SSJ
V. 2.6.

umontreal.iro.lecuyer.hups
Class RadicalInverse

java.lang.Object
  extended by umontreal.iro.lecuyer.hups.RadicalInverse

public class RadicalInverse
extends Object

This class implements basic methods for working with radical inverses of integers in an arbitrary basis b. These methods are used in classes that implement point sets and sequences based on the van der Corput sequence (the Hammersley nets and the Halton sequence, for example).

We recall that for a k-digit integer i whose digital b-ary expansion is

i = a0 + a1b + ... + ak-1bk-1,

the radical inverse in base b is

ψb(i) = a0b-1 + a1b-2 + ... + ak-1b-k.

The van der Corput sequence in base b is the sequence ψb(0), ψb(1), ψb(2),...

Note that ψb(i) cannot always be represented exactly as a floating-point number on the computer (e.g., if b is not a power of two). For an exact representation, one can use the integer

bkψb(i) = ak-1 + ... + a1bk-2 + a0bk-1,

which we called the integer radical inverse representation. This representation is simply a mirror image of the digits of the usual b-ary representation of i.

It is common practice to permute locally the values of the van der Corput sequence. One way of doing this is to apply a permutation to the digits of i before computing ψb(i). That is, for a permutation π of the digits {0,..., b - 1},

ψb(i) = ∑r=0k-1arb-r-1

is replaced by

r=0k-1π(ar)b-r-1.

Applying such a permutation only changes the order in which the values of ψb(i) are enumerated. For every integer k, the first bk values that are enumerated remain the same (they are the values of ψb(i) for i = 0,..., bk - 1), but they are enumerated in a different order. Often, different permutations π will be applied for different coordinates of a point set.

The permutation π can be deterministic or random. One (deterministic) possibility implemented here is the Faure permutation σb of {0,..., b - 1} defined as follows. For b = 2, take σ = I, the identical permutation. For even b = 2c > 2, take

σ[i] = 2τ[i]        i = 0, 1,…, c - 1  
σ[i + c] = 2τ[i] + 1        i = 0, 1,…, c - 1  

where τ[i] is the Faure permutation for base c. For odd b = 2c + 1, take
σ[c] = c  
σ[i] = τ[i],         if 0 <= τ[i] < c  
σ[i] = τ[i] + 1,         if c <= τ[i] < 2c  

for 0 <= i < c, and take
σ[i] = τ[i - 1],         if 0 <= τ[i - 1] < c  
σ[i] = τ[i - 1] + 1,         if c <= τ[i - 1] < 2c  

for c < i <= 2c, and where τ[i] is the Faure permutation for base c. The Faure permutations give very small discrepancies (amongst the best known today) for small bases.


Constructor Summary
RadicalInverse(int b, double x0)
          Initializes the base of this object to b and its first value of x to x0.
 
Method Summary
static void getFaureLemieuxPermutation(int coordinate, int[] pi)
          Computes the permutations as proposed in σb of the set {0,…, b - 1} and puts it in array pi.
static void getFaurePermutation(int b, int[] pi)
          Computes the Faure permutation σb of the set {0,…, b - 1} and puts it in array pi.
static int[] getPrimes(int n)
          Provides an elementary method for obtaining the first n prime numbers larger than 1.
static int integerRadicalInverse(int b, int i)
          Computes the integer radical inverse of i in base b, equal to bkψb(i) if i has k b-ary digits.
 double nextRadicalInverse()
          A fast method that incrementally computes the radical inverse xi+1 in base b from xi = ψb(i), using addition with rigthward carry as described in Wang and Hickernell.
static double nextRadicalInverse(double invb, double x)
          A fast method that incrementally computes the radical inverse xi+1 in base b from xi = x = ψb(i), using addition with rigthward carry.
static int nextRadicalInverseDigits(int b, int k, int[] idigits)
          Given the k digits of the integer radical inverse of i in bdigits, in base b, this method replaces them by the digits of the integer radical inverse of i + 1 and returns their number.
static double permutedRadicalInverse(int b, int[] pi, long i)
          Computes the radical inverse of i in base b, where the digits are permuted using the permutation π.
static double radicalInverse(int b, long i)
          Computes the radical inverse of i in base b.
static int radicalInverseInteger(int b, double x)
          Computes the radical inverse of x in base b.
static long radicalInverseLong(int b, double x)
          Computes the radical inverse of x in base b.
static void reverseDigits(int k, int[] bdigits, int[] idigits)
          Given the k b-ary digits of i in bdigits, returns the k digits of the integer radical inverse of i in idigits.
 
Methods inherited from class java.lang.Object
equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

RadicalInverse

public RadicalInverse(int b,
                      double x0)
Initializes the base of this object to b and its first value of x to x0.

