Van der Corput sequence
You are encouraged to solve this task according to the task description, using any language you may know.
When counting integers in binary, if you put a (binary) point to the right of the count then the column immediately to the left denotes a digit with a multiplier of ; the digit in the next column to the left has a multiplier of ; and so on.
So in the following table:
0. 1. 10. 11. ...
the binary number "10" is .
You can also have binary digits to the right of the “point”, just as in the decimal number system. In that case, the digit in the place immediately to the right of the point has a weight of , or . The weight for the second column to the right of the point is or . And so on.
If you take the integer binary count of the first table, and reflect the digits about the binary point, you end up with the van der Corput sequence of numbers in base 2.
.0 .1 .01 .11 ...
The third member of the sequence, binary 0.01, is therefore or .
Members of the sequence lie within the interval
. Points within the sequence tend to be evenly distributed which is a useful trait to have for Monte Carlo simulations.
This sequence is also a superset of the numbers representable by the "fraction" field of an old IEEE floating point standard. In that standard, the "fraction" field represented the fractional part of a binary number beginning with "1." e.g. 1.101001101.
Hint
A hint at a way to generate members of the sequence is to modify a routine used to change the base of an integer:
>>> def base10change(n, base):
digits = []
while n:
n,remainder = divmod(n, base)
digits.insert(0, remainder)
return digits
>>> base10change(11, 2)
[1, 0, 1, 1]
the above showing that 11 in decimal is .
Reflected this would become .1101 or
- Task description
- Create a function/method/routine that given n, generates the n'th term of the van der Corput sequence in base 2.
- Use the function to compute and display the first ten members of the sequence. (The first member of the sequence is for n=0).
- As a stretch goal/extra credit, compute and show members of the sequence for bases other than 2.
- See also
11l
F vdc(=n, base = 2)
V (vdc, denom) = (0.0, 1)
L n != 0
denom *= base
(n, V remainder) = divmod(n, base)
vdc += Float(remainder) / denom
R vdc
print((0.<10).map(i -> vdc(i)))
print((0.<10).map(i -> vdc(i, 3)))- Output:
[0, 0.5, 0.25, 0.75, 0.125, 0.625, 0.375, 0.875, 0.0625, 0.5625] [0, 0.333333, 0.666667, 0.111111, 0.444444, 0.777778, 0.222222, 0.555556, 0.888889, 0.037037]
360 Assembly
The program uses two ASSIST macros (XDECO,XPRNT) to keep the code as short as possible.
* Van der Corput sequence 31/01/2017
VDCS CSECT
USING VDCS,R13 base register
B 72(R15) skip savearea
DC 17F'0' savearea
STM R14,R12,12(R13) prolog
ST R13,4(R15) " <-
ST R15,8(R13) " ->
LR R13,R15 " addressability
ZAP B,=P'2' b=2 (base)
ZAP M,=P'-1' m=-1
SR R6,R6 i=0
LOOPI CH R6,=H'10' do i=0 to 10
BH ELOOPI
AP M,=P'1' w=m+1
ZAP V,=P'0' v=0
ZAP S,=P'1' s=1
ZAP N,M n=m
WHILE CP N,=P'0' do while n<>0
BE EWHILE
MP S,B s=s*b
ZAP PL16,N n
DP PL16,B n/b
ZAP W,PL16+8(8) w=n mod b
MP W,=P'100000' *100000
ZAP PL16,W w
DP PL16,S w/s
ZAP W,PL16(8) w=w/s
AP V,W v=v+(n mod b)*100000/s
ZAP PL16,N n
DP PL16,B n/b
ZAP N,PL16(8) n=n/b
B WHILE
EWHILE XDECO R6,XDEC edit i
MVC PG+0(3),XDEC+9 output i
MVC PG+3(3),=C' 0.'
UNPK Z,V unpack v
OI Z+L'Z-1,X'F0' edit v
MVC PG+6(5),Z+11 output v (v/100000)
XPRNT PG,L'PG print buffer
LA R6,1(R6) i=i+1
B LOOPI
ELOOPI L R13,4(0,R13) epilog
LM R14,R12,12(R13) " restore
XR R15,R15 " rc=0
BR R14 exit
B DS PL8
M DS PL8
V DS PL8
S DS PL8
N DS PL8
W DS PL8 packed
Z DS ZL16 zoned
PL16 DS PL16 packed max
PG DC CL80' ' buffer
XDEC DS CL12 work area for xdeco
YREGS
END VDCS- Output:
0 0.00000 1 0.50000 2 0.25000 3 0.75000 4 0.12500 5 0.62500 6 0.37500 7 0.87500 8 0.06250 9 0.56250 10 0.31250
Action!
INCLUDE "D2:REAL.ACT" ;from the Action! Tool Kit
PROC Generate(INT value,base REAL POINTER res)
REAL denom,rbase,r1,r2
IntToReal(0,res)
IntToReal(1,denom)
IntToReal(base,rbase)
WHILE value#0
DO
RealMult(denom,rbase,r1)
RealAssign(r1,denom)
IntToReal(value MOD base,r1)
RealDiv(r1,denom,r2)
RealAdd(res,r2,r1)
RealAssign(r1,res)
value==/base
OD
RETURN
PROC Main()
INT value,base
REAL res
Put(125) PutE() ;clear the screen
FOR base=2 TO 5
DO
PrintF("Base %I:%E",base)
FOR value=0 TO 9
DO
Generate(value,base,res)
PrintR(res) Put(32)
OD
PutE() PutE()
OD
RETURN- Output:
Screenshot from Atari 8-bit computer
Base 2: 0 .5 .25 .75 .125 .625 .375 .875 .0625 .5625 Base 3: 0 .3333333333 .6666666666 .1111111111 .4444444444 .7777777777 .2222222222 .5555555555 .8888888888 .037037037 Base 4: 0 .25 .5 .75 .0625 .3125 .5625 .8125 .125 .375 Base 5: 0 .2 .4 .6 .8 .04 .24 .44 .64 .84
ActionScript
This implementation uses logarithms to computes the nth term of the sequence at any base. Numbers in the output are rounded to 6 decimal places to hide any floating point inaccuracies.
package {
import flash.display.Sprite;
import flash.events.Event;
public class VanDerCorput extends Sprite {
public function VanDerCorput():void {
if (stage) init();
else addEventListener(Event.ADDED_TO_STAGE, init);
}
private function init(e:Event = null):void {
removeEventListener(Event.ADDED_TO_STAGE, init);
var base2:Vector.<Number> = new Vector.<Number>(10, true);
var base3:Vector.<Number> = new Vector.<Number>(10, true);
var base4:Vector.<Number> = new Vector.<Number>(10, true);
var base5:Vector.<Number> = new Vector.<Number>(10, true);
var base6:Vector.<Number> = new Vector.<Number>(10, true);
var base7:Vector.<Number> = new Vector.<Number>(10, true);
var base8:Vector.<Number> = new Vector.<Number>(10, true);
var i:uint;
for ( i = 0; i < 10; i++ ) {
base2[i] = Math.round( _getTerm(i, 2) * 1000000 ) / 1000000;
base3[i] = Math.round( _getTerm(i, 3) * 1000000 ) / 1000000;
base4[i] = Math.round( _getTerm(i, 4) * 1000000 ) / 1000000;
base5[i] = Math.round( _getTerm(i, 5) * 1000000 ) / 1000000;
base6[i] = Math.round( _getTerm(i, 6) * 1000000 ) / 1000000;
base7[i] = Math.round( _getTerm(i, 7) * 1000000 ) / 1000000;
base8[i] = Math.round( _getTerm(i, 8) * 1000000 ) / 1000000;
}
trace("Base 2: " + base2.join(', '));
trace("Base 3: " + base3.join(', '));
trace("Base 4: " + base4.join(', '));
trace("Base 5: " + base5.join(', '));
trace("Base 6: " + base6.join(', '));
trace("Base 7: " + base7.join(', '));
trace("Base 8: " + base8.join(', '));
}
private function _getTerm(n:uint, base:uint = 2):Number {
var r:Number = 0, p:uint, digit:uint;
var baseLog:Number = Math.log(base);
while ( n > 0 ) {
p = Math.pow( base, uint(Math.log(n) / baseLog) );
digit = n / p;
n %= p;
r += digit / (p * base);
}
return r;
}
}
}
- Output:
Base 2: 0, 0.5, 0.25, 0.75, 0.125, 0.625, 0.375, 0.875, 0.0625, 0.5625 Base 3: 0, 0.333333, 0.666667, 0.111111, 0.444444, 0.777778, 0.222222, 0.555556, 0.888889, 0.037037 Base 4: 0, 0.25, 0.5, 0.75, 0.0625, 0.3125, 0.5625, 0.8125, 0.125, 0.375 Base 5: 0, 0.2, 0.4, 0.6, 0.8, 0.04, 0.24, 0.44, 0.64, 0.84 Base 6: 0, 0.166667, 0.333333, 0.5, 0.666667, 0.833333, 0.027778, 0.194444, 0.361111, 0.527778 Base 7: 0, 0.142857, 0.285714, 0.428571, 0.571429, 0.714286, 0.857143, 0.020408, 0.163265, 0.306122 Base 8: 0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 0.015625, 0.140625
Ada
with Ada.Text_IO;
procedure Main is
package Float_IO is new Ada.Text_IO.Float_IO (Float);
function Van_Der_Corput (N : Natural; Base : Positive := 2) return Float is
Value : Natural := N;
Result : Float := 0.0;
Exponent : Positive := 1;
begin
while Value > 0 loop
Result := Result +
Float (Value mod Base) / Float (Base ** Exponent);
Value := Value / Base;
Exponent := Exponent + 1;
end loop;
return Result;
end Van_Der_Corput;
begin
for Base in 2 .. 5 loop
Ada.Text_IO.Put ("Base" & Integer'Image (Base) & ":");
for N in 1 .. 10 loop
Ada.Text_IO.Put (' ');
Float_IO.Put (Item => Van_Der_Corput (N, Base), Exp => 0);
end loop;
Ada.Text_IO.New_Line;
end loop;
end Main;
- Output:
Base 2: 0.50000 0.25000 0.75000 0.12500 0.62500 0.37500 0.87500 0.06250 0.56250 0.31250 Base 3: 0.33333 0.66667 0.11111 0.44444 0.77778 0.22222 0.55556 0.88889 0.03704 0.37037 Base 4: 0.25000 0.50000 0.75000 0.06250 0.31250 0.56250 0.81250 0.12500 0.37500 0.62500 Base 5: 0.20000 0.40000 0.60000 0.80000 0.04000 0.24000 0.44000 0.64000 0.84000 0.08000
ALGOL 68
BEGIN # show members of the van der Corput sequence in various bases #
# translated from the C sample #
# sets num and denom to the numerator and denominator of the nth member #
# of the van der Corput sequence in the specified base #
PROC vc = ( INT nth, base, REF INT num, denom )VOID:
BEGIN
INT p := 0, q := 1, n := nth;
WHILE n /= 0 DO
p *:= base +:= n MOD base;
q *:= base;
n OVERAB base
OD;
num := p;
denom := q;
# reduce the numerrator and denominator by their gcd #
WHILE p /= 0 DO n := p; p := q MOD p; q := n OD;
num OVERAB q;
denom OVERAB q
END # vc # ;
# task #
FOR b FROM 2 TO 5 DO
print( ( "base ", whole( b, 0 ), ":" ) );
FOR i FROM 0 TO 9 DO
INT d, n;
vc( i, b, n, d );
IF n /= 0
THEN print( ( " ", whole( n, 0 ), "/", whole( d, 0 ) ) )
ELSE print( ( " 0" ) )
FI
OD;
print( ( newline ) )
OD
END- Output:
base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
Arturo
corput: function [num, base][
result: to :rational 0
b: 1 // base
n: num
while [not? zero? n][
result: result + b * n % base
n: n / base
b: b // base
]
return result
]
loop 2..5 'bs ->
print ["Base" bs ":" join.with:", " to [:string] map 1..10 'z -> corput z bs]
- Output:
Base 2 : 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16, 5/16 Base 3 : 1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27, 10/27 Base 4 : 1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8, 5/8 Base 5 : 1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25, 2/25
AutoHotkey
SetFormat, FloatFast, 0.5
for i, v in [2, 3, 4, 5, 6] {
seq .= "Base " v ": "
Loop, 10
seq .= VanDerCorput(A_Index - 1, v) (A_Index = 10 ? "`n" : ", ")
}
MsgBox, % seq
VanDerCorput(n, b, r=0) {
while n
r += Mod(n, b) * b ** -A_Index, n := n // b
return, r
}
- Output:
Base 2: 0, 0.50000, 0.25000, 0.75000, 0.12500, 0.62500, 0.37500, 0.87500, 0.06250, 0.56250 Base 3: 0, 0.33333, 0.66667, 0.11111, 0.44444, 0.77778, 0.22222, 0.55555, 0.88889, 0.03704 Base 4: 0, 0.25000, 0.50000, 0.75000, 0.06250, 0.31250, 0.56250, 0.81250, 0.12500, 0.37500 Base 5: 0, 0.20000, 0.40000, 0.60000, 0.80000, 0.04000, 0.24000, 0.44000, 0.64000, 0.84000 Base 6: 0, 0.16667, 0.33333, 0.50000, 0.66667, 0.83333, 0.02778, 0.19445, 0.36111, 0.52778
AWK
# syntax: GAWK -f VAN_DER_CORPUT_SEQUENCE.AWK
# converted from BBC BASIC
BEGIN {
printf("base")
for (i=0; i<=9; i++) {
printf(" %7d",i)
}
printf("\n")
for (base=2; base<=5; base++) {
printf("%-4s",base)
for (i=0; i<=9; i++) {
printf(" %7.5f",vdc(i,base))
}
printf("\n")
}
exit(0)
}
function vdc(n,b, s,v) {
s = 1
while (n) {
s *= b
v += (n % b) / s
n /= b
n = int(n)
}
return(v)
}
Output:
base 0 1 2 3 4 5 6 7 8 9 2 0.00000 0.50000 0.25000 0.75000 0.12500 0.62500 0.37500 0.87500 0.06250 0.56250 3 0.00000 0.33333 0.66667 0.11111 0.44444 0.77778 0.22222 0.55556 0.88889 0.03704 4 0.00000 0.25000 0.50000 0.75000 0.06250 0.31250 0.56250 0.81250 0.12500 0.37500 5 0.00000 0.20000 0.40000 0.60000 0.80000 0.04000 0.24000 0.44000 0.64000 0.84000
BASIC
10 DEFINT A-Z
20 FOR B=2 TO 5
30 PRINT USING "BASE #:";B;
40 FOR I=0 TO 9
50 P=0: Q=1: N=I
60 IF N=0 GOTO 110
70 P=P*B+N MOD B
80 Q=Q*B
90 N=N\B
100 GOTO 60
110 X=P: Y=Q
120 IF P=0 GOTO 150
130 N=P: P=Q MOD P: Q=N
140 GOTO 120
150 X=X\Q
160 Y=Y\Q
170 IF X=0 THEN PRINT " 0"; ELSE PRINT USING " ##/##";X;Y;
180 NEXT I
190 PRINT
200 NEXT B
- Output:
BASE 2: 0 1/ 2 1/ 4 3/ 4 1/ 8 5/ 8 3/ 8 7/ 8 1/16 9/16 BASE 3: 0 1/ 3 2/ 3 1/ 9 4/ 9 7/ 9 2/ 9 5/ 9 8/ 9 1/27 BASE 4: 0 1/ 4 1/ 2 3/ 4 1/16 5/16 9/16 13/16 1/ 8 3/ 8 BASE 5: 0 1/ 5 2/ 5 3/ 5 4/ 5 1/25 6/25 11/25 16/25 21/25
BASIC256
function num_base$(number, base)
if base > 9 then
print "base not handled by function"
pause 5
return ""
end if
ans$ = ""
while number <> 0
n = (number mod base)
ans$ = string(n) + ans$
number = number \ base
end while
if ans$ = "" then ans$ = "0"
return "." + ans$
end function
for k = 2 to 5
print "Base = "; k
for l = 0 to 12
print ljust(num_base$(l, k), 6);
next l
print : print
next k
endYabasic
sub num_base$(number, base)
if _base_ > 9 then
print "base not handled by function"
sleep 5000
return ""
end if
while number <> 0
n = mod(number, base)
ans$ = str$(n) + ans$
number = int(number / base)
wend
if ans$ = "" then ans$ = "0" : fi
return "." + ans$
end sub
for k = 2 to 5
print "Base = ", k
for l = 0 to 12
print left$(num_base$(l, k), 7), " ";
next l
print : print
next k
end
BBC BASIC
@% = &20509
FOR base% = 2 TO 5
PRINT "Base " ; STR$(base%) ":"
FOR number% = 0 TO 9
PRINT FNvdc(number%, base%);
NEXT
PRINT
NEXT
END
DEF FNvdc(n%, b%)
LOCAL v, s%
s% = 1
WHILE n%
s% *= b%
v += (n% MOD b%) / s%
n% DIV= b%
ENDWHILE
= v
- Output:
Base 2: 0.00000 0.50000 0.25000 0.75000 0.12500 0.62500 0.37500 0.87500 0.06250 0.56250 Base 3: 0.00000 0.33333 0.66667 0.11111 0.44444 0.77778 0.22222 0.55556 0.88889 0.03704 Base 4: 0.00000 0.25000 0.50000 0.75000 0.06250 0.31250 0.56250 0.81250 0.12500 0.37500 Base 5: 0.00000 0.20000 0.40000 0.60000 0.80000 0.04000 0.24000 0.44000 0.64000 0.84000
bc
This solution hardcodes the literal 10 because numeric literals in bc can use any base from 2 to 16. This solution only works with integer bases from 2 to 16.
