (a1, a0)
printf("%2d %.8f\n", i, δ)
}</lang>
- Output:
i δ 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917156 10 4.66919516 11 4.66920023 12 4.66920131 13 4.66920155 14 4.66920160 15 4.66920161
Swift
<lang swift>import Foundation
func feigenbaum(iterations: Int = 13) {
var a = 0.0 var a1 = 1.0 var a2 = 0.0 var d = 0.0 var d1 = 3.2
print(" i d")
for i in 2...iterations {
a = a1 + (a1 - a2) / d1
for _ in 1...10 {
var x = 0.0
var y = 0.0
for _ in 1...1<<i {
y = 1.0 - 2.0 * y * x
x = a - x * x
}
a -= x / y }
d = (a1 - a2) / (a - a1) d1 = d (a1, a2) = (a, a1)
print(String(format: "%2d %.8f", i, d)) }
}
feigenbaum()</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Visual Basic .NET
<lang vbnet>Module Module1
Sub Main()
Dim maxIt = 13
Dim maxItJ = 10
Dim a1 = 1.0
Dim a2 = 0.0
Dim d1 = 3.2
Console.WriteLine(" i d")
For i = 2 To maxIt
Dim a = a1 + (a1 - a2) / d1
For j = 1 To maxItJ
Dim x = 0.0
Dim y = 0.0
For k = 1 To 1 << i
y = 1.0 - 2.0 * y * x
x = a - x * x
Next
a -= x / y
Next
Dim d = (a1 - a2) / (a - a1)
Console.WriteLine("{0,2:d} {1:f8}", i, d)
d1 = d
a2 = a1
a1 = a
Next
End Sub
End Module</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
Wren
<lang ecmascript>import "/fmt" for Fmt
var feigenbaum = Fn.new {
var maxIt = 13
var maxItJ = 10
var a1 = 1
var a2 = 0
var d1 = 3.2
System.print(" i d")
for (i in 2..maxIt) {
var a = a1 + (a1 - a2)/d1
for (j in 1..maxItJ) {
var x = 0
var y = 0
for (k in 1..(1<<i)) {
y = 1 - 2*y*x
x = a - x*x
}
a = a - x/y
}
var d = (a1 - a2)/(a - a1)
System.print("%(Fmt.d(2, i)) %(Fmt.f(0, d, 8))")
d1 = d
a2 = a1
a1 = a
}
}
feigenbaum.call()</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537
zkl
<lang zkl>fcn feigenbaum{
maxIt,maxItJ,a1,a2,d1,a,d := 13, 10, 1.0, 0.0, 3.2, 0, 0;
println(" i d");
foreach i in ([2..maxIt]){
a=a1 + (a1 - a2)/d1;
foreach j in ([1..maxItJ]){
x,y := 0.0, 0.0;
foreach k in ([1..(1).shiftLeft(i)]){ y,x = 1.0 - 2.0*y*x, a - x*x; } a-=x/y
}
d=(a1 - a2)/(a - a1);
println("%2d %.8f".fmt(i,d));
d1,a2,a1 = d,a1,a;
}
}();</lang>
- Output:
i d 2 3.21851142 3 4.38567760 4 4.60094928 5 4.65513050 6 4.66611195 7 4.66854858 8 4.66906066 9 4.66917155 10 4.66919515 11 4.66920026 12 4.66920098 13 4.66920537