Parameters:
b - Base
x0 - Initial value of x
Method Detail

getPrimes

public static int[] getPrimes(int n)
Provides an elementary method for obtaining the first n prime numbers larger than 1. Creates and returns an array that contains these numbers. This is useful for determining the prime bases for the different coordinates of the Halton sequence and Hammersley nets.

Parameters:
n - number of prime numbers to return
Returns:
an array with the first n prime numbers

radicalInverse

public static double radicalInverse(int b,
                                    long i)
Computes the radical inverse of i in base b. If i = ∑r=0k-1arbr, the method computes and returns

x = ∑r=0k-1arb-r-1.

Parameters:
b - base used for the operation
i - the value for which the radical inverse will be computed
Returns:
the radical inverse of i in base b

radicalInverseInteger

public static int radicalInverseInteger(int b,
                                        double x)
Computes the radical inverse of x in base b. If x has more decimals in base b than logb(Long.MAX_VALUE), it is truncated to its minimum precision in base b. If x = ∑r=0k-1arb-r-1, the method computes and returns

i = ∑r=0k-1arbr.

Parameters:
b - base used for the operation
x - the value for which the radical inverse will be computed
Returns:
the radical inverse of x in base b

radicalInverseLong

public static long radicalInverseLong(int b,
                                      double x)
Computes the radical inverse of x in base b. If x has more decimals in base b than logb(Long.MAX_VALUE), it is truncated to its minimum precision in base b. If x = ∑r=0k-1arb-r-1, the method computes and returns

i = ∑r=0k-1arbr.

Parameters:
b - base used for the operation
x - the value for which the radical inverse will be computed
Returns:
the radical inverse of x in base b

nextRadicalInverse

public static double nextRadicalInverse(double invb,
                                        double x)
A fast method that incrementally computes the radical inverse xi+1 in base b from xi = x = ψb(i), using addition with rigthward carry. The parameter invb is equal to 1/b. Using long incremental streams (i.e., calling this method several times in a row) cause increasing inaccuracy in x. Thus the user should recompute the radical inverse directly by calling radicalInverse every once in a while (i.e. in every few thousand calls).

Parameters:
invb - 1/b where b is the base
x - the inverse xi
Returns:
the radical inverse xi+1

nextRadicalInverse

public double nextRadicalInverse()
A fast method that incrementally computes the radical inverse xi+1 in base b from xi = ψb(i), using addition with rigthward carry as described in Wang and Hickernell. Since using long incremental streams (i.e., calling this method several times in a row) cause increasing inaccuracy in x, the method recomputes the radical inverse directly from i by calling radicalInverse once in every 1000 calls.

Returns:
the radical inverse xi+1

reverseDigits

public static void reverseDigits(int k,
                                 int[] bdigits,
                                 int[] idigits)
Given the k b-ary digits of i in bdigits, returns the k digits of the integer radical inverse of i in idigits. This simply reverses the order of the digits.

Parameters:
k - number of digits in arrays
bdigits - digits in original order
idigits - digits in reverse order

integerRadicalInverse

public static int integerRadicalInverse(int b,
                                        int i)
Computes the integer radical inverse of i in base b, equal to bkψb(i) if i has k b-ary digits.

Parameters:
b - base used for the operation
i - the value for which the integer radical inverse will be computed
Returns:
the integer radical inverse of i in base b

nextRadicalInverseDigits

public static int nextRadicalInverseDigits(int b,
                                           int k,
                                           int[] idigits)
Given the k digits of the integer radical inverse of i in bdigits, in base b, this method replaces them by the digits of the integer radical inverse of i + 1 and returns their number. The array must be large enough to hold this new number of digits.

Parameters:
b - base
k - initial number of digits in arrays
idigits - digits of integer radical inverse
Returns:
new number of digits in arrays

getFaureLemieuxPermutation

public static void getFaureLemieuxPermutation(int coordinate,
                                              int[] pi)
Computes the permutations as proposed in σb of the set {0,…, b - 1} and puts it in array pi.

Parameters:
coordinate - the coordinate
pi - an array of size at least b, to be filled with the permutation

getFaurePermutation

public static void getFaurePermutation(int b,
                                       int[] pi)
Computes the Faure permutation σb of the set {0,…, b - 1} and puts it in array pi. See the description in the introduction above.

Parameters:
b - the base
pi - an array of size at least b, to be filled with the permutation

permutedRadicalInverse

public static double permutedRadicalInverse(int b,
                                            int[] pi,
                                            long i)
Computes the radical inverse of i in base b, where the digits are permuted using the permutation π. If i = ∑r=0k-1arbr, the method will compute and return

x = ∑r=0k-1π[ar]b-r-1.

Parameters:
b - base b used for the operation
pi - an array of length at least b containing the permutation used during the computation
i - the value for which the radical inverse will be computed
Returns:
the radical inverse of i in base b

SSJ
V. 2.6.

To submit a bug or ask questions, send an e-mail to Pierre L'Ecuyer.