/*
* Return the _n_th term of the van der Corput sequence.
* Uses the current _ibase_.
*/
define v(n) {
auto c, r, s
s = scale
scale = 0 /* to use integer division */
/*
* c = count digits of n
* r = reverse the digits of n
*/
for (0; n != 0; n /= 10) {
c += 1
r = (10 * r) + (n % 10)
}
/* move radix point to left of digits */
scale = length(r) + 6
r /= 10 ^ c
scale = s
return r
}
t = 10
for (b = 2; b <= 4; b++) {
"base "; b
obase = b
for (i = 0; i < 10; i++) {
ibase = b
" "; v(i)
ibase = t
}
obase = t
}
quit
Some of the calculations are not exact, because bc performs calculations using base 10. So the program prints a result like .202222221 (base 3) when the exact result would be .21 (base 3).
- Output:
base 2 0.00000000000000 .10000000000000 .01000000000000 .11000000000000 .00100000000000 .10100000000000 .01100000000000 .11100000000000 .00010000000000 .10010000000000 base 3 0.000000000 .022222222 .122222221 .002222222 .102222222 .202222221 .012222222 .112222221 .212222221 .000222222 base 4 0.0000000 .1000000 .2000000 .3000000 .0100000 .1100000 .2100000 .310000000 .0200000 .1200000
BCPL
get "libhdr"
let corput(n, base, num, denom) be
$( let p = 0 and q = 1
until n=0
$( p := p * base + n rem base
q := q * base
n := n / base
$)
!num := p
!denom := q
until p=0
$( n := p
p := q rem p
q := n
$)
!num := !num / q
!denom := !denom / q
$)
let writefrac(num, denom) be
test num=0
do writes(" 0")
or writef(" %N/%N", num, denom)
let start() be
$( let num = ? and denom = ?
for base=2 to 5
$( writef("base %N:", base)
for i=0 to 9
$( corput(i, base, @num, @denom)
writefrac(num, denom)
$)
wrch('*N')
$)
$)- Output:
base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
C
#include <stdio.h>
void vc(int n, int base, int *num, int *denom)
{
int p = 0, q = 1;
while (n) {
p = p * base + (n % base);
q *= base;
n /= base;
}
*num = p;
*denom = q;
while (p) { n = p; p = q % p; q = n; }
*num /= q;
*denom /= q;
}
int main()
{
int d, n, i, b;
for (b = 2; b < 6; b++) {
printf("base %d:", b);
for (i = 0; i < 10; i++) {
vc(i, b, &n, &d);
if (n) printf(" %d/%d", n, d);
else printf(" 0");
}
printf("\n");
}
return 0;
}
- Output:
base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
C#
This is based on the C version.
It uses LINQ and enumeration over a collection
to package the sequence and make it easy to use.
Note that the iterator returns a generic Tuple
whose items are the numerator and denominator for the item.
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace VanDerCorput
{
/// <summary>
/// Computes the Van der Corput sequence for any number base.
/// The numbers in the sequence vary from zero to one, including zero but excluding one.
/// The sequence possesses low discrepancy.
/// Here are the first ten terms for bases 2 to 5:
///
/// base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16
/// base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27
/// base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8
/// base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
/// </summary>
/// <see cref="http://rosettacode.org/wiki/Van_der_Corput_sequence"/>
public class VanDerCorputSequence: IEnumerable<Tuple<long,long>>
{
/// <summary>
/// Number base for the sequence, which must bwe two or more.
/// </summary>
public int Base { get; private set; }
/// <summary>
/// Maximum number of terms to be returned by iterator.
/// </summary>
public long Count { get; private set; }
/// <summary>
/// Construct a sequence for the given base.
/// </summary>
/// <param name="iBase">Number base for the sequence.</param>
/// <param name="count">Maximum number of items to be returned by the iterator.</param>
public VanDerCorputSequence(int iBase, long count = long.MaxValue) {
if (iBase < 2)
throw new ArgumentOutOfRangeException("iBase", "must be two or greater, not the given value of " + iBase);
Base = iBase;
Count = count;
}
/// <summary>
/// Compute nth term in the Van der Corput sequence for the base specified in the constructor.
/// </summary>
/// <param name="n">The position in the sequence, which may be zero or any positive number.</param>
/// This number is always an integral power of the base.</param>
/// <returns>The Van der Corput sequence value expressed as a Tuple containing a numerator and a denominator.</returns>
public Tuple<long,long> Compute(long n)
{
long p = 0, q = 1;
long numerator, denominator;
while (n != 0)
{
p = p * Base + (n % Base);
q *= Base;
n /= Base;
}
numerator = p;
denominator = q;
while (p != 0)
{
n = p;
p = q % p;
q = n;
}
numerator /= q;
denominator /= q;
return new Tuple<long,long>(numerator, denominator);
}
/// <summary>
/// Compute nth term in the Van der Corput sequence for the given base.
/// </summary>
/// <param name="iBase">Base to use for the sequence.</param>
/// <param name="n">The position in the sequence, which may be zero or any positive number.</param>
/// <returns>The Van der Corput sequence value expressed as a Tuple containing a numerator and a denominator.</returns>
public static Tuple<long, long> Compute(int iBase, long n)
{
var seq = new VanDerCorputSequence(iBase);
return seq.Compute(n);
}
/// <summary>
/// Iterate over the Van Der Corput sequence.
/// The first value in the sequence is always zero, regardless of the base.
/// </summary>
/// <returns>A tuple whose items are the Van der Corput value given as a numerator and denominator.</returns>
public IEnumerator<Tuple<long, long>> GetEnumerator()
{
long iSequenceIndex = 0L;
while (iSequenceIndex < Count)
{
yield return Compute(iSequenceIndex);
iSequenceIndex++;
}
}
System.Collections.IEnumerator System.Collections.IEnumerable.GetEnumerator()
{
return GetEnumerator();
}
}
class Program
{
static void Main(string[] args)
{
TestBasesTwoThroughFive();
Console.WriteLine("Type return to continue...");
Console.ReadLine();
}
static void TestBasesTwoThroughFive()
{
foreach (var seq in Enumerable.Range(2, 5).Select(x => new VanDerCorputSequence(x, 10))) // Just the first 10 elements of the each sequence
{
Console.Write("base " + seq.Base + ":");
foreach(var vc in seq)
Console.Write(" " + vc.Item1 + "/" + vc.Item2);
Console.WriteLine();
}
}
}
}
- Output:
base 2: 0/1 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 base 3: 0/1 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 base 4: 0/1 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 base 5: 0/1 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25 base 6: 0/1 1/6 1/3 1/2 2/3 5/6 1/36 7/36 13/36 19/36 Type return to continue...
C++
#include <cmath>
#include <iostream>
double vdc(int n, double base = 2)
{
double vdc = 0, denom = 1;
while (n)
{
vdc += fmod(n, base) / (denom *= base);
n /= base; // note: conversion from 'double' to 'int'
}
return vdc;
}
int main()
{
for (double base = 2; base < 6; ++base)
{
std::cout << "Base " << base << "\n";
for (int n = 0; n < 10; ++n)
{
std::cout << vdc(n, base) << " ";
}
std::cout << "\n\n";
}
}
- Output:
Base 2 0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 Base 3 0 0.333333 0.666667 0.111111 0.444444 0.777778 0.222222 0.555556 0.888889 0.037037 Base 4 0 0.25 0.5 0.75 0.0625 0.3125 0.5625 0.8125 0.125 0.375 Base 5 0 0.2 0.4 0.6 0.8 0.04 0.24 0.44 0.64 0.84
Clojure
(defn van-der-corput
"Get the nth element of the van der Corput sequence."
([n]
;; Default base = 2
(van-der-corput n 2))
([n base]
(let [s (/ 1 base)] ;; A multiplicand to shift to the right of the decimal.
;; We essentially want to reverse the digits of n and put them after the
;; decimal point. So, we repeatedly pull off the lowest digit of n, scale
;; it to the right of the decimal point, and accumulate that.
(loop [sum 0
n n
scale s]
(if (zero? n)
sum ;; Base case: no digits left, so we're done.
