Generate the van der Corput sequence by reflecting the digits of successive integers about the radix point. Provide a generator that works for arbitrary bases.
Description
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
-
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
- 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
#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;
}
{{out}}
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}}
#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";
}
}
{{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
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="https://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();
}
}
}
}
{{out}}
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
(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}}
| :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
(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}}
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
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}}
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
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..]
{{out}}
[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}}
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}}
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" ).
{{out}}
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
{{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
Euphoria
{{trans|D}}
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}}
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#}}==
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
{{out}}
[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}}
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
{{out}}
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 ;
{{out}}
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
{{out}}
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
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}}
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.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:
$ ./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].)
$ ./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:
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 Perl 6 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
{{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.
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}}
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.
# 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) ] )"
{{out}}
$ 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
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}}
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}}
// 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()
}
}
{{out}}
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
Mathematica
VanDerCorput[n_,base_:2]:=Table[
FromDigits[{Reverse[IntegerDigits[k,base]],0},base],
{k,n}]
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}}==
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}}
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"
""" """ """ """ """ """ """ """
*/
=={{header|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.
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))
{{out}}
[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
{{trans|Perl6}}
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 $base: ", join ' ', map { vdc($_, $base) } 0 .. 10;
print "\n";
}
Perl 6
{{Works with|rakudo|2016.08}} First a cheap implementation in base 2, using string operations.
constant VdC = map { :2("0." ~ .base(2).flip) }, ^Inf;
.say for VdC[^16];
Here is a more elaborate version using the polymod built-in integer method:
sub VdC($base = 2) {
map {
[+] $_ && .polymod($base xx *) Z/ [\*] $base xx *
}, ^Inf
}
.say for VdC[^10];
{{out}}
0
0.5
0.25
0.75
0.125
0.625
0.375
0.875
0.0625
0.5625
Here is a fairly standard imperative version in which we mutate three variables in parallel:
sub vdc($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 $b";
say (vdc($_,$b) for ^10).perl;
say '';
}
{{out}}
Base 2
(0, 1/2, 1/4, 3/4, 1/8, 5/8, 3/8, 7/8, 1/16, 9/16)
Base 3
(0, 1/3, 2/3, 1/9, 4/9, 7/9, 2/9, 5/9, 8/9, 1/27)
Base 4
(0, 1/4, 1/2, 3/4, 1/16, 5/16, 9/16, 13/16, 1/8, 3/8)
Base 5
(0, 1/5, 2/5, 3/5, 4/5, 1/25, 6/25, 11/25, 16/25, 21/25)
Here is a functional version that produces the same output:
sub vdc($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 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.
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}}
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)) ) )
{{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/I
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
% 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):
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
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}}
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]
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 |#
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. */
else return '.'reverse(y) /*heavy lifting is performed by REXX. */
'''output''' 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 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 }}
.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 }}
.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 }}
.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 }}
.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 }}
.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 vdc
Output:
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
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
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}}
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}}
$ 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}}
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}}
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}}
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}}
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}}
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 '''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))
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 {, }]"
}
{{out}}
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.
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}}
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
'https://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}}
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}}
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}}
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
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}}
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}}
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}}
fcn vdc(n,base=2){
str:=n.toString(base).reverse();
str.toInt(base).toFloat()/(base.toFloat().pow(str.len()))
}
{{out}}
[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