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{{task}} 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 .
[[File:Van der corput distribution.png|400|thumb|right|Distribution of 2500 points each: Van der Corput (top) vs pseudorandom]] 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 [[wp:Monte Carlo method|Monte Carlo simulations]]. This sequence is also a superset of the numbers representable by the "fraction" field of [[wp:IEEE 754-1985|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:
- [http://www.puc-rio.br/marco.ind/quasi_mc.html#low_discrep The Basic Low Discrepancy Sequences]
- [[Non-decimal radices/Convert]]
- [[wp:Van der Corput sequence|Van der Corput sequence]]
360 Assembly
{{trans|BBC BASIC}} 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
{{out}}
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
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;
}
}
}
{{out}}
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;
{{out}}
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
AutoHotkey
{{works with|AutoHotkey_L}}
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
}
{{out}}
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
BBC BASIC
{{works with|BBC BASIC for Windows}}
@% = &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
{{out}}
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 [[Literals/Integer#bc|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).
{{out}}
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 ``` ## C ```c #includevoid 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; } ``` {{out}} ```txt 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++ {{trans|Perl 6}} ```cpp #include #include 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"; } } ``` {{out}} ```txt 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 ``` ## 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. ```c# using System; using System.Collections.Generic; using System.Linq; using System.Text; namespace VanDerCorput { ////// 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 /// ///public class VanDerCorputSequence: IEnumerable > { /// /// Number base for the sequence, which must bwe two or more. /// public int Base { get; private set; } ////// Maximum number of terms to be returned by iterator. /// public long Count { get; private set; } ////// Construct a sequence for the given base. /// /// Number base for the sequence. /// Maximum number of items to be returned by the iterator. 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; } ////// Compute nth term in the Van der Corput sequence for the base specified in the constructor. /// /// The position in the sequence, which may be zero or any positive number. /// This number is always an integral power of the base. ///The Van der Corput sequence value expressed as a Tuple containing a numerator and a denominator. public TupleCompute(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 (numerator, denominator); } /// /// Compute nth term in the Van der Corput sequence for the given base. /// /// Base to use for the sequence. /// The position in the sequence, which may be zero or any positive number. ///The Van der Corput sequence value expressed as a Tuple containing a numerator and a denominator. public static TupleCompute(int iBase, long n) { var seq = new VanDerCorputSequence(iBase); return seq.Compute(n); } /// /// Iterate over the Van Der Corput sequence. /// The first value in the sequence is always zero, regardless of the base. /// ///A tuple whose items are the Van der Corput value given as a numerator and denominator. public IEnumerator> 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(); } } } } ``` {{out}} ```txt 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... ``` ## Clojure ```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)])))) ``` {{out}} ```txt | :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 | ``` ## Common Lisp ```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)))) ``` {{out}} ```txt 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 ``` ## D ```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))); } ``` {{out}} ```txt 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] ``` ## Ela ```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): ```ela take 10 <| map' (vdc 2.0) [1..] ``` {{out}} ```txt [0.5,0.25,0.75,0.125,0.625,0.375,0.875,0.0625,0.5625,0.3125] ``` ## Elixir {{works with|Elixir|1.1}} ```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) ``` {{out}} ```txt 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. ```Erlang -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" ). ``` {{out}} ```txt 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 ```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 ``` {{out}} ```txt 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 {{trans|D}} ```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 ``` {{out}} ```txt 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 ``` =={{header|F_Sharp|F#}}== ```fsharp 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 [ ] let main argv = printfn "%A" [ for n in 0 .. 9 -> (vdc n 2) ] printfn "%A" [ for n in 0 .. 