(recur (+ sum (* (rem n base) scale)) ;; Accumulate the least digit
(quot n base) ;; Drop a digit of n
(* scale s))))))) ;; Move farther past the decimal
(clojure.pprint/print-table
(cons :base (range 10)) ;; column headings
(for [base (range 2 6)] ;; rows
(into {:base base}
(for [n (range 10)] ;; table entries
[n (van-der-corput n base)]))))
- Output:
| :base | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | |-------+---+-----+-----+-----+------+------+------+-------+-------+-------| | 2 | 0 | 1/2 | 1/4 | 3/4 | 1/8 | 5/8 | 3/8 | 7/8 | 1/16 | 9/16 | | 3 | 0 | 1/3 | 2/3 | 1/9 | 4/9 | 7/9 | 2/9 | 5/9 | 8/9 | 1/27 | | 4 | 0 | 1/4 | 1/2 | 3/4 | 1/16 | 5/16 | 9/16 | 13/16 | 1/8 | 3/8 | | 5 | 0 | 1/5 | 2/5 | 3/5 | 4/5 | 1/25 | 6/25 | 11/25 | 16/25 | 21/25 |
CLU
vc = proc (n, base: int) returns (int, int)
p: int := 0
q: int := 1
while n ~= 0 do
p := p * base + n // base
q := q * base
n := n / base
end
num: int := p
denom: int := q
while p ~= 0 do
p, q := q // p, p
end
return(num/q, denom/q)
end vc
print_frac = proc (po: stream, num, denom: int)
if num=0 then
stream$puts(po, " 0")
else
stream$puts(po, " ")
stream$putright(po, int$unparse(num), 2)
stream$puts(po, "/")
stream$putright(po, int$unparse(denom), 2)
end
end print_frac
start_up = proc ()
po: stream := stream$primary_output()
for base: int in int$from_to(2,5) do
stream$puts(po, "base " || int$unparse(base) || ":")
for i: int in int$from_to(0, 9) do
n, d: int := vc(i, base)
print_frac(po, n, d)
end
stream$putl(po, "")
end
end start_up- Output:
base 2: 0 1/ 2 1/ 4 3/ 4 1/ 8 5/ 8 3/ 8 7/ 8 1/16 9/16 base 3: 0 1/ 3 2/ 3 1/ 9 4/ 9 7/ 9 2/ 9 5/ 9 8/ 9 1/27 base 4: 0 1/ 4 1/ 2 3/ 4 1/16 5/16 9/16 13/16 1/ 8 3/ 8 base 5: 0 1/ 5 2/ 5 3/ 5 4/ 5 1/25 6/25 11/25 16/25 21/25
Common Lisp
(defun van-der-Corput (n base)
(loop for d = 1 then (* d base) while (<= d n)
finally
(return (/ (parse-integer
(reverse (write-to-string n :base base))
:radix base)
d))))
(loop for base from 2 to 5 do
(format t "Base ~a: ~{~6a~^~}~%" base
(loop for i to 10 collect (van-der-Corput i base))))
- Output:
Base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 5/16 Base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 10/27 Base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 5/8 Base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25 2/25
Cowgol
include "cowgol.coh";
sub vc(n: uint16, base: uint16): (num: uint16, denom: uint16) is
var p: uint16 := 0;
var q: uint16 := 1;
while n != 0 loop
p := p * base + n % base;
q := q * base;
n := n / base;
end loop;
num := p;
denom := q;
while p != 0 loop
n := p;
p := q % p;
q := n;
end loop;
num := num / q;
denom := denom / q;
end sub;
sub printfrac(num: uint16, denom: uint16) is
if num == 0 then
print(" 0");
else
print(" ");
print_i16(num);
print("/");
print_i16(denom);
end if;
end sub;
var i: uint16;
var base: uint16;
var num: uint16;
var denom: uint16;
base := 2;
while base < 6 loop
print("base ");
print_i16(base);
print(":");
i := 0;
while i < 10 loop
(num, denom) := vc(i, base);
printfrac(num, denom);
i := i + 1;
end loop;
print_nl();
base := base + 1;
end loop;- Output:
base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
D
double vdc(int n, in double base=2.0) pure nothrow @safe @nogc {
double vdc = 0.0, denom = 1.0;
while (n) {
denom *= base;
vdc += (n % base) / denom;
n /= base;
}
return vdc;
}
void main() {
import std.stdio, std.algorithm, std.range;
foreach (immutable b; 2 .. 6)
writeln("\nBase ", b, ": ", 10.iota.map!(n => vdc(n, b)));
}
- Output:
Base 2: [0, 0.5, 0.25, 0.75, 0.125, 0.625, 0.375, 0.875, 0.0625, 0.5625] Base 3: [0, 0.333333, 0.666667, 0.111111, 0.444444, 0.777778, 0.222222, 0.555556, 0.888889, 0.037037] Base 4: [0, 0.25, 0.5, 0.75, 0.0625, 0.3125, 0.5625, 0.8125, 0.125, 0.375] Base 5: [0, 0.2, 0.4, 0.6, 0.8, 0.04, 0.24, 0.44, 0.64, 0.84]
Delphi
function VanDerCorput(N,Base: integer): double;
{Calculate binary value for numbers right of decimal}
var Value,Exponent,Digit: integer;
begin
Value:= N; Result:= 0; Exponent:= -1;
{D1 * Base^-1 + D2 * Base^-2 + D3 * Base^-3}
while Value > 0 do
begin
{Get digit in specified base}
Digit:=Value mod Base;
{Digit * Base^-Exponent}
Result:=Result + Digit * Power(Base,Exponent);
{Divide by base to put next digit in place}
Value:= Value div Base;
{Next exponent}
Dec(Exponent);
end;
end;
procedure ShowVanDerCorput(Memo: TMemo);
{Show Vander Coput numbers for bases 2..8 and items 1..9 }
var Base,N: integer;
var V: double;
var S: string;
begin
S:='';
for Base:=2 to 8 do
begin
S:=S+Format('Base %D:',[Base]);
for N:=1 to 10 do
begin
V:=VanDerCorput(N,Base);
S:=S+Format(' %1.5f',[V]);
end;
S:=S+CRLF;
end;
Memo.Lines.Add(S);
end;
- Output:
Base 2: 0.50000 0.25000 0.75000 0.12500 0.62500 0.37500 0.87500 0.06250 0.56250 0.31250 Base 3: 0.33333 0.66667 0.11111 0.44444 0.77778 0.22222 0.55556 0.88889 0.03704 0.37037 Base 4: 0.25000 0.50000 0.75000 0.06250 0.31250 0.56250 0.81250 0.12500 0.37500 0.62500 Base 5: 0.20000 0.40000 0.60000 0.80000 0.04000 0.24000 0.44000 0.64000 0.84000 0.08000 Base 6: 0.16667 0.33333 0.50000 0.66667 0.83333 0.02778 0.19444 0.36111 0.52778 0.69444 Base 7: 0.14286 0.28571 0.42857 0.57143 0.71429 0.85714 0.02041 0.16327 0.30612 0.44898 Base 8: 0.12500 0.25000 0.37500 0.50000 0.62500 0.75000 0.87500 0.01563 0.14063 0.26563 Elapsed Time: 1.344 ms.
EasyLang
func vdc b n .
s = 1
while n > 0
s *= b
m = n mod b
v += m / s
n = n div b
.
return v
.
for b = 2 to 5
write "base " & b & ":"
for n range0 10
write " " & vdc b n
.
print ""
.
- Output:
base 2: 0 0.50 0.25 0.75 0.12 0.62 0.38 0.88 0.06 0.56 base 3: 0 0.33 0.67 0.11 0.44 0.78 0.22 0.56 0.89 0.04 base 4: 0 0.25 0.50 0.75 0.06 0.31 0.56 0.81 0.12 0.38 base 5: 0 0.20 0.40 0.60 0.80 0.04 0.24 0.44 0.64 0.84
EDSAC order code
Base 2 only, extra credit not attempted.
EDSAC's fixed-point arithmetic makes this task in base 2 fairly straightforward. If, as usual, a 17-bit integer n is stored as n/(2^16), then to find the n'th term in the van der Corput sequence we just reverse the order of bits after the binary point. E.g. 13 is stored as 0.0000000000001101, so the 13th term is 0.1011000000000000 = 11/16. Similarly for 35-bit numbers. The demo program contains subroutines for both formats.
[Van der Corput sequence for Rosetta Code.
EDSAC solution, Initial Orders 2.]
[Library subroutine M3 - prints header at load time and is then overwritten.
Here, the last character sets the teleprinter to figures.]
PFGKIFAFRDLFUFOFE@A6FG@E8FEZPF
*VAN!DER!CORPUT!SEQUENCE@A*BIT!!!#35A*BIT@&#
..PZ [blank tape then re-sync]
[Define load addresses]
T55K P100F [V parameter: van der Corput subroutines]
T51K P64F [G parameter: print subroutine]
T47K P400F [M parameter: main routine]
[Subroutines to return n'th element of van der Corput sequence.
17-bit version: Call by GV, pass n in 0F (not preserved), result in 4F.
35-bit version: Call by G1V, pass n in 0D (not preserved), result in 4D.]
E25K TV GK
G2@ [jump to 17-bit version]
G25@ [jump to 35-bit version]
[17-bit version.
On EDSAC, it's a matter of reversing the bits after the binary point
To save time, we use a table to reverse the 16 bits in groups of 4.]
[2] A3F T24@ [plant return link as usual]
H5 6@ [set mult reg to 0...01111 binary]
A55@ T4F [set marker bit 0...01 in result]
[7] A4F L4F T4F [shift result 4 left]
CF [acc := next 4 bits of n]
LD [shift into address field]
A58@ T14@ [plant A order to load from table]
[14] AF [{planted) load bits from table]
A4F [add to result]
G22@ [jump out if marker bit has reached sign bit]
T4F [update result]
AF R4F TF [shift n 4 right]
E7@ [always loop back]
[22] S57@ [done, remove marker bit]
T4F [store final result]
[24] ZF [(planted) jump to return to caller]
[35-bit version. Very similar to the 17-bit version, except that
after reversing 8 groups of 4, there are 2 bits left over,
which require separate treatment.]
[25] A3F T54@ [plant return link as usual]
H56@ [set mult reg to 0...01111 binary]
YF L2F [set marker bit 0...0100 in result]
[30] L4F T4D [shift result 4 left]
CF LD A58@ T36@ AF A4F T4F [update from table as in 17-bit version]
ADR4FTD [shift n 4 right]
A4D [load result]
E30@ [if marker bit hasn't reached sign bit, loop back]
[Last 2 bits]
[44] L1FT4D [shift result 2 right]
CF LD A58@ T50@ [plant A order as in 17-bit version]
[50] AF [Planted) load bits from table]
R1F A4F T4F [shift table entry 2 right and add to result]
[54] ZF [(planted) jump to return to caller]
[Constants]
[55] PD [17-bit 1]
[56] P7D [17-bit 15]
[57] K4096F [17-bit 10...0 binary]
[58] A59@ [order to load from table{0}]
[Table to reverse group of 4 bits, e.g. table{0010b} = 0100b]
[59] PFP4FP2FP6FP1FP5FP3FP7FPDP4DP2DP6DP1DP5DP3DP7D
[Library subroutine P1 to print number in range 0 <= x < 1.
Caller must print leading '0.' if required. 21 storage locations.]
E25K TG
GKA18@U17@S20@T5@H19@PFT5@VDUFOFFFSFL4FTDA5@A2FG6@EFU3FJFM1F
[Main routine]
E25K TM GK
[0] PF PF [n, 35 bits, must be at even address]
[2] PF [negative count of terms]
[3] P10F [<=== EDIT number of terms, in address field]
[4] PD [17-bit integer 1]
[5] MF [dot (in figures mode)]
[6] @F [carriage return]
[7] &F [line feed]
[8] !F [space character]
[9] K4096F [null character]
[Enter with acc = 0]
[10] T#@ [n := 0]
S3@ T2@ [initialize negative count]
[13] A@ TF [pass 17-bit n in 0F]
[15] A15@ GV [call 17-bit van der Corput routine]
TD [clear 0D, including sandwich bit]
A4F T1F [extend 17-bit result to 35 bits in 0D]
O4@ O5@ [print '0.']
[22] A22@ GG P5F [print result to 5 decimals]
O8@ O8@ [print 2 spaces]
A#@ TD [pass 35-bit n in 0D]
[29] A29@ G1V [call 35-bit van der Corput routine]
A4D TD [pass result in 0D]
O4@ O5@ [print '0.']
[35] A35@ GG P10F [print result to 10 decimals]
O6@ O7@ [print CR LF]
A2@ A2F [inc negative count]
E48@ [jump out if count = 0]
T2@ [update count]
A@ A4@ T@ [inc n]
E13@ [loop back]
[48] O9@ [print null to flush teleprinter buffer]
ZF [stop]
E10Z [define entry point]
PF [acc = 0 on entry]
[end]- Output:
VAN DER CORPUT SEQUENCE 17-BIT 35-BIT 0.00000 0.0000000000 0.50000 0.5000000000 0.25000 0.2500000000 0.75000 0.7500000000 0.12500 0.1250000000 0.62500 0.6250000000 0.37500 0.3750000000 0.87500 0.8750000000 0.06250 0.0625000000 0.56250 0.5625000000
Ela
open random number list
vdc bs n = vdc' 0.0 1.0 n
where vdc' v d n
| n > 0 = vdc' v' d' n'
| else = v
where
d' = d * bs
rem = n % bs
n' = truncate (n / bs)
v' = v + rem / d'Test (with base 2.0, using non-strict map function on infinite list):
take 10 <| map' (vdc 2.0) [1..]- Output:
[0.5,0.25,0.75,0.125,0.625,0.375,0.875,0.0625,0.5625,0.3125]
Elixir
defmodule Van_der_corput do
def sequence( n, base \\ 2 ) do
"0." <> (Integer.to_string(n, base) |> String.reverse )
end
def float( n, base \\ 2 ) do
Integer.digits(n, base) |> Enum.reduce(0, fn i,acc -> (i + acc) / base end)
end
def fraction( n, base \\ 2 ) do
str = Integer.to_string(n, base) |> String.reverse
denominator = Enum.reduce(1..String.length(str), 1, fn _,acc -> acc*base end)
reduction( String.to_integer(str, base), denominator )
end
defp reduction( 0, _ ), do: "0"
defp reduction( numerator, denominator ) do
gcd = gcd( numerator, denominator )
"#{ div(numerator, gcd) }/#{ div(denominator, gcd) }"
end
defp gcd( a, 0 ), do: a
defp gcd( a, b ), do: gcd( b, rem(a, b) )
end
funs = [ {"Float(Base):", &Van_der_corput.sequence/2},
{"Float(Decimal):", &Van_der_corput.float/2 },
{"Fraction:", &Van_der_corput.fraction/2} ]
Enum.each(funs, fn {title, fun} ->
IO.puts title
Enum.each(2..5, fn base ->
IO.puts " Base #{ base }: #{ Enum.map_join(0..9, ", ", &fun.(&1, base)) }"
end)
end)
- Output:
Float(Base): Base 2: 0.0, 0.1, 0.01, 0.11, 0.001, 0.101, 0.011, 0.111, 0.0001, 0.1001 Base 3: 0.0, 0.1, 0.2, 0.01, 0.11, 0.21, 0.02, 0.12, 0.22, 0.001 Base 4: 0.0, 0.1, 0.2, 0.3, 0.01, 0.11, 0.21, 0.31, 0.02, 0.12 Base 5: 0.0, 0.1, 0.2, 0.3, 0.4, 0.01, 0.11, 0.21, 0.31, 0.41 Float(Decimal): Base 2: 0.0, 0.5, 0.25, 0.75, 0.125, 0.625, 0.375, 0.875, 0.0625, 0.5625 Base 3: 0.0, 0.3333333333333333, 0.6666666666666666, 0.1111111111111111, 0.4444444444444444, 0.7777777777777778, 0.2222222222222222, 0.5555555555555555, 0.8888888888888888, 0.037037037037037035 Base 4: 0.0, 0.25, 0.5, 0.75, 0.0625, 0.3125, 0.5625, 0.8125, 0.125, 0.375 Base 5: 0.0, 0.2, 0.4, 0.6, 0.8, 0.04, 0.24, 0.44000000000000006, 0.64, 0.8400000000000001 Fraction: Base 2: 0, 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16 Base 3: 0, 1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27 Base 4: 0, 1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8 Base 5: 0, 1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25
Erlang
I liked the bc output-in-same-base, but think this is the way it should look.
