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SSJ is a Java library for stochastic simulation, developed under the direction of Pierre L'Ecuyer,
in the Département d'Informatique et de Recherche Opérationnelle (DIRO), at the Université de Montréal.
It provides facilities for generating uniform and nonuniform random variates, computing different
measures related to probability distributions, performing goodness-of-fit tests, applying quasi-Monte
Carlo methods, collecting (elementary) statistics, and programming discrete-event simulations with both
events and processes.
The newest version!
/*
* Class: SobolSequence
* Description: Sobol sequences
* Environment: Java
* Software: SSJ
* Copyright (C) 2001 Pierre L'Ecuyer and Université de Montréal
* Organization: DIRO, Université de Montréal
* @author
* @since
* SSJ is free software: you can redistribute it and/or modify it under
* the terms of the GNU General Public License (GPL) as published by the
* Free Software Foundation, either version 3 of the License, or
* any later version.
* SSJ is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
* A copy of the GNU General Public License is available at
GPL licence site.
*/
package umontreal.iro.lecuyer.hups;
import umontreal.iro.lecuyer.util.PrintfFormat;
import java.io.*;
import java.net.MalformedURLException;
/**
* This class implements digital nets or digital sequences in base 2 formed by
* the first n = 2k points of a Sobol' sequence.
* Values of n up to 230 are allowed.
*
* In Sobol's proposal, the generator matrices
* Cj are upper triangular
* matrices defined by a set of direction numbers
*
*
*
* vj, c = mj, c2-c = ∑l=1cvj, c, l2-l,
*
* where each mj, c is an odd integer smaller than 2c,
* for
* c = 1,..., k and
* j = 0,..., s - 1.
* The digit vj, c, l is the element (l, c) of
* Cj,
* so vj, c represents column c of
* Cj.
* One can also write
*
*
*
* mj, c = ∑l=1cvj, c, l2c-l,
*
* so column c of
* Cj contains the c digits of the binary expansion
* of mj, c, from the most to the least significant,
* followed by w - c zeros, where w is the number of output digits.
* Since each mj, c is odd, the first k rows of each
* Cj form
* a non-singular upper triangular matrix whose diagonal elements
* are all ones.
*
*
* For each dimension j, the integers mj, c are defined by
* selecting a primitive polynomial over
*
* F2 of degree cj,
*
*
*
* fj(z) = zcj + aj, 1zcj-1 + ... + aj, cj,
*
* and the first cj integers
* mj, 0,..., mj, cj-1.
* Then the following integers
* mj, cj, mj, cj+1,... are
* determined by the recurrence
*
*
*
* mj, c = 2aj, 1mj, c-1⊕ ... ⊕2cj-1aj, cj-1mj, c-cj+1⊕2cjmj, c-cj⊕mj, c-cj
*
* for c >= cj, or equivalently,
*
*
*
* vj, c, l = aj, 1vj, c-1, l⊕ ... ⊕aj, cj-1vj, c-cj+1, l⊕vj, c-cj, l⊕vj, c-cj, l+cj
*
* for l >= 0, where ⊕ means bitwise exclusive or
* (i.e., bitwise addition modulo 2).
* Sobol' has shown that with this construction, if the
* primitive polynomials fj(z) are all distinct, one obtains
* a (t, s)-sequence whose t-value does not exceed
*
* c0 + ... + cs-1 + 1 - s.
* He then suggested to list the set of all primitive polynomials over
*
* F2 by increasing order of degree, starting with
* f0(z)≡1
* (whose corresponding matrix
* C0 is the identity), and take fj(z)
* as the (j + 1)th polynomial in the list, for j >= 0.
*
*
* This list of primitive polynomials, as well as default choices
* for the direction numbers, are stored in precomputed tables.
* The ordered list of primitive polynomials was taken from Florent Chabaud's web site,
* at http://fchabaud.free.fr/.
* Each polynomial fj(z) is stored in the form of the integer
*
* 2cj + aj, 12cj-1 + ... + aj, cj, whose binary
* representation gives the polynomial coefficients.
*
*
* For the set of direction numbers, there are several possibilities
* based on different selection criteria.
* The original values proposed by Sobol' and implemented in the code of
* Bratley and Fox for j <= 40
* were selected in terms of his properties A and A',
* which are equivalent to s-distribution with one and
* two bits of accuracy, respectively.
* The default direction numbers used here have been taken from
* Lemieux et al.