9 -> (vdc n 5) ] 0 ``` {{out}} ```txt [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 {{works with|Factor|0.98}} ```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 ] [ nip '[ 1 + neg _ swap ^ * ] ] 2bi map-index sum ; : vdc-demo ( -- ) 2 5 [a,b] [ dup "Base %d: " printf 10 [ swap vdc "%-5u " printf ] with each nl ] each ; MAIN: vdc-demo ``` {{out}} ```txt 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 ```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 ; ``` {{out}} ```txt 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. ```Fortran 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. ```txt 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 ```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 ``` {{out}} ```txt 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 ``` ## Go ```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)) } } ``` {{out}} ```txt 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.) ```haskell import Data.List import Data.Ratio import System.Environment import Text.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) -- Print the base followed by a sequence of van der Corput numbers. printVdc :: (Integer,[Rat]) -> IO () printVdc (b,ns) = putStrLn $ printf "Base %d:" b ++ concatMap (printf " %5s" . show) ns -- To print the n'th van der Corput numbers for n in [2,3,4,5] call the program -- with no arguments. Otherwise, passing the base b, first n, next n and -- maximum n will print the base b numbers for n in [firstN, nextN, ..., maxN]. main :: IO () main = do args <- getArgs let (bases, nums) = case args of [b, f, s, m] -> ([read b], [read f, read s..read m]) _ -> ([2,3,4,5], [0..9]) mapM_ printVdc [(b,rs) | b <- bases, let rs = map (vdc b) nums] ``` {{out}} for small bases: ```txt $ ./vandercorput 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 ``` {{out}} for a larger base. (Base 123 for n ∈ [50, 100, …, 300].) ```txt $ ./vandercorput 123 50 100 300 Base 123: 50/123 100/123 3322/15129 9472/15129 494/15129 6644/15129 ``` =={{header|Icon}} and {{header|Unicon}}== The following solution works in both Icon and Unicon: ```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: ```txt ->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 Perl 6 solution is: ```Unicon 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:''' ```j vdc=: ([ %~ %@[ #. #.inv)"0 _ ``` '''Examples:''' ```j 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 {{trans|Perl 6}} 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 Perl 6 (doubling and assignment first). It was kept the Perl 6 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. ```java 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)); } } } ``` {{out}} ```txt 0.0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 0.3125 ``` ## jq {{ works with|jq|1.4}} 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. ```jq # 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:''' ```jq 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) ] )" ``` {{out}} ```sh $ 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 ```julia vandercorput(num::Integer, base::Integer) = sum(d * Float64(base) ^ -ex for (ex, d) in enumerate(digits(num, 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 ``` {{out}} ```txt 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 {{trans|C}} ```scala // 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) { 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() } } ``` {{out}} ```txt 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 ```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 ``` ## Mathematica ```Mathematica VanDerCorput[n_,base_:2]:=Table[ FromDigits[{Reverse[IntegerDigits[k,base]],0},base], {k,n}] ``` ```txt VanDerCorput[10,2] ->{1/2,1/4,3/4,1/8,5/8,3/8,7/8,1/16,9/16,5/16} VanDerCorput[10,3] ->{1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27, 10/27} VanDerCorput[10,4] ->{1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8, 5/8} VanDerCorput[10,5] ->{1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25, 2/25} ``` =={{header|MATLAB}} / {{header|Octave}}== ```Matlab 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; ``` {{out}} ```txt 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 ```Maxima /* 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] */ ``` ```Maxima /* 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 ```Maxima 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 ```Maxima /* 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" """ """ """ """ """ """ """ """ */ ``` =={{header|Modula-2}}== ```modula2 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. ``` ## PARI/GP ```parigp 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)) ``` {{out}} ```txt [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 ```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: ```txt 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 {{trans|Perl6}} ```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 $base: ", join ' ', map { vdc($_, $base) } 0 .. 10; print "\n"; } ``` ## Perl 6 {{Works with|rakudo|2016.08}} First a cheap implementation in base 2, using string operations. ```perl6 constant VdC = map { :2("0." ~ .base(2).flip) }, ^Inf; .say for VdC[^16]; ``` Here is a more elaborate version using the polymod built-in integer method: ```perl6 sub VdC($base = 2) { map { [+] $_ && .polymod($base xx *) Z/ [\*] $base xx * }, ^Inf } .say for VdC[^10]; ``` {{out}} ```txt 0 0.5 0.25 0.75 0.125 0.625 0.375 0.875 0.0625 0.5625 ``` Here is a fairly standard imperative version in which we mutate three variables in parallel: ```perl6 sub vdc($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 $b"; say (vdc($_,$b) for ^10).perl; say ''; } ``` {{out}} ```txt Base 2 (0, 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16) Base 3 (0, 1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27) Base 4 (0, 1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8) Base 5 (0, 1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25) ``` Here is a functional version that produces the same output: ```perl6 sub vdc($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 Perl 6 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 