-module( van_der_corput ).
-export( [sequence/1, sequence/2, task/0] ).
sequence( N ) -> sequence( N, 2 ).
sequence( 0, _Base ) -> 0.0;
sequence( N, Base ) -> erlang:list_to_float( "0." ++ lists:flatten([erlang:integer_to_list(X) || X <- sequence_loop(N, Base)]) ).
task() -> [task(X) || X <- lists:seq(2, 5)].
sequence_loop( 0, _Base ) -> [];
sequence_loop( N, Base ) ->
New_n = N div Base,
Digit = N rem Base,
[Digit | sequence_loop( New_n, Base )].
task( Base ) ->
io:fwrite( "Base ~p:", [Base] ),
[io:fwrite( " ~p", [sequence(X, Base)] ) || X <- lists:seq(0, 9)],
io:fwrite( "~n" ).
- Output:
34> van_der_corput:task(). Base 2: 0.0 0.1 0.01 0.11 0.001 0.101 0.011 0.111 0.0001 0.1001 Base 3: 0.0 0.1 0.2 0.01 0.11 0.21 0.02 0.12 0.22 0.001 Base 4: 0.0 0.1 0.2 0.3 0.01 0.11 0.21 0.31 0.02 0.12 Base 5: 0.0 0.1 0.2 0.3 0.4 0.01 0.11 0.21 0.31 0.41
ERRE
PROGRAM VAN_DER_CORPUT
!
! for rosettacode.org
!
PROCEDURE VDC(N%,B%->RES)
LOCAL V,S%
S%=1
WHILE N%>0 DO
S%*=B%
V+=(N% MOD B%)/S%
N%=N% DIV B%
END WHILE
RES=V
END PROCEDURE
BEGIN
FOR BASE%=2 TO 5 DO
PRINT("Base";STR$(BASE%);":")
FOR NUMBER%=0 TO 9 DO
VDC(NUMBER%,BASE%->RES)
WRITE("#.##### ";RES;)
END FOR
PRINT
END FOR
END PROGRAM- Output:
Base 2: 0.00000 0.50000 0.25000 0.75000 0.12500 0.62500 0.37500 0.87500 0.06250 0.56250 Base 3: 0.00000 0.33333 0.66667 0.11111 0.44444 0.77778 0.22222 0.55556 0.88889 0.03704 Base 4: 0.00000 0.25000 0.50000 0.75000 0.06250 0.31250 0.56250 0.81250 0.12500 0.37500 Base 5: 0.00000 0.20000 0.40000 0.60000 0.80000 0.04000 0.24000 0.44000 0.64000 0.84000
Euphoria
function vdc(integer n, atom base)
atom vdc, denom, rem
vdc = 0
denom = 1
while n do
denom *= base
rem = remainder(n,base)
n = floor(n/base)
vdc += rem / denom
end while
return vdc
end function
for i = 2 to 5 do
printf(1,"Base %d\n",i)
for j = 0 to 9 do
printf(1,"%g ",vdc(j,i))
end for
puts(1,"\n\n")
end for- Output:
Base 2 0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 Base 3 0 0.333333 0.666667 0.111111 0.444444 0.777778 0.222222 0.555556 0.888889 0.037037 Base 4 0 0.25 0.5 0.75 0.0625 0.3125 0.5625 0.8125 0.125 0.375 Base 5 0 0.2 0.4 0.6 0.8 0.04 0.24 0.44 0.64 0.84
F#
open System
let vdc n b =
let rec loop n denom acc =
if n > 0l then
let m, remainder = Math.DivRem(n, b)
loop m (denom * b) (acc + (float remainder) / (float (denom * b)))
else acc
loop n 1 0.0
[<EntryPoint>]
let main argv =
printfn "%A" [ for n in 0 .. 9 -> (vdc n 2) ]
printfn "%A" [ for n in 0 .. 9 -> (vdc n 5) ]
0
- Output:
[0.0; 0.5; 0.25; 0.75; 0.125; 0.625; 0.375; 0.875; 0.0625; 0.5625] [0.0; 0.2; 0.4; 0.6; 0.8; 0.04; 0.24; 0.44; 0.64; 0.84]
Factor
USING: formatting fry io kernel math math.functions math.parser
math.ranges sequences ;
IN: rosetta-code.van-der-corput
: vdc ( n base -- x )
[ >base string>digits <reversed> ]
[ nip '[ 1 + neg _ swap ^ * ] ] 2bi map-index sum ;
: vdc-demo ( -- )
2 5 [a,b] [
dup "Base %d: " printf 10 <iota>
[ swap vdc "%-5u " printf ] with each nl
] each ;
MAIN: vdc-demo
- Output:
Base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 Base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 Base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 Base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
Forth
: fvdc ( base n -- f )
0e 1e ( F: vdc denominator )
begin dup while
over s>d d>f f*
over /mod ( base rem n )
swap s>d d>f fover f/
frot f+ fswap
repeat 2drop fdrop ;
: test 10 0 do 2 i fvdc cr f. loop ;
- Output:
test 0. 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 ok
Fortran
This is straightforward once one remembers that the obvious scheme for extracting digits from a number produces them from the low-order end to the high-order end. This reversal is normally annoying, but here a "reflection" is desired. The source is old-style, except for using F90's ability to have a function (or subroutine) name appear on its END statement with this checked by the compiler. Because the MODULE protocol introduced by F90 is not bothered with, the type of the function has to be declared in all routines invoking it if the default type based on the form of the name does not suffice. Single precision suffices, but the F90 compiler moans that the type of the function itself has not been explicitly declared. Ah well.
FUNCTION VDC(N,BASE) !Calculates a Van der Corput number...
Converts 1234 in decimal to 4321 in V, and P = 10000.
INTEGER N !For this integer,
INTEGER BASE !In this base.
INTEGER I !A copy of N that can be damaged.
INTEGER P !Successive powers of BASE.
INTEGER V !Accumulates digits.
P = 1 ! = BASE**0
V = 0 !Start with no digits, as if N = 0.
I = N !Here we go.
DO WHILE (I .NE. 0) !While something remains,
V = V*BASE + MOD(I,BASE) !Extract its low-order digit.
I = I/BASE !Reduce it by a power.
P = P*BASE !And track the power.
END DO !Thus extract the digits in reverse order: right-to-left.
VDC = V/FLOAT(P) !The power is one above the highest digit.
END FUNCTION VDC !Numerology is weird.
PROGRAM POKE
INTEGER FIRST,LAST !Might as well document some constants.
PARAMETER (FIRST = 0,LAST = 9) !Thus, the first ten values.
INTEGER I,BASE !Steppers.
REAL VDC !Stop the compiler moaning about undeclared items.
WRITE (6,1) FIRST,LAST,(I, I = FIRST,LAST) !Announce.
1 FORMAT ("Calculates values ",I0," to ",I0," of the ",
1 "Van der Corput sequence, in various bases."/
2 "Base",666I9)
DO BASE = 2,13 !A selection of bases.
WRITE (6,2) BASE,(VDC(I,BASE), I = FIRST,LAST) !Show the specified span.
2 FORMAT (I4,666F9.6) !Aligns with FORMAT 1.
END DO !On to the next base.
END
Output: six-digit precision is about the most that single precision offers.
Calculates values 0 to 9 of the Van der Corput sequence, in various bases. Base 0 1 2 3 4 5 6 7 8 9 2 0.000000 0.500000 0.250000 0.750000 0.125000 0.625000 0.375000 0.875000 0.062500 0.562500 3 0.000000 0.333333 0.666667 0.111111 0.444444 0.777778 0.222222 0.555556 0.888889 0.037037 4 0.000000 0.250000 0.500000 0.750000 0.062500 0.312500 0.562500 0.812500 0.125000 0.375000 5 0.000000 0.200000 0.400000 0.600000 0.800000 0.040000 0.240000 0.440000 0.640000 0.840000 6 0.000000 0.166667 0.333333 0.500000 0.666667 0.833333 0.027778 0.194444 0.361111 0.527778 7 0.000000 0.142857 0.285714 0.428571 0.571429 0.714286 0.857143 0.020408 0.163265 0.306122 8 0.000000 0.125000 0.250000 0.375000 0.500000 0.625000 0.750000 0.875000 0.015625 0.140625 9 0.000000 0.111111 0.222222 0.333333 0.444444 0.555556 0.666667 0.777778 0.888889 0.012346 10 0.000000 0.100000 0.200000 0.300000 0.400000 0.500000 0.600000 0.700000 0.800000 0.900000 11 0.000000 0.090909 0.181818 0.272727 0.363636 0.454545 0.545455 0.636364 0.727273 0.818182 12 0.000000 0.083333 0.166667 0.250000 0.333333 0.416667 0.500000 0.583333 0.666667 0.750000 13 0.000000 0.076923 0.153846 0.230769 0.307692 0.384615 0.461538 0.538462 0.615385 0.692308
FreeBASIC
' version 03-12-2016
' compile with: fbc -s console
Function num_base(number As ULongInt, _base_ As UInteger) As String
If _base_ > 9 Then
Print "base not handled by function"
Sleep 5000
Return ""
End If
Dim As ULongInt n
Dim As String ans
While number <> 0
n = number Mod _base_
ans = Str(n) + ans
number = number \ _base_
Wend
If ans = "" Then ans = "0"
Return "." + ans
End Function
' ------=< MAIN >=------
Dim As ULong k, l
For k = 2 To 5
Print "Base = "; k
For l = 0 To 12
Print left(num_base(l, k) + " ",6);
Next
Print : print
Next
' empty keyboard buffer
While Inkey <> "" : Wend
Print : Print "hit any key to end program"
Sleep
End
- Output:
Base = 2 .0 .1 .10 .11 .100 .101 .110 .111 .1000 .1001 .1010 .1011 .1100 Base = 3 .0 .1 .2 .10 .11 .12 .20 .21 .22 .100 .101 .102 .110 Base = 4 .0 .1 .2 .3 .10 .11 .12 .13 .20 .21 .22 .23 .30 Base = 5 .0 .1 .2 .3 .4 .10 .11 .12 .13 .14 .20 .21 .22
Fōrmulæ
Fōrmulæ programs are not textual, visualization/edition of programs is done showing/manipulating structures but not text. 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 storage and transfer purposes more than visualization and edition.
Programs in Fōrmulæ are created/edited online in its website.
In this page you can see and run the program(s) related to this task and their results. You can also change either the programs or the parameters they are called with, for experimentation, but remember that these programs were created with the main purpose of showing a clear solution of the task, and they generally lack any kind of validation.
Solution
Case 1. Van der Corput sequences for numbers 0 .. 25, in bases 2 to 10
Case 2. Numerical values
Case 3. A plot of Van der Corput sequence for values 0 to 500, in base 10
Go
package main
import "fmt"
func v2(n uint) (r float64) {
p := .5
for n > 0 {
if n&1 == 1 {
r += p
}
p *= .5
n >>= 1
}
return
}
func newV(base uint) func(uint) float64 {
invb := 1 / float64(base)
return func(n uint) (r float64) {
p := invb
for n > 0 {
r += p * float64(n%base)
p *= invb
n /= base
}
return
}
}
func main() {
fmt.Println("Base 2:")
for i := uint(0); i < 10; i++ {
fmt.Println(i, v2(i))
}
fmt.Println("Base 3:")
v3 := newV(3)
for i := uint(0); i < 10; i++ {
fmt.Println(i, v3(i))
}
}
- Output:
Base 2: 0 0 1 0.5 2 0.25 3 0.75 4 0.125 5 0.625 6 0.375 7 0.875 8 0.0625 9 0.5625 Base 3: 0 0 1 0.3333333333333333 2 0.6666666666666666 3 0.1111111111111111 4 0.4444444444444444 5 0.7777777777777777 6 0.2222222222222222 7 0.5555555555555556 8 0.8888888888888888 9 0.037037037037037035
Haskell
The function vdc returns the nth exact, arbitrary precision van der Corput number for any base ≥ 2 and any n. (A reasonable value is returned for negative values of n.)
import Data.Ratio (Rational(..), (%), numerator, denominator)
import Data.List (unfoldr)
import Text.Printf (printf)
-- A wrapper type for Rationals to make them look nicer when we print them.
newtype Rat =
Rat Rational
instance Show Rat where
show (Rat n) = show (numerator n) <> ('/' : show (denominator n))
-- Convert a list of base b digits to its corresponding number.
-- We assume the digits are valid base b numbers and that
-- their order is from least to most significant.
digitsToNum :: Integer -> [Integer] -> Integer
digitsToNum b = foldr1 (\d acc -> b * acc + d)
-- Convert a number to the list of its base b digits.
-- The order will be from least to most significant.
numToDigits :: Integer -> Integer -> [Integer]
numToDigits _ 0 = [0]
numToDigits b n = unfoldr step n
where
step 0 = Nothing
step m =
let (q, r) = m `quotRem` b
in Just (r, q)
-- Return the n'th element in the base b van der Corput sequence.