* For j <= 40, they are the same as in
* Bratley and Fox.
* Several files of parameters for Sobol sequences are given on F. Kuo's
* Web site at http://web.maths.unsw.edu.au/~fkuo/sobol/index.html.
*
*/
public class SobolSequence extends DigitalSequenceBase2 {
// Maximal dimension for primitive polynomials included in this file
protected static final int MAXDIM = 360;
protected static final int MAXDEGREE = 18; // Of primitive polynomial
private String filename = null;
/**
* Constructs a new digital net with n = 2k points and w
* output digits, in dimension dim, formed by taking the first
* n points of the Sobol' sequence.
* The predefined generator matrices
* Cj are w×k. Restrictions:
* 0 <= k <= 30, k <= w and dim <= 360.
*
* @param k there will be 2^k points
*
* @param w number of output digits
*
* @param dim dimension of the point set
*
*
*/
public SobolSequence (int k, int w, int dim) {
init (k, w, w, dim);
}
private void init (int k, int r, int w, int dim) {
if (filename == null)
if ((dim < 1) || (dim > MAXDIM))
throw new IllegalArgumentException
("Dimension for SobolSequence must be > 0 and <= " + MAXDIM);
else
if (dim < 1)
throw new IllegalArgumentException
("Dimension for SobolSequence must be > 0");
if (r < k || w < r || w > MAXBITS || k >= MAXBITS)
throw new IllegalArgumentException
("One must have k < 31 and k <= r <= w <= 31 for SobolSequence");
numCols = k;
numRows = r; // Not used!
outDigits = w;
numPoints = (1 << k);
this.dim = dim;
normFactor = 1.0 / ((double) (1L << (outDigits)));
genMat = new int[dim * numCols];
initGenMat();
}
/**
* Constructs a Sobol point set with at least n points and 31
* output digits, in dimension dim. Equivalent to
* SobolSequence (k, 31, dim) with
* k = ceil(log2n).
*
* @param dim dimension of the point set
*
* @param n minimal number of points
*
*
*/
public SobolSequence (int n, int dim) {
numCols = MAXBITS; // Defined in PointSet.
while (((n >> numCols) & 1) == 0)
numCols--;
if (1 << numCols != n)
numCols++;
init (numCols, MAXBITS, MAXBITS, dim);
}
/**
* Constructs a new digital net using the direction numbers provided in file
* filename. The net has n = 2k points, w output digits and
* dimension dim. The file can be either on the user's host, or
* somewhere on the Internet: in that case, the full url address must
* be given using either the http or ftp protocol. For
* example:
*
*
* net = new SobolSequence(
*
"http://web.maths.unsw.edu.au/˜fkuo/sobol/joe-kuo-6.16900", k, w, dim);
*
*
* The file must have the following format
* (the first line is treated as a comment by the program and discarded):
*
*
* dim
* s
* a
* mi
* 2
* 1
* 0
* 1
* 3
* 2
* 1
* 1 3
* 4
* 3
* 1
* 1 3 1
* 5
* 3
* 2
* 1 1 1
* 6
* 4
* 1
* 1 1 3 3
* 7
* 4
* 4
* 1 3 5 13
*
* ⋮
* ⋮
*
*
*
* dim is the
* dimension, s the degree of the polynomial, the binary representation of a
* gives the inner coefficients of the polynomial
* (the first and the last coefficients are always 1), and mi are the
* direction numbers. Thus if
* a = (a1a2…as-1)2 for a given s,
* then the polynomial is
* xs + a1xs-1 + a2xs-2 + ... + as-1x + 1.
* For example, if s = 4 and
* a = 4 = 1002, then the
* polynomial is
* x4 + x3 + 1.
*
* @param k number of points is 2k
*
* @param w number of output digits
*
* @param dim dimension of the point set
*
* @param filename file containing the direction numbers
*
*
*/
public SobolSequence (String filename, int k, int w, int dim) {
poly_from_file = new int[dim];
for (int i = 0; i < dim; i++)
poly_from_file[i] = 0;
minit_from_file = new int[dim][MAXDEGREE];
for (int i = 0; i < dim; i++) {
for (int j = 0; j < MAXDEGREE; j++) {
minit_from_file[i][j] = 0;
}
}
// Dimension d = 1
int d = 1;
poly_from_file[d - 1] = 1;
// Read the direction number file up to a certain number of dimension dim
try {
// If filename can be found starting from the program's directory,
// it will be used; otherwise, the filename in the Jar archive will
// be used.