Perl 6 distinguishes "sentences" from "noun phrases" as a natural language might. ## 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. ```Phix 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 ") ``` {{out}} ```txt 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 ```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)) ) ) ``` {{out}} 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/Ivdcb: 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; ``` {{out}} ```txt 0.0000000000000000000000000000000 0.1000000000000000000000000000000 0.0100000000000000000000000000000 0.1100000000000000000000000000000 0.0010000000000000000000000000000 0.1010000000000000000000000000000 0.0110000000000000000000000000000 0.1110000000000000000000000000000 0.0001000000000000000000000000000 0.1001000000000000000000000000000 0.0101000000000000000000000000000 ``` ## Prolog ```prolog % 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 ). ``` {{out}} (result of test): ```txt 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 . ``` ## PureBasic ```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() ``` {{out}} ```txt 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 ```python 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: ```python>>> [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): ```python>>> 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: ```python>>> 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] ``` ## Racket Following the suggestion. ```racket #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. ```racket #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. ```racket (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 |# ``` ## 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 /*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. */ else return '.'reverse(y) /*heavy lifting is performed by REXX. */ ``` '''output''' when using the default input of: 0 10 ```txt 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 /*REXX program 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.) {{out|output|text= when using the input of: 0 30 -2 }} ```txt .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 ``` {{out|output|text= when using the input of: 1 30 -3 }} ```txt .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 ``` {{out|output|text= when using the input of: 1 30 -4 }} ```txt .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 ``` {{out|output|text= when using the input of: 1 30 -5 }} ```txt .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 ``` {{out|output|text= when using the input of: 55582777 55582804 -80 }} ```txt .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 ```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 vdc ``` Output: ```txt 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 ``` ## Ruby The multi-base sequence generator ```ruby 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''' ```txt 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 ``` ## Scala ```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))) } ``` {{out}} ```txt 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 {{trans|D}} ```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; ``` {{out}} ```txt 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 {{trans|Perl}} ```ruby 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(', ')) } ``` {{out}} ```txt 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 ``` ## Swift {{trans|C}} ```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() } ``` {{out}} ```txt 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 ``` ## Stata Stata has builtin functions in Mata to compute '''[https://en.wikipedia.org/wiki/Halton_sequence Halton sequences]''', which are generalizations of the Van der Corput sequence. See '''[https://www.stata.com/help.cgi?mf_halton halton]''' in Stata help, and two articles in the Stata Journal: '''[http://www.stata-journal.com/article.html?article=st0244 Scrambled Halton sequences in Mata]''' by Stanislav Kolenikov and '''[http://www.stata-journal.com/article.html?article=st0103 Generating Halton sequences using Mata]''' by David M. Drukker and Richard Gates. ```stata 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: ```stata 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)) ``` ## Tcl The core of this is code to handle digit reversing. Note that this also tackles negative numbers (by preserving the sign independently). ```tcl 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 : $result}] } ``` Note that the above procedure will produce terms of the Van der Corput sequence by default. ```tcl # Print the first 10 terms of the Van der Corput sequence for {set i 1} {$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 {, }]" } ``` {{out}} ```txt 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 {{trans|Phix}}Base only. ```vb 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 ``` {{out}} ```txt 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 ```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 ``` {{out}} ```txt 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 {{trans|C}} ```vbnet 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 ``` {{out}} ```txt 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 ``` ## XPL0 ```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)] ``` {{out}} ```txt 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 {{trans|Python}} ```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 } ``` {{trans|Ruby}} ```zkl fcn vdc(n,base=2){ str:=n.toString(base).reverse(); str.toInt(base).toFloat()/(base.toFloat().pow(str.len())) } ``` {{out}} ```txt [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 ```