-- The base must be ≥ 2.
vdc :: Integer -> Integer -> Rat
vdc b n
| b < 2 = error "vdc: base must be ≥ 2"
| otherwise =
let ds = reverse $ numToDigits b n
in Rat (digitsToNum b ds % b ^ length ds)
-- Each base followed by a specified range of van der Corput numbers.
printVdcRanges :: ([Integer], [Integer]) -> IO ()
printVdcRanges (bases, nums) =
mapM_
putStrLn
[ printf "Base %d:" b <> concatMap (printf " %5s" . show) rs
| b <- bases
, let rs = map (vdc b) nums ]
main :: IO ()
main = do
-- Small bases:
printVdcRanges ([2, 3, 4, 5], [0 .. 9])
putStrLn []
-- Base 123:
printVdcRanges ([123], [50,100 .. 300])
- Output:
Base 2: 0/1 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 Base 3: 0/1 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 Base 4: 0/1 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 Base 5: 0/1 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25 Base 123: 50/123 100/123 3322/15129 9472/15129 494/15129 6644/15129
Icon and Unicon
The following solution works in both Icon and Unicon:
procedure main(A)
base := integer(get(A)) | 2
every writes(round(vdc(0 to 9,base),10)," ")
write()
end
procedure vdc(n, base)
e := 1.0
x := 0.0
while x +:= 1(((0 < n) % base) / (e *:= base), n /:= base)
return x
end
procedure round(n,d)
places := 10 ^ d
return real(integer(n*places + 0.5)) / places
end
and a sample run is:
->vdc 0.0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 ->vdc 3 0.0 0.3333333333 0.6666666667 0.1111111111 0.4444444444 0.7777777778 0.2222222222 0.5555555556 0.8888888889 0.037037037 ->vdc 5 0.0 0.2 0.4 0.6 0.8 0.04 0.24 0.44 0.64 0.84 ->vdc 123 0.0 0.0081300813 0.0162601626 0.0243902439 0.0325203252 0.0406504065 0.0487804878 0.0569105691 0.0650406504 0.07317073170000001 ->
An alternate, Unicon-specific implementation of vdc patterned after the functional Raku solution is:
procedure vdc(n, base)
s1 := create |((0 < 1(.n, n /:= base)) % base)
s2 := create 2(e := 1.0, |(e *:= base))
every (result := 0) +:= |s1() / s2()
return result
end
It produces the same output as shown above.
J
Solution:
vdc=: ([ %~ %@[ #. #.inv)"0 _
Examples:
2 vdc i.10 NB. 1st 10 nums of Van der Corput sequence in base 2
0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625
2x vdc i.10 NB. as above but using rational nums
0 1r2 1r4 3r4 1r8 5r8 3r8 7r8 1r16 9r16
2 3 4 5x vdc i.10 NB. 1st 10 nums of Van der Corput sequence in bases 2 3 4 5
0 1r2 1r4 3r4 1r8 5r8 3r8 7r8 1r16 9r16
0 1r3 2r3 1r9 4r9 7r9 2r9 5r9 8r9 1r27
0 1r4 1r2 3r4 1r16 5r16 9r16 13r16 1r8 3r8
0 1r5 2r5 3r5 4r5 1r25 6r25 11r25 16r25 21r25
In other words: use the left argument as the "base" to structure the sequence numbers into digits ("base 2", etc.). Then use the reciprocal of the left argument as the "base" to re-represent this sequence and divide that result by the left argument to get the Van der Corput sequence number.
Java
Using (denom *= 2) as the denominator is not a recommended way of doing things since it is not clear when the multiplication and assignment happen.
Comparing this to the "++" operator, it looks like it should do the doubling and assignment second. Comparing it to the "++" operator used as a preincrement operator, it looks like it should do the doubling and assignment first.
Comparing it to the behavior of parentheses, it looks like it should do the doubling and assignment first. Luckily for us, it works the same in Java as in Raku (doubling and assignment first). It was kept the Raku way to help with the comparison.
Normally, we would initialize denom to 2 (since that is the denominator of the leftmost digit), use it alone in the vdc sum, and then double it after.
public class VanDerCorput{
public static double vdc(int n){
double vdc = 0;
int denom = 1;
while(n != 0){
vdc += n % 2.0 / (denom *= 2);
n /= 2;
}
return vdc;
}
public static void main(String[] args){
for(int i = 0; i <= 10; i++){
System.out.println(vdc(i));
}
}
}
- Output:
0.0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 0.3125
jq
The neat thing about the following implementation of vdc(base) is that it shows how the task can be accomplished in two separate steps without the need to construct an intermediate array.
# vdc(base) converts an input decimal integer to a decimal number based on the van der
# Corput sequence using base 'base', e.g. (4 | vdc(2)) is 0.125.
#
def vdc(base):
# The helper function converts a stream of residuals to a decimal,
# e.g. if base is 2, then decimalize( (0,0,1) ) yields 0.125
def decimalize(stream):
reduce stream as $d # state: [accumulator, power]
( [0, 1/base];
.[1] as $power | [ .[0] + ($d * $power), $power / base] )
| .[0];
if . == 0 then 0
else decimalize(recurse( if . == 0 then empty else ./base | floor end ) % base)
end ;Example:
def round(n):
(if . < 0 then -1 else 1 end) as $s
| $s*10*.*n | if (floor%10)>4 then (.+5) else . end | ./10 | floor/n | .*$s;
range(2;6) | . as $base | "Base \(.): \( [ range(0;11) | vdc($base)|round(1000) ] )"- Output:
$ jq -n -f -c -r van_der_corput_sequence.jq
Base 2: [0,0.5,0.25,0.75,0.125,0.625,0.375,0.875,0.063,0.563,0.313]
Base 3: [0,0.333,0.667,0.111,0.444,0.778,0.222,0.556,0.889,0.037,0.37]
Base 4: [0,0.25,0.5,0.75,0.063,0.313,0.563,0.813,0.125,0.375,0.625]
Base 5: [0,0.2,0.4,0.6,0.8,0.04,0.24,0.44,0.64,0.84,0.08]
Julia
using Printf
vandercorput(num::Integer, base::Integer) = sum(d * Float64(base) ^ -ex for (ex, d) in enumerate(digits(num, base = base)))
for base in 2:9
@printf("%10s %i:", "Base", base)
for num in 0:9 @printf("%7.3f", vandercorput(num, base)) end
println(" [...]")
end
- Output:
Base 2: 0.000 0.500 0.250 0.750 0.125 0.625 0.375 0.875 0.063 0.563...
Base 3: 0.000 0.333 0.667 0.111 0.444 0.778 0.222 0.556 0.889 0.037...
Base 4: 0.000 0.250 0.500 0.750 0.063 0.313 0.563 0.813 0.125 0.375...
Base 5: 0.000 0.200 0.400 0.600 0.800 0.040 0.240 0.440 0.640 0.840...
Base 6: 0.000 0.167 0.333 0.500 0.667 0.833 0.028 0.194 0.361 0.528...
Base 7: 0.000 0.143 0.286 0.429 0.571 0.714 0.857 0.020 0.163 0.306...
Base 8: 0.000 0.125 0.250 0.375 0.500 0.625 0.750 0.875 0.016 0.141...
Base 9: 0.000 0.111 0.222 0.333 0.444 0.556 0.667 0.778 0.889 0.012...
Kotlin
// version 1.1.2
data class Rational(val num: Int, val denom: Int)
fun vdc(n: Int, base: Int): Rational {
var p = 0
var q = 1
var nn = n
while (nn != 0) {
p = p * base + nn % base
q *= base
nn /= base
}
val num = p
val denom = q
while (p != 0) {
nn = p
p = q % p
q = nn
}
return Rational(num / q, denom / q)
}
fun main(args: Array<String>) {
for (b in 2..5) {
print("base $b:")
for (i in 0..9) {
val(num, denom) = vdc(i, b)
if (num != 0) print(" $num/$denom")
else print(" 0")
}
println()
}
}
- Output:
base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
Lua
function vdc(n, base)
local digits = {}
while n ~= 0 do
local m = math.floor(n / base)
table.insert(digits, n - m * base)
n = m
end
m = 0
for p, d in pairs(digits) do
m = m + math.pow(base, -p) * d
end
return m
end
Alternative version, prints the sequence elements as fractions - based on the Algol 68 sample.
function vdc( nth, base ) -- returns the numerator & denominator of the sequence element n in base
local p, q, n = 0, 1, nth
while n ~= 0 do
p = p * base
p = p + n % base;
q = q * base;
n = math.floor( n / base )
end
local num, denom = p, q;
-- reduce the numerator and denominator by their gcd
while p ~= 0 do
n = p
p = q % p
q = n
end
num = math.floor( num / q )
denom = math.floor( denom / q )
return num, denom
end
for b = 2,5 do
io.write( "base ", b, ": " )
for n = 0,9 do
local num, denom = vdc( n, b )
io.write( " ", num ) if num ~= 0 then io.write( "/", denom ) end
end
io.write( "\n" )
end
- Output:
base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
Maple
Halton:=proc(n,b)
local i:=n,k:=1,s:=0,r;
while i>0 do
k/=b;
i:=iquo(i,b,'r');
s+=k*r
od;
s
end;
map(Halton,[$1..10],2);
# [1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16, 5/16]
map(Halton,[$1..10],3);
# [1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27, 10/27]
map(Halton,[$1..10],4);
# [1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8, 5/8]
map(Halton,[$1..10],5);
[1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25, 2/25]Mathematica /Wolfram Language
VanDerCorput[n_,base_:2]:=Table[
FromDigits[{Reverse[IntegerDigits[k,base]],0},base],
{k,n}]
VanDerCorput[10,2]
VanDerCorput[10,3]
VanDerCorput[10,4]
VanDerCorput[10,5]
- Output:
{1/2,1/4,3/4,1/8,5/8,3/8,7/8,1/16,9/16,5/16}
{1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27, 10/27}
{1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8, 5/8}
{1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25, 2/25}
MATLAB / Octave
function x = corput (n)
b = dec2bin(1:n)-'0'; % generate sequence of binary numbers from 1 to n
l = size(b,2); % get number of binary digits
w = (1:l)-l-1; % 2.^w are the weights
x = b * ( 2.^w'); % matrix times vector multiplication for
end;
- Output:
corput(10) ans = 0.500000 0.250000 0.750000 0.125000 0.625000 0.375000 0.875000 0.062500 0.562500 0.312500
Maxima
Define two helper functions
/* convert a decimal integer to a list of digits in base `base' */
dec2digits(d, base):= block([digits: []],
while (d>0) do block([newdi: mod(d, base)],
digits: cons(newdi, digits),
d: round( (d - newdi) / base)),
digits)$
dec2digits(123, 10);
/* [1, 2, 3] */
dec2digits( 8, 2);
/* [1, 0, 0, 0] */
/* convert a list of digits in base `base' to a decimal integer */
digits2dec(l, base):= block([s: 0, po: 1],
for di in reverse(l) do (s: di*po + s, po: po*base),
s)$
digits2dec([1, 2, 3], 10);
/* 123 */
digits2dec([1, 0, 0, 0], 2);
/* 8 */
The main function
vdc(n, base):= makelist(
digits2dec(
dec2digits(k, base),
1/base) / base,
k, n);
vdc(10, 2);
/*
1 1 3 1 5 3 7 1 9 5
(%o123) [-, -, -, -, -, -, -, --, --, --]
2 4 4 8 8 8 8 16 16 16
*/
vdc(10, 5);
/*
1 2 3 4 1 6 11 16 21 2
(%o124) [-, -, -, -, --, --, --, --, --, --]
5 5 5 5 25 25 25 25 25 25
*/
digits2dec can by used with symbols to produce the same example as in the task description
/* 11 in decimal is */
digits: digits2dec([box(1), box(0), box(1), box(1)], box(2));
aux: expand(digits2dec(digits, 1/base) / base)$
simp: false$
/* reflected this would become ... */
subst(box(2), base, aux);
simp: true$
/*
3 2
""" """ """ """ """ """ """
(%o126) "2" "1" + "2" "0" + "2" "1" + "1"
""" """ """ """ """ """ """
- 4 - 3 - 2 - 1
""" """ """ """ """ """ """ """
(%o129) "1" "2" + "0" "2" + "1" "2" + "1" "2"
""" """ """ """ """ """ """ """
*/
Modula-2
MODULE Sequence;
FROM FormatString IMPORT FormatString;
FROM Terminal IMPORT WriteString,WriteLn,ReadChar;
PROCEDURE vc(n,base : INTEGER; VAR num,denom : INTEGER);
VAR p,q : INTEGER;
BEGIN
p := 0;
q := 1;
WHILE n#0 DO
p := p * base + (n MOD base);
q := q * base;
n := n DIV base
END;
num := p;
denom := q;
WHILE p#0 DO
n := p;
p := q MOD p;
q := n
END;
num := num DIV q;
denom := denom DIV q
END vc;
VAR
buf : ARRAY[0..31] OF CHAR;
d,n,i,b : INTEGER;
BEGIN
FOR b:=2 TO 5 DO
FormatString("base %i:", buf, b);
WriteString(buf);
FOR i:=0 TO 9 DO
vc(i,b,n,d);
IF n#0 THEN
FormatString(" %i/%i", buf, n, d);
WriteString(buf)
ELSE
WriteString(" 0")
END
END;
WriteLn
END;
ReadChar
END Sequence.