int prev_d = 1;
BufferedReader reader;
if (filename.startsWith("http") || filename.startsWith("ftp"))
reader = DigitalNetFromFile.openURL(filename);
else
reader = new BufferedReader(new FileReader(filename));
// First line of file is a comment; discard it
String line = reader.readLine();
String[] tokens;
while ((line = reader.readLine()) != null) {
tokens = line.split("[\t ]+");
// Direction number lines from dimension 2 and up
if (tokens.length < 4) {
System.err.println("\nBad direction number file format!\n");
System.exit(1);
}
// Parse dim d, polynomial degree s and coefficients a
d = Integer.parseInt(tokens[0]);
int s = Integer.parseInt(tokens[1]);
int a = Integer.parseInt(tokens[2]);
if (s + 3 != tokens.length) {
System.err.println("\nBad direction number file format!\n");
System.exit(1);
}
if (d != prev_d + 1) {
System.err.println("Dimension in file shall be in ");
System.err.println("increasing order, one per line!");
System.exit(1);
}
prev_d = d;
// If d in the file exceeds dim, stop reading!
if (d > dim)
break;
poly_from_file[d - 1] = (1 << s) ^ (a << 1) ^ 1;
// Parse the s direction numbers
for (int i = 0; i < s; i++)
minit_from_file[d - 2][i] = Integer.parseInt(tokens[i + 3]);
} // end while
} catch (MalformedURLException e) {
System.err.println (" Invalid URL address: " + filename);
System.exit(1);
} catch (IOException e) {
System.err.println("Error: " + e);
System.exit(1);
}
if (dim > d) {
System.err.printf("\n\nNot enough dimension in file: %s", filename);
System.exit(1);
}
this.filename = filename;
init(k, w, w, dim);
}
public String toString() {
StringBuffer sb = new StringBuffer ("Sobol sequence:" +
PrintfFormat.NEWLINE);
sb.append (super.toString());
return sb.toString();
}
public void extendSequence (int k) {
int start, degree, nextCol;
int i, j, c;
int oldNumCols = numCols;
int[] oldGenMat = genMat; // Save old generating matrix.
numCols = k;
numPoints = (1 << k);
genMat = new int[dim * numCols];
// the first dimension, j = 0.
for (c = 0; c < numCols; c++)
genMat[c] = (1 << (outDigits-c-1));
// the other dimensions j > 0.
for (j = 1; j < dim; j++) {
// find the degree of primitive polynomial f_j
for (degree = MAXDEGREE; ((poly[j] >> degree) & 1) == 0; degree--)
;
// Get initial direction numbers m_{j,0},..., m_{j,degree-1}.
start = j * numCols;
for (c = 0; (c < degree && c < numCols); c++)
genMat[start+c] = minit[j-1][c] << (outDigits-c-1);
// Compute the following ones via the recursion.
for (c = degree; c < numCols; c++) {
if (c < oldNumCols)
genMat[start+c] = oldGenMat[j*oldNumCols + c];
else {
nextCol = genMat[start+c-degree] >> degree;
for (i = 0; i < degree; i++)
if (((poly[j] >> i) & 1) == 1)
nextCol ^= genMat[start+c-degree+i];
genMat[start+c] = nextCol;
}
}
}
}
// Initializes the generator matrices for a sequence.
private void initGenMat() {
int start, degree, nextCol;
int i, j, c;
// the first dimension, j = 0.
for (c = 0; c < numCols; c++)
genMat[c] = (1 << (outDigits-c-1));
// the other dimensions j > 0.
for (j = 1; j < dim; j++) {
// if a direction number file was provided, use it
int polynomial = (filename != null ? poly_from_file[j] : poly[j]);
// find the degree of primitive polynomial f_j
for (degree = MAXDEGREE; ((polynomial >> degree) & 1) == 0; degree--)
;
// Get initial direction numbers m_{j,0},..., m_{j,degree-1}.
start = j * numCols;
for (c = 0; (c < degree && c < numCols); c++) {
int m_i = (filename != null ?
minit_from_file[j-1][c] : minit[j-1][c]);
genMat[start+c] = m_i << (outDigits-c-1);
}
// Compute the following ones via the recursion.
for (c = degree; c < numCols; c++) {
nextCol = genMat[start+c-degree] >> degree;
for (i = 0; i < degree; i++)
if (((polynomial >> i) & 1) == 1)
nextCol ^= genMat[start+c-degree+i];
genMat[start+c] = nextCol;
}
}
}
/*
// Initializes the generator matrices for a net.
