Feigenbaum constant calculation
Calculate the Feigenbaum constant.
- Task
- 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
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
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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
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
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
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
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>
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
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
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