Nim
Using the “rationals” module of the standard library.
import rationals, strutils, sugar
type Fract = Rational[int]
proc corput(n: int; base: Positive): Fract =
result = 0.toRational
var b = 1 // base
var n = n
while n != 0:
result += n mod base * b
n = n div base
b /= base
for base in 2..5:
let list = collect(newSeq, for n in 1..10: corput(n, base))
echo "Base $#: ".format(base), list.join(" ")
- Output:
Base 2: 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 5/16 Base 3: 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 10/27 Base 4: 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 5/8 Base 5: 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25 2/25
PARI/GP
VdC(n)=n=binary(n);sum(i=1,#n,if(n[i],1.>>(#n+1-i)));
VdC(n)=sum(i=1,#binary(n),if(bittest(n,i-1),1.>>i)); \\ Alternate approach
vector(10,n,VdC(n))- Output:
[0.500000000, 0.250000000, 0.750000000, 0.125000000, 0.625000000, 0.375000000, 0.875000000, 0.0625000000, 0.562500000, 0.312500000]
Pascal
Tested with Free Pascal
Program VanDerCorput;
{$IFDEF FPC}
{$MODE DELPHI}
{$ELSE}
{$APPTYPE CONSOLE}
{$ENDIF}
type
tvdrCallback = procedure (nom,denom: NativeInt);
{ Base=2
function rev2(n,Pot:NativeUint):NativeUint;
var
r : Nativeint;
begin
r := 0;
while Pot > 0 do
Begin
r := r shl 1 OR (n AND 1);
n := n shr 1;
dec(Pot);
end;
rev2 := r;
end;
}
function reverse(n,base,Pot:NativeUint):NativeUint;
var
r,c : Nativeint;
begin
r := 0;
//No need to test n> 0 in this special case, n starting in upper half
while Pot > 0 do
Begin
c := n div base;
r := n+(r-c)*base;
n := c;
dec(Pot);
end;
reverse := r;
end;
procedure VanDerCorput(base,count:NativeUint;f:tvdrCallback);
//calculates count nominater and denominater of Van der Corput sequence
// to base
var
Pot,
denom,nom,
i : NativeUint;
Begin
denom := 1;
Pot := 0;
while count > 0 do
Begin
IF Pot = 0 then
f(0,1);
//start in upper half
i := denom;
inc(Pot);
denom := denom *base;
repeat
nom := reverse(i,base,Pot);
IF count > 0 then
f(nom,denom)
else
break;
inc(i);
dec(count);
until i >= denom;
end;
end;
procedure vdrOutPut(nom,denom: NativeInt);
Begin
write(nom,'/',denom,' ');
end;
var
i : NativeUint;
Begin
For i := 2 to 5 do
Begin
write(' Base ',i:2,' :');
VanDerCorput(i,9,@vdrOutPut);
writeln;
end;
end.
- output
Base 2 :0/1 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 Base 3 :0/1 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 Base 4 :0/1 1/4 2/4 3/4 1/16 5/16 9/16 13/16 2/16 6/16 Base 5 :0/1 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
Perl
sub vdc {
my @value = shift;
my $base = shift // 2;
use integer;
push @value, $value[-1] / $base while $value[-1] > 0;
my ($x, $sum) = (1, 0);
no integer;
$sum += ($_ % $base) / ($x *= $base) for @value;
return $sum;
}
for my $base ( 2 .. 5 ) {
print "base $base: ", join ' ', map { vdc($_, $base) } 0 .. 10;
print "\n";
}
Phix
Not entirely sure what to print, so decided to print in three different ways.
It struck me straightaway that the VdC of say 123 is 321/1000, which seems trivial in any base or desired format.
enum BASE, FRAC, DECIMAL constant DESC = {"Base","Fraction","Decimal"} function vdc(integer n, atom base, integer flag) object res = "" atom num = 0, denom = 1, digit, g while n do denom *= base digit = remainder(n,base) n = floor(n/base) if flag=BASE then res &= digit+'0' else num = num*base+digit end if end while if flag=FRAC then g = gcd(num,denom) return {num/g,denom/g} elsif flag=DECIMAL then return num/denom end if return {iff(length(res)=0?"0":"0."&res)} end function procedure show_vdc(integer flag, string fmt) object v for i=2 to 5 do printf(1,"%s %d: ",{DESC[flag],i}) for j=0 to 9 do v = vdc(j,i,flag) if flag=FRAC and v[1]=0 then printf(1,"0 ") else printf(1,fmt,v) end if end for puts(1,"\n") end for end procedure show_vdc(BASE,"%s ") show_vdc(FRAC,"%d/%d ") show_vdc(DECIMAL,"%g ")
- Output:
Base 2: 0 0.1 0.01 0.11 0.001 0.101 0.011 0.111 0.0001 0.1001 Base 3: 0 0.1 0.2 0.01 0.11 0.21 0.02 0.12 0.22 0.001 Base 4: 0 0.1 0.2 0.3 0.01 0.11 0.21 0.31 0.02 0.12 Base 5: 0 0.1 0.2 0.3 0.4 0.01 0.11 0.21 0.31 0.41 Fraction 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 Fraction 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 Fraction 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 Fraction 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25 Decimal 2: 0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 Decimal 3: 0 0.333333 0.666667 0.111111 0.444444 0.777778 0.222222 0.555556 0.888889 0.037037 Decimal 4: 0 0.25 0.5 0.75 0.0625 0.3125 0.5625 0.8125 0.125 0.375 Decimal 5: 0 0.2 0.4 0.6 0.8 0.04 0.24 0.44 0.64 0.84
PicoLisp
(scl 6)
(de vdc (N B)
(default B 2)
(let (R 0 A 1.0)
(until (=0 N)
(inc 'R (* (setq A (/ A B)) (% N B)))
(setq N (/ N B)) )
R ) )
(for B (2 3 4)
(prinl "Base: " B)
(for N (range 0 9)
(prinl N ": " (round (vdc N B) 4)) ) )- Output:
Base: 2 0: 0.0000 1: 0.5000 2: 0.2500 3: 0.7500 4: 0.1250 5: 0.6250 6: 0.3750 7: 0.8750 8: 0.0625 9: 0.5625 Base: 3 0: 0.0000 1: 0.3333 2: 0.6667 3: 0.1111 4: 0.4444 5: 0.7778 6: 0.2222 7: 0.5556 8: 0.8889 9: 0.0370 Base: 4 0: 0.0000 1: 0.2500 2: 0.5000 3: 0.7500 4: 0.0625 5: 0.3125 6: 0.5625 7: 0.8125 8: 0.1250 9: 0.3750
PL/I
vdcb: procedure (an) returns (bit (31)); /* 6 July 2012 */
declare an fixed binary (31);
declare (n, i) fixed binary (31);
declare v bit (31) varying;
n = an; v = ''b;
do i = 1 by 1 while (n > 0);
if iand(n, 1) = 1 then v = v || '1'b; else v = v || '0'b;
n = isrl(n, 1);
end;
return (v);
end vdcb;
declare i fixed binary (31);
do i = 0 to 10;
put skip list ('0.' || vdcb(i));
end;
- Output:
0.0000000000000000000000000000000 0.1000000000000000000000000000000 0.0100000000000000000000000000000 0.1100000000000000000000000000000 0.0010000000000000000000000000000 0.1010000000000000000000000000000 0.0110000000000000000000000000000 0.1110000000000000000000000000000 0.0001000000000000000000000000000 0.1001000000000000000000000000000 0.0101000000000000000000000000000
Prolog
Example solution
% vdc( N, Base, Out )
% Out = the Van der Corput representation of N in given Base
vdc( 0, _, [] ).
vdc( N, Base, Out ) :-
Nr is mod(N, Base),
Nq is N // Base,
vdc( Nq, Base, Tmp ),
Out = [Nr|Tmp].
% Writes every element of a list to stdout; no newlines
write_list( [] ).
write_list( [H|T] ) :-
write( H ),
write_list( T ).
% Writes the Nth Van der Corput item.
print_vdc( N, Base ) :-
vdc( N, Base, Lst ),
write('0.'),
write_list( Lst ).
print_vdc( N ) :-
print_vdc( N, 2 ).
% Prints the first N+1 elements of the Van der Corput
% sequence, each to its own line
print_some( 0, _ ) :-
write( '0.0' ).
print_some( N, Base ) :-
M is N - 1,
print_some( M, Base ),
nl,
print_vdc( N, Base ).
print_some( N ) :-
print_some( N, 2 ).
test :-
writeln('First 10 members in base 2:'),
print_some( 9 ),
nl,
write('7th member in base 4 (stretch goal) => '),
print_vdc( 7, 4 ).
- Output:
(result of test)
First 10 members in base 2: 0.0 0.1 0.01 0.11 0.001 0.101 0.011 0.111 0.0001 0.1001 7th member in base 4 (stretch goal) => 0.31 true .
Solution with generator
% g(B,N,X):- consecutively generate in X the first N elements of the sequence based on {0, 1, ..., B}
g(_,N,[L|_]-_,X):- N > 1, atomic_list_concat(['0.'|L],X).
g(B,N,[L|Ls]-Xs,X):- N > 2, M is N-1, findall([I|L], between(0,B,I), T), append(T,Ys,Xs), g(B,M,Ls-Ys,X).
g(_,N,'0.0'):- N > 0.
g(B,N,X):- N > 0, findall([I], between(1,B,I), T), T \= [], append(T,Ys,Xs), g(B,N,Xs-Ys,X).
- Output:
?- g(2,10,X). X = '0.0' ; X = '0.1' ; X = '0.2' ; X = '0.01' ; ... X = '0.001' ; false. ?- time(findall(X, g(1,1000000,X), T)). % 23,000,011 inferences, 5.938 CPU in 6.083 seconds (98% CPU, 3873686 Lips) T = ['0.0', '0.1', '0.01', '0.11', '0.001', '0.101', '0.011', '0.111', '0.0001'|...].
PureBasic
Procedure.d nBase(n.i,b.i)
Define r.d,s.i=1
While n
s*b
r+(Mod(n,b)/s)
n=Int(n/b)
Wend
ProcedureReturn r
EndProcedure
Define.i b,c
OpenConsole("van der Corput - Sequence")
For b=2 To 5
Print("Base "+Str(b)+": ")
For c=0 To 9
Print(StrD(nBase(c,b),5)+~"\t")
Next
PrintN("")
Next
Input()
- Output:
Base 2: 0.00000 0.50000 0.25000 0.75000 0.12500 0.62500 0.37500 0.87500 0.06250 0.56250 Base 3: 0.00000 0.33333 0.66667 0.11111 0.44444 0.77778 0.22222 0.55556 0.88889 0.03704 Base 4: 0.00000 0.25000 0.50000 0.75000 0.06250 0.31250 0.56250 0.81250 0.12500 0.37500 Base 5: 0.00000 0.20000 0.40000 0.60000 0.80000 0.04000 0.24000 0.44000 0.64000 0.84000
Python
(Python3.x)
The multi-base sequence generator
def vdc(n, base=2):
vdc, denom = 0,1
while n:
denom *= base
n, remainder = divmod(n, base)
vdc += remainder / denom
return vdc
Sample output
Base 2 and then 3:
>>> [vdc(i) for i in range(10)]
[0, 0.5, 0.25, 0.75, 0.125, 0.625, 0.375, 0.875, 0.0625, 0.5625]
>>> [vdc(i, 3) for i in range(10)]
[0, 0.3333333333333333, 0.6666666666666666, 0.1111111111111111, 0.4444444444444444, 0.7777777777777777, 0.2222222222222222, 0.5555555555555556, 0.8888888888888888, 0.037037037037037035]
>>>
As fractions
We can get the output as rational numbers if we use the fraction module (and change its string representation to look like a fraction):
>>> from fractions import Fraction
>>> Fraction.__repr__ = lambda x: '%i/%i' % (x.numerator, x.denominator)
>>> [vdc(i, base=Fraction(2)) for i in range(10)]
[0, 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16]
Stretch goal
Sequences for different bases:
>>> for b in range(3,6):
print('\nBase', b)
print([vdc(i, base=Fraction(b)) for i in range(10)])
Base 3
[0, 1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27]
Base 4
[0, 1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8]
Base 5
[0, 1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25]
Quackery
[ $ "bigrat.qky" loadfile ] now!
[ [] swap
[ dup while
base share /mod
rot join swap
again ]
drop ] is digits ( n --> [ )
[ base put
digits reverse
dup 0 swap
witheach
[ base share rot * + ]
base take rot size **
reduce ] is corput ( n n --> n/d )
5 times
[ say "base "
i^ 2 + dup echo
say ": "
10 times
[ i^ over corput
vulgar$ echo$ sp sp ]
cr drop ]- Output:
base 2: 0/1 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 base 3: 0/1 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 base 4: 0/1 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 base 5: 0/1 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25 base 6: 0/1 1/6 1/3 1/2 2/3 5/6 1/36 7/36 13/36 19/36
Racket
Following the suggestion.
#lang racket
(define (van-der-Corput n base)
(if (zero? n)
0
(let-values ([(q r) (quotient/remainder n base)])
(/ (+ r (van-der-Corput q base))
base))))
By digits, extracted arithmetically.
#lang racket
(define (digit-length n base)
(if (< n base) 1 (add1 (digit-length (quotient n base) base))))
(define (digit n i base)
(remainder (quotient n (expt base i)) base))
(define (van-der-Corput n base)
(for/sum ([i (digit-length n base)]) (/ (digit n i base) (expt base (+ i 1)))))
Output.
(for ([base (in-range 2 (add1 5))])
(printf "Base ~a: " base)
(for ([n (in-range 0 10)])
(printf "~a " (van-der-Corput n base)))
(newline))
#| Base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16
Base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27
Base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8
Base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25 |#
Raku
(formerly Perl 6)
First a cheap implementation in base 2, using string operations.
constant VdC = map { :2("0." ~ .base(2).flip) }, ^Inf;
.say for VdC[^16];
Here is a more elaborate version using the polymod built-in integer method:
sub VdC($base = 2) {
map {
[+] $_ && .polymod($base xx *) Z/ [\*] $base xx *
}, ^Inf
}
.say for VdC[^10];
- Output:
0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625
Here is a fairly standard imperative version in which we mutate three variables in parallel:
sub vdc($num, $base = 2) {
my $n = $num;
my $vdc = 0;
my $denom = 1;
while $n {
$vdc += $n mod $base / ($denom *= $base);
$n div= $base;
}
$vdc;
}
for 2..5 -> $b {
say "Base $b";
say ( vdc($_,$b).Rat.nude.join('/') for ^10 ).join(', ');
say '';
}
- Output:
Base 2 0, 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16 Base 3 0, 1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27 Base 4 0, 1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8 Base 5 0, 1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25
Here is a functional version that produces the same output:
sub vdc($value, $base = 2) {
my @values = $value, { $_ div $base } ... 0;
my @denoms = $base, { $_ * $base } ... *;
[+] do for (flat @values Z @denoms) -> $v, $d {
$v mod $base / $d;
}
}
We first define two sequences, one finite, one infinite. When we zip those sequences together, the finite sequence terminates the loop (which, since a Raku loop returns all its values, is merely another way of writing a map). We then sum with [+], a reduction of the + operator. (We could have in-lined the sequences or used a traditional map operator, but this way seems more readable than the typical FP solution.) The do is necessary to introduce a statement where a term is expected, since Raku distinguishes "sentences" from "noun phrases" as a natural language might.