protected void initGenMatNet() {
int start, degree, nextCol;
int i, j, c;
// the first dimension, j = 0.
for (c = 0; c < numCols; c++)
genMat[c] = (1 << (outDigits-numCols+c));
// the second dimension, j = 1.
for (c = 0; c < numCols; c++)
genMat[numCols+c] = (1 << (outDigits-c-1));
// the other dimensions j > 1.
for (j = 2; j < dim; j++) {
// find the degree of primitive polynomial f_j
for (degree = MAXDEGREE; ((poly[j-1] >> degree) & 1) == 0; degree--);
// Get initial direction numbers m_{j,0},..., m_{j,degree-1}.
start = j * numCols;
for (c = 0; (c < degree && c < numCols); c++)
genMat[start+c] = minit[j-2][c] << (outDigits-c-1);
// Compute the following ones via the recursion.
for (c = degree; c < numCols; c++) {
nextCol = genMat[start+c-degree] >> degree;
for (i = 0; i < degree; i++)
if (((poly[j-1] >> i) & 1) == 1)
nextCol ^= genMat[start+c-degree+i];
genMat[start+c] = nextCol;
}
}
}
*/
// *******************************************************
// The ordered list of first MAXDIM primitive polynomials.
protected int[] poly_from_file;
protected static final int[] poly = {
1, 3, 7, 11, 13, 19, 25, 37, 59,
47, 61, 55, 41, 67, 97, 91, 109, 103,
115, 131, 193, 137, 145, 143, 241, 157, 185,
167, 229, 171, 213, 191, 253, 203, 211, 239,
247, 285, 369, 299, 425, 301, 361, 333, 357,
351, 501, 355, 397, 391, 451, 463, 487, 529,
545, 539, 865, 557, 721, 563, 817, 601, 617,
607, 1001, 623, 985, 631, 953, 637, 761, 647,
901, 661, 677, 675, 789, 687, 981, 695, 949,
701, 757, 719, 973, 731, 877, 787, 803, 799,
995, 827, 883, 847, 971, 859, 875, 895, 1019,
911, 967, 1033, 1153, 1051, 1729, 1063, 1825, 1069,
1441, 1125, 1329, 1135, 1969, 1163, 1673, 1221, 1305,
1239, 1881, 1255, 1849, 1267, 1657, 1279, 2041, 1293,
1413, 1315, 1573, 1341, 1509, 1347, 1557, 1367, 1877,
1387, 1717, 1423, 1933, 1431, 1869, 1479, 1821, 1527,
1917, 1531, 1789, 1555, 1603, 1591, 1891, 1615, 1939,
1627, 1747, 1663, 2035, 1759, 2011, 1815, 1863, 2053,
2561, 2071, 3713, 2091, 3393, 2093, 2881, 2119, 3617,
2147, 3169, 2149, 2657, 2161, 2273, 2171, 3553, 2189,
2833, 2197, 2705, 2207, 3985, 2217, 2385, 2225, 2257,
2255, 3889, 2279, 3697, 2283, 3441, 2293, 2801, 2317,
2825, 2323, 3209, 2341, 2633, 2345, 2377, 2363, 3529,
2365, 3017, 2373, 2601, 2395, 3497, 2419, 3305, 2421,
2793, 2431, 4073, 2435, 3097, 2447, 3865, 2475, 3417,
2477, 2905, 2489, 2521, 2503, 3641, 2533, 2681, 2551,
3833, 2567, 3589, 2579, 3205, 2581, 2693, 2669, 2917,
2687, 4069, 2717, 2965, 2727, 3669, 2731, 3413, 2739,
3285, 2741, 2773, 2783, 4021, 2799, 3957, 2811, 3573,
2819, 3085, 2867, 3277, 2879, 4045, 2891, 3373, 2911,
4013, 2927, 3949, 2941, 3053, 2951, 3613, 2955, 3357,
2963, 3229, 2991, 3933, 2999, 3805, 3005, 3037, 3035,
3517, 3047, 3709, 3083, 3331, 3103, 3971, 3159, 3747,
3179, 3427, 3187, 3299, 3223, 3731, 3227, 3475, 3251,
3283, 3263, 4051, 3271, 3635, 3319, 3827, 3343, 3851,
3367, 3659, 3399, 3627, 3439, 3947, 3487, 3995, 3515,
3547, 3543, 3771, 3559, 3707, 3623, 3655, 3679, 4007,
3743, 3991, 3791, 3895, 4179, 6465, 4201, 4801, 4219,
7105, 4221, 6081, 4249, 4897, 4305, 4449, 4331, 6881,
4359, 7185, 4383, 7953, 4387, 6289, 4411, 7057, 4431
};
// The direction numbers. For j > 0 and c < c_j,
// minit[j-1][c] contains the integer m_{j,c}.