REXX
binary version
This REXX version only handles binary (base 2).
Virtually any integer (including negative) is allowed and is accurate (no rounding).
A range of integers (for output) is also supported.
/*REXX program converts an integer (or a range) ──► a Van der Corput number in base 2.*/
numeric digits 1000 /*handle almost anything the user wants*/
parse arg a b . /*obtain the optional arguments from CL*/
if a=='' then parse value 0 10 with a b /*Not specified? Then use the defaults*/
if b=='' then b= a /*assume a range for a single number.*/
do j=a to b /*traipse through the range of numbers.*/
_= VdC( abs(j) ) /*convert absolute value of an integer.*/
leading= substr('-', 2 + sign(j) ) /*if needed, elide the leading sign. */
say leading || _ /*show number, with leading minus sign?*/
end /*j*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
VdC: procedure; y= x2b( d2x( arg(1) ) ) + 0 /*convert to hexadecimal, then binary.*/
if y==0 then return 0 /*handle the special case of zero. */
return '.'reverse(y) /*heavy lifting is performed by REXX. */
{{out|output|text= when using the default input of: 0 10
0 .1 .01 .11 .001 .101 .011 .111 .0001 .1001 .0101
any radix up to 90
This version handles what the first version does, plus any radix up to (and including) base 90.
It can also support a list (enabled when the base is negative).
/*REXX pgm converts an integer (or a range) ──► a Van der Corput number, in base 2, or */
/*────────────────────────────── optionally, any other base up to and including base 90.*/
numeric digits 1000 /*handle almost anything the user wants*/
parse arg a b r . /*obtain optional arguments from the CL*/
if a=='' | a=="," then parse value 0 10 with a b /*Not specified? Then use the defaults*/
if b=='' | b=="," then b= a /* " " " " " " */
if r=='' | r=="," then r= 2 /* " " " " " " */
z= /*a placeholder for a list of numbers. */
do j=a to b /*traipse through the range of integers*/
_= VdC( abs(j), abs(r) ) /*convert the ABSolute value of integer*/
_= substr('-', 2 + sign(j) )_ /*if needed, keep the leading - sign.*/
if r>0 then say _ /*if positive base, then just show it. */
else z=z _ /* ··· else append (build) a list. */
end /*j*/
if z\=='' then say strip(z) /*if a list is wanted, then display it.*/
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
base: procedure; parse arg x, toB, inB /*get a number, toBase, and inBase. */
/*╔══════════════════════════════════════════════════════════════════════════════════╗
║ Input to this function: x (X is required and it must be an integer). ║
║ toBase the base to convert X to (default=10). ║
║ inBase the base X is expressed in (default=10). ║
║ ║
║ toBase & inBase have a limit of: 2 ──► 90 ║
╚══════════════════════════════════════════════════════════════════════════════════╝*/
@abc= 'abcdefghijklmnopqrstuvwxyz' /*the lowercase Latin alphabet letters.*/
@abcU= @abc; upper @abcU /*go whole hog & extend with uppercase.*/
@@@= 0123456789 || @abc || @abcU /*prefix them with the decimal digits. */
@@@= @@@'<>[]{}()?~!@#$%^&*_+-=|\/;:`' /*add some special characters as well, */
/*──those chars should all be viewable.*/
numeric digits 1000 /*what the hey, support bigun' numbers.*/
maxB= length(@@@) /*maximum base (radix) supported here. */
if toB=='' then toB= 10 /*if omitted, then assume default (10)*/
if inB=='' then inB= 10 /* " " " " " " */
#=0 /* [↓] convert base inB X ──► base 10*/
do j=1 for length(x) /*process each "numeral" in the string.*/
_= substr(x, j, 1) /*pick off a "digit" (numeral) from X.*/
v= pos(_, @@@) /*get the value of this "digit"/numeral*/
if v==0 | v>inB then call erd /*is it an illegal "digit" (numeral) ? */
#= # * inB + v - 1 /*construct new number, digit by digit.*/
end /*j*/
y= /* [↓] convert base 10 # ──► base toB.*/
do while #>=toB /*deconstruct the new number (#). */
y= substr(@@@, # // toB + 1, 1)y /* construct the output number, ··· */
#= # % toB /* ··· and also whittle down #. */
end /*while*/
return substr(@@@, # + 1, 1)y /*return a constructed "numeric" string*/
/*──────────────────────────────────────────────────────────────────────────────────────*/
erd: say 'the character ' v " isn't a legal numeral for base " inB'.'; exit 13
VdC: return '.'reverse( base( arg(1), arg(2) )) /*convert the #, reverse the #, append.*/
(A negative base indicates to show numbers as a list.)
- output when using the input of: 0 30 -2
.0 .1 .01 .11 .001 .101 .011 .111 .0001 .1001 .0101 .1101 .0011 .1011 .0111 .1111 .00001 .10001 .01001 .11001 .00101 .10101 .01101 .11101 .00011 .10011 .01011 .11011 .00111 .10111 .01111
- output when using the input of: 1 30 -3
.1 .2 .01 .11 .21 .02 .12 .22 .001 .101 .201 .011 .111 .211 .021 .121 .221 .002 .102 .202 .012 .112 .212 .022 .122 .222 .0001 .1001 .2001 .0101
- output when using the input of: 1 30 -4
.1 .2 .3 .01 .11 .21 .31 .02 .12 .22 .32 .03 .13 .23 .33 .001 .101 .201 .301 .011 .111 .211 .311 .021 .121 .221 .321 .031 .131 .231
- output when using the input of: 1 30 -5
.1 .2 .3 .4 .01 .11 .21 .31 .41 .02 .12 .22 .32 .42 .03 .13 .23 .33 .43 .04 .14 .24 .34 .44 .001 .101 .201 .301 .401 .011
- output when using the input of: 55582777 55582804 -80
.V[Is1 .W[Is1 .X[Is1 .Y[Is1 .Z[Is1 .<[Is1 .>[Is1 .[[Is1 .][Is1 .{[Is1 .}[Is1 .([Is1 .)[Is1 .?[Is1 .~[Is1 .![Is1 .@[Is1 .#[Is1 .$[Is1 .%[Is1 .^[Is1 .&[Is1 .*[Is1 .0]Is1 .1]Is1 .2]Is1 .3]Is1 .4]Is1
Ring
decimals(4)
for base = 2 to 5
see "base " + string(base) + " : "
for number = 0 to 9
see "" + corput(number, base) + " "
next
see nl
next
func corput n, b
vdc = 0
denom = 1
while n
denom *= b
rem = n % b
n = floor(n/b)
vdc += rem / denom
end
return vdcOutput:
base 2 : 0 0.5000 0.2500 0.7500 0.1250 0.6250 0.3750 0.8750 0.0625 0.5625 base 3 : 0 0.3333 0.6667 0.1111 0.4444 0.7778 0.2222 0.5556 0.8889 0.0370 base 4 : 0 0.2500 0.5000 0.7500 0.0625 0.3125 0.5625 0.8125 0.1250 0.3750 base 5 : 0 0.2000 0.4000 0.6000 0.8000 0.0400 0.2400 0.4400 0.6400 0.8400
RPL
| RPL code | Comment |
|---|---|
≪ → base
≪ 0 1 ROT
WHILE DUP REPEAT
SWAP base * SWAP
base / LAST MOD IP
3 PICK / 4 ROLL + ROT ROT
END DROP2
≫ ≫ 'VDC' STO
|
VDC ( n base -- vdc )
vdc, denom = 0,1
while n:
denom *= base
n, remainder = divmod(n, base)
vdc += remainder / denom
return vdc
|
- Input:
≪ {} 0 9 FOR j j 2 VDC + NEXT ≫ EVAL
≪ {} 0 9 FOR j j 3 VDC + NEXT ≫ EVAL
- Output:
2: { 0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 }
1: { 0 0.333333333333 0.666666666667 0.111111111111 0.444444444444 0.777777777778 0.222222222222 0.555555555556 0.888888888889 3.7037037037E-02 }
Ruby
The multi-base sequence generator
def vdc(n, base=2)
str = n.to_s(base).reverse
str.to_i(base).quo(base ** str.length)
end
(2..5).each do |base|
puts "Base #{base}: " + Array.new(10){|i| vdc(i,base)}.join(", ")
end
Sample output
Base 2: 0/1, 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16 Base 3: 0/1, 1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27 Base 4: 0/1, 1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8 Base 5: 0/1, 1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25
Rust
/// Van der Corput sequence for any base, based on C languange example from Wikipedia.
pub fn corput(nth: usize, base: usize) -> f64 {
let mut n = nth;
let mut q: f64 = 0.0;
let mut bk: f64 = 1.0 / (base as f64);
while n > 0_usize {
q += ((n % base) as f64)*bk;
n /= base;
bk /= base as f64;
}
q
}
fn main() {
for base in 2_usize..=5_usize {
print!("Base {}:", base);
for i in 1_usize..=10_usize {
let c = corput(i, base);
print!(" {:.6}", c)
}
println!("");
}
}
- Output:
Base 2: 0.500000 0.250000 0.750000 0.125000 0.625000 0.375000 0.875000 0.062500 0.562500 0.312500 Base 3: 0.333333 0.666667 0.111111 0.444444 0.777778 0.222222 0.555556 0.888889 0.037037 0.370370 Base 4: 0.250000 0.500000 0.750000 0.062500 0.312500 0.562500 0.812500 0.125000 0.375000 0.625000 Base 5: 0.200000 0.400000 0.600000 0.800000 0.040000 0.240000 0.440000 0.640000 0.840000 0.080000
Scala
object VanDerCorput extends App {
def compute(n: Int, base: Int = 2) =
Iterator.from(0).
scanLeft(1)((a, _) => a * base).
map(b => (n - 1) / b -> b).
takeWhile(_._1 != 0).
foldLeft(0d)((a, b) => a + (b._1 % base).toDouble / b._2 / base)
val n = scala.io.StdIn.readInt
val b = scala.io.StdIn.readInt
(1 to n).foreach(x => println(compute(x, b)))
}
- Output:
n: 30 base: 2 0.0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 0.3125 0.8125 0.1875 0.6875 0.4375 0.9375 0.03125 0.53125 0.28125 0.78125 0.15625 0.65625 0.40625 0.90625 0.09375 0.59375 0.34375 0.84375 0.21875 0.71875
Seed7
$ include "seed7_05.s7i";
include "float.s7i";
const func float: vdc (in var integer: number, in integer: base) is func
result
var float: vdc is 0.0;
local
var integer: denom is 1;
var integer: remainder is 0;
begin
while number <> 0 do
denom *:= base;
remainder := number rem base;
number := number div base;
vdc +:= flt(remainder) / flt(denom);
end while;
end func;
const proc: main is func
local
var integer: base is 0;
var integer: number is 0;
begin
for base range 2 to 5 do
writeln;
writeln("Base " <& base);
for number range 0 to 9 do
write(vdc(number, base) digits 6 <& " ");
end for;
writeln;
end for;
end func;- Output:
Base 2 0.000000 0.500000 0.250000 0.750000 0.125000 0.625000 0.375000 0.875000 0.062500 0.562500 Base 3 0.000000 0.333333 0.666667 0.111111 0.444444 0.777778 0.222222 0.555556 0.888889 0.037037 Base 4 0.000000 0.250000 0.500000 0.750000 0.062500 0.312500 0.562500 0.812500 0.125000 0.375000 Base 5 0.000000 0.200000 0.400000 0.600000 0.800000 0.040000 0.240000 0.440000 0.640000 0.840000
Sidef
func vdc(value, base=2) {
while (value[-1] > 0) {
value.append(value[-1] / base -> int)
}
var (x, sum) = (1, 0)
value.each { |i|
sum += ((i % base) / (x *= base))
}
return sum
}
for base in (2..5) {
var seq = 10.of {|i| vdc([i], base) }
"base %d: %s\n".printf(base, seq.map{|n| "%.4f" % n}.join(', '))
}
- Output:
base 2: 0.0000, 0.5000, 0.2500, 0.7500, 0.1250, 0.6250, 0.3750, 0.8750, 0.0625, 0.5625 base 3: 0.0000, 0.3333, 0.6667, 0.1111, 0.4444, 0.7778, 0.2222, 0.5556, 0.8889, 0.0370 base 4: 0.0000, 0.2500, 0.5000, 0.7500, 0.0625, 0.3125, 0.5625, 0.8125, 0.1250, 0.3750 base 5: 0.0000, 0.2000, 0.4000, 0.6000, 0.8000, 0.0400, 0.2400, 0.4400, 0.6400, 0.8400
Stata
Stata has builtin functions in Mata to compute Halton sequences, which are generalizations of the Van der Corput sequence. See halton in Stata help, and two articles in the Stata Journal: Scrambled Halton sequences in Mata by Stanislav Kolenikov and Generating Halton sequences using Mata by David M. Drukker and Richard Gates.
mata
// 5th term of Van der Corput sequence
halton(1,1,5)
.625
// the first 10 terms of Van der Corput sequence
halton(10,1)
1
+---------+
1 | .5 |
2 | .25 |
3 | .75 |
4 | .125 |
5 | .625 |
6 | .375 |
7 | .875 |
8 | .0625 |
9 | .5625 |
10 | .3125 |
+---------+
// the first 10 terms of Van der Corput sequence in base 3
ghalton(10,3,0)
1
+---------------+
1 | .3333333333 |
2 | .6666666667 |
3 | .1111111111 |
4 | .4444444444 |
5 | .7777777778 |
6 | .2222222222 |
7 | .5555555556 |
8 | .8888888889 |
9 | .037037037 |
10 | .3703703704 |
+---------------+
end
Reproduce the plot in the task description:
clear
mata
st_addobs(2500)
st_addvar("double","x")
st_addvar("double","y")
st_addvar("double","z")
k=1::2500
st_store(k,1,k)
st_store(k,2,0.5*runiform(2500,1))
st_store(k,3,0.5:+0.5*halton(2500,1))
end
twoway scatter y x, msize(tiny) color(blue) ///
|| scatter z x, msize(tiny) color(green) legend(off) xtitle("") ///
title(Distribution: Van der Corput (top) vs pseudorandom) ///
ylabel(, angle(0) format(%3.1f))
Swift
func vanDerCorput(n: Int, base: Int, num: inout Int, denom: inout Int) {
var n = n, p = 0, q = 1
while n != 0 {
p = p * base + (n % base)
q *= base
n /= base
}
num = p
denom = q
while p != 0 {
n = p
p = q % p
q = n
}
num /= q
denom /= q
}
var num = 0
var denom = 0
for base in 2...5 {
print("base \(base): 0 ", terminator: "")
for n in 1..<10 {
vanDerCorput(n: n, base: base, num: &num, denom: &denom)
print("\(num)/\(denom) ", terminator: "")
}
print()
}
- Output:
base 2: 0 1/2 1/4 3/4 1/8 5/8 3/8 7/8 1/16 9/16 base 3: 0 1/3 2/3 1/9 4/9 7/9 2/9 5/9 8/9 1/27 base 4: 0 1/4 1/2 3/4 1/16 5/16 9/16 13/16 1/8 3/8 base 5: 0 1/5 2/5 3/5 4/5 1/25 6/25 11/25 16/25 21/25
Tcl
The core of this is code to handle digit reversing. Note that this also tackles negative numbers (by preserving the sign independently).
proc digitReverse {n {base 2}} {
set n [expr {[set neg [expr {$n < 0}]] ? -$n : $n}]
set result 0.0
set bit [expr {1.0 / $base}]
for {} {$n > 0} {set n [expr {$n / $base}]} {
set result [expr {$result + $bit * ($n % $base)}]
set bit [expr {$bit / $base}]
}
return [expr {$neg ? -$result : $result}]
}
Note that the above procedure will produce terms of the Van der Corput sequence by default.
# Print the first 10 terms of the Van der Corput sequence
for {set i 1} {$i <= 10} {incr i} {
puts "vanDerCorput($i) = [digitReverse $i]"
}
# In other bases
foreach base {3 4 5} {
set seq {}
for {set i 1} {$i <= 10} {incr i} {
lappend seq [format %.5f [digitReverse $i $base]]
}
puts "${base}: [join $seq {, }]"
}
- Output:
vanDerCorput(1) = 0.5 vanDerCorput(2) = 0.25 vanDerCorput(3) = 0.75 vanDerCorput(4) = 0.125 vanDerCorput(5) = 0.625 vanDerCorput(6) = 0.375 vanDerCorput(7) = 0.875 vanDerCorput(8) = 0.0625 vanDerCorput(9) = 0.5625 vanDerCorput(10) = 0.3125 3: 0.33333, 0.66667, 0.11111, 0.44444, 0.77778, 0.22222, 0.55556, 0.88889, 0.03704, 0.37037 4: 0.25000, 0.50000, 0.75000, 0.06250, 0.31250, 0.56250, 0.81250, 0.12500, 0.37500, 0.62500 5: 0.20000, 0.40000, 0.60000, 0.80000, 0.04000, 0.24000, 0.44000, 0.64000, 0.84000, 0.08000
VBA
Base only.
Private Function vdc(ByVal n As Integer, BASE As Variant) As Variant
Dim res As String
Dim digit As Integer, g As Integer, denom As Integer
denom = 1
Do While n
denom = denom * BASE
digit = n Mod BASE
n = n \ BASE
res = res & CStr(digit) '+ "0"
Loop
vdc = IIf(Len(res) = 0, "0", "0." & res)
End Function
Public Sub show_vdc()
Dim v As Variant, j As Integer
For i = 2 To 5
Debug.Print "Base "; i; ": ";
For j = 0 To 9
v = vdc(j, i)
Debug.Print v; " ";
Next j
Debug.Print
Next i
End Sub
- Output:
Base 2 : 0 0.1 0.01 0.11 0.001 0.101 0.011 0.111 0.0001 0.1001 Base 3 : 0 0.1 0.2 0.01 0.11 0.21 0.02 0.12 0.22 0.001 Base 4 : 0 0.1 0.2 0.3 0.01 0.11 0.21 0.31 0.02 0.12 Base 5 : 0 0.1 0.2 0.3 0.4 0.01 0.11 0.21 0.31 0.41
VBScript
'http://rosettacode.org/wiki/Van_der_Corput_sequence
'Van der Corput Sequence fucntion call = VanVanDerCorput(number,base)
Base2 = "0" : Base3 = "0" : Base4 = "0" : Base5 = "0"
Base6 = "0" : Base7 = "0" : Base8 = "0" : Base9 = "0"
l = 1
h = 1
Do Until l = 9
'Set h to the value of l after each function call
'as it sets it to 0 - see lines 37 to 40.
Base2 = Base2 & ", " & VanDerCorput(h,2) : h = l
Base3 = Base3 & ", " & VanDerCorput(h,3) : h = l
Base4 = Base4 & ", " & VanDerCorput(h,4) : h = l
Base5 = Base5 & ", " & VanDerCorput(h,5) : h = l
Base6 = Base6 & ", " & VanDerCorput(h,6) : h = l
l = l + 1
Loop
WScript.Echo "Base 2: " & Base2
WScript.Echo "Base 3: " & Base3
WScript.Echo "Base 4: " & Base4
WScript.Echo "Base 5: " & Base5
WScript.Echo "Base 6: " & Base6
'Van der Corput Sequence
Function VanDerCorput(n,b)
k = RevString(Dec2BaseN(n,b))
For i = 1 To Len(k)
VanDerCorput = VanDerCorput + (CLng(Mid(k,i,1)) * b^-i)
Next
End Function
'Decimal to Base N Conversion
Function Dec2BaseN(q,c)
Dec2BaseN = ""
Do Until q = 0
Dec2BaseN = CStr(q Mod c) & Dec2BaseN
q = Int(q / c)
Loop
End Function
'Reverse String
Function RevString(s)
For j = Len(s) To 1 Step -1
RevString = RevString & Mid(s,j,1)
Next
End Function
- Output:
Base 2: 0, 0.5, 0.5, 0.25, 0.75, 0.125, 0.625, 0.375, 0.875 Base 3: 0, 0.333333333333333, 0.666666666666667, 0.111111111111111, 0.444444444444444, 0.777777777777778, 0.222222222222222, 0.555555555555556, 0.888888888888889 Base 4: 0, 0.25, 0.5, 0.75, 0.0625, 0.3125, 0.5625, 0.8125, 0.125 Base 5: 0, 0.2, 0.4, 0.6, 0.8, 0.04, 0.24, 0.44, 0.64 Base 6: 0, 0.166666666666667, 0.333333333333333, 0.5, 0.666666666666667, 0.833333333333333, 2.77777777777778E-02, 0.194444444444444, 0.361111111111111
Visual Basic .NET
Module Module1
Function ToBase(n As Integer, b As Integer) As String
Dim result = ""
If b < 2 Or b > 16 Then
Throw New ArgumentException("The base is out of range")
End If
Do
Dim remainder = n Mod b
result = "0123456789ABCDEF"(remainder) + result
n = n \ b
Loop While n > 0
Return result
End Function
Sub Main()
For b = 2 To 5
Console.WriteLine("Base = {0}", b)
For i = 0 To 12
Dim s = "." + ToBase(i, b)
Console.Write("{0,6} ", s)
Next
Console.WriteLine()
Console.WriteLine()
Next
End Sub
End Module
- Output:
Base = 2
.0 .1 .10 .11 .100 .101 .110 .111 .1000 .1001 .1010 .1011 .1100
Base = 3
.0 .1 .2 .10 .11 .12 .20 .21 .22 .100 .101 .102 .110
Base = 4
.0 .1 .2 .3 .10 .11 .12 .13 .20 .21 .22 .23 .30
Base = 5
.0 .1 .2 .3 .4 .10 .11 .12 .13 .14 .20 .21 .22
V (Vlang)
fn v2(nn u32) f64 {
mut n:=nn
mut r := f64(0)
mut p := .5
for n > 0 {
if n&1 == 1 {
r += p
}
p *= .5
n >>= 1
}
return r
}
fn new_v(base u32) fn(u32) f64 {
invb := 1 / f64(base)
return fn[base,invb](nn u32) f64 {
mut n:=nn
mut r := f64(0)
mut p := invb
for n > 0 {
r += p * f64(n%base)
p *= invb
n /= base
}
return r
}
}
fn main() {
println("Base 2:")
for i := u32(0); i < 10; i++ {
println('$i ${v2(i)}')
}
println("Base 3:")
v3 := new_v(3)
for i := u32(0); i < 10; i++ {
println('$i ${v3(i)}')
}
}- Output:
Base 2: 0 0 1 0.5 2 0.25 3 0.75 4 0.125 5 0.625 6 0.375 7 0.875 8 0.0625 9 0.5625 Base 3: 0 0 1 0.3333333333333333 2 0.6666666666666666 3 0.1111111111111111 4 0.4444444444444444 5 0.7777777777777777 6 0.2222222222222222 7 0.5555555555555556 8 0.8888888888888888 9 0.037037037037037035
Wren
var v2 = Fn.new { |n|
var p = 0.5
var r = 0
while (n > 0) {
if (n%2 == 1) r = r + p
p = p / 2
n = (n/2).floor
}
return r
}
var newV = Fn.new { |base|
var invb = 1 / base
return Fn.new { |n|
var p = invb
var r = 0
while (n > 0) {
r = r + p*(n%base)
p = p * invb
n = (n/base).floor
}
return r
}
}
System.print("Base 2:")
for (i in 0..9) System.print("%(i) -> %(v2.call(i))")
System.print("\nBase 3:")
var v3 = newV.call(3)
for (i in 0..9) System.print("%(i) -> %(v3.call(i))")
- Output:
Base 2: 0 -> 0 1 -> 0.5 2 -> 0.25 3 -> 0.75 4 -> 0.125 5 -> 0.625 6 -> 0.375 7 -> 0.875 8 -> 0.0625 9 -> 0.5625 Base 3: 0 -> 0 1 -> 0.33333333333333 2 -> 0.66666666666667 3 -> 0.11111111111111 4 -> 0.44444444444444 5 -> 0.77777777777778 6 -> 0.22222222222222 7 -> 0.55555555555556 8 -> 0.88888888888889 9 -> 0.037037037037037
XPL0
include c:\cxpl\codes; \intrinsic 'code' declarations
func real VdC(N); \Return Nth term of van der Corput sequence in base 2
int N;
real V, U;
[V:= 0.0; U:= 0.5;
repeat N:= N/2;
if rem(0) then V:= V+U;
U:= U/2.0;
until N=0;
return V;
];
int N;
for N:= 0 to 10-1 do
[IntOut(0, N); RlOut(0, VdC(N)); CrLf(0)]- Output:
0 0.00000 1 0.50000 2 0.25000 3 0.75000 4 0.12500 5 0.62500 6 0.37500 7 0.87500 8 0.06250 9 0.56250
zkl
fcn vdc(n,base=2){
vdc:=0.0; denom:=1;
while(n){ reg remainder;
denom *= base;
n, remainder = n.divr(base);
vdc += (remainder.toFloat() / denom);
}
vdc
}fcn vdc(n,base=2){
str:=n.toString(base).reverse();
str.toInt(base).toFloat()/(base.toFloat().pow(str.len()))
}- Output:
[0..10].apply(vdcR).println("base 2");
L(0,0.5,0.25,0.75,0.125,0.625,0.375,0.875,0.0625,0.5625,0.3125)base 2
[0..10].apply(vdc.fp1(3)).println("base 3");
L(0,0.333333,0.666667,0.111111,0.444444,0.777778,0.222222,0.555556,0.888889,0.037037,0.37037)base 3
Categories:
- Programming Tasks
- Solutions by Programming Task
- 11l
- 360 Assembly
- Action!
- Action! Tool Kit
- ActionScript
- Ada
- ALGOL 68
- Arturo
- AutoHotkey
- AWK
- BASIC
- BASIC256
- Yabasic
- BBC BASIC
- Bc
- BCPL
- C
- C sharp
- C++
- Clojure
- CLU
- Common Lisp
- Cowgol
- D
- Delphi
- SysUtils,StdCtrls
- EasyLang
- EDSAC order code
- Ela
- Elixir
- Erlang
- ERRE
- Euphoria
- F Sharp
- Factor
- Forth
- Fortran
- FreeBASIC
- Fōrmulæ
- Go
- Haskell
- Icon
- Unicon
- J
- Java
- Jq
- Julia
- Kotlin
- Lua
- Maple
- Mathematica
- Wolfram Language
- MATLAB
- Octave
- Maxima
- Modula-2
- Nim
- PARI/GP
- Pascal
- Perl
- Phix
- PicoLisp
- PL/I
- Prolog
- PureBasic
- Python
- Quackery
- Racket
- Raku
- REXX
- Ring
- RPL
- Ruby
- Rust
- Scala
- Seed7
- Sidef
- Stata
- Swift
- Tcl
- VBA
- VBScript
- Visual Basic .NET
- V (Vlang)
- Wren
- XPL0
- Zkl