// The values for j=0 are not stored, since C_0 is the identity matrix.
protected int minit_from_file[][];
protected static final int minit[][] = {
{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{1, 3, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{1, 1, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0},
{1, 3, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0},
{1, 1, 3, 7, 0, 0, 0, 0, 0, 0, 0, 0},
{1, 3, 3, 9, 9, 0, 0, 0, 0, 0, 0, 0},
{1, 3, 7, 13, 3, 0, 0, 0, 0, 0, 0, 0},
{1, 1, 5, 11, 27, 0, 0, 0, 0, 0, 0, 0},
{1, 3, 5, 1, 15, 0, 0, 0, 0, 0, 0, 0},
{1, 1, 7, 3, 29, 0, 0, 0, 0, 0, 0, 0},
{1, 3, 7, 7, 21, 0, 0, 0, 0, 0, 0, 0},
{1, 1, 1, 9, 23, 37, 0, 0, 0, 0, 0, 0},
{1, 3, 3, 5, 19, 33, 0, 0, 0, 0, 0, 0},
{1, 1, 3, 13, 11, 7, 0, 0, 0, 0, 0, 0},
{1, 1, 7, 13, 25, 5, 0, 0, 0, 0, 0, 0},
{1, 3, 5, 11, 7, 11, 0, 0, 0, 0, 0, 0},
{1, 1, 1, 3, 13, 39, 0, 0, 0, 0, 0, 0},
{1, 3, 1, 15, 17, 63, 13, 0, 0, 0, 0, 0},
{1, 1, 5, 5, 1, 27, 33, 0, 0, 0, 0, 0},
{1, 3, 3, 3, 25, 17, 115, 0, 0, 0, 0, 0},
{1, 1, 3, 15, 29, 15, 41, 0, 0, 0, 0, 0},
{1, 3, 1, 7, 3, 23, 79, 0, 0, 0, 0, 0},
{1, 3, 7, 9, 31, 29, 17, 0, 0, 0, 0, 0},
{1, 1, 5, 13, 11, 3, 29, 0, 0, 0, 0, 0},
{1, 3, 1, 9, 5, 21, 119, 0, 0, 0, 0, 0},
{1, 1, 3, 1, 23, 13, 75, 0, 0, 0, 0, 0},
{1, 3, 3, 11, 27, 31, 73, 0, 0, 0, 0, 0},
{1, 1, 7, 7, 19, 25, 105, 0, 0, 0, 0, 0},
{1, 3, 5, 5, 21, 9, 7, 0, 0, 0, 0, 0},
{1, 1, 1, 15, 5, 49, 59, 0, 0, 0, 0, 0},
{1, 1, 1, 1, 1, 33, 65, 0, 0, 0, 0, 0},
{1, 3, 5, 15, 17, 19, 21, 0, 0, 0, 0, 0},
{1, 1, 7, 11, 13, 29, 3, 0, 0, 0, 0, 0},
{1, 3, 7, 5, 7, 11, 113, 0, 0, 0, 0, 0},
{1, 1, 5, 3, 15, 19, 61, 0, 0, 0, 0, 0},
{1, 3, 1, 1, 9, 27, 89, 7, 0, 0, 0, 0},
{1, 1, 3, 7, 31, 15, 45, 23, 0, 0, 0, 0},
{1, 3, 3, 9, 9, 25, 107, 39, 0, 0, 0, 0},
{1, 1, 3, 13, 7, 35, 61, 91, 0, 0, 0, 0},
{1, 1, 7, 11, 5, 35, 55, 75, 0, 0, 0, 0},
{1, 3, 5, 5, 11, 23, 29, 139, 0, 0, 0, 0},
{1, 1, 1, 7, 11, 15, 17, 81, 0, 0, 0, 0},
{1, 1, 7, 9, 5, 57, 79, 103, 0, 0, 0, 0},
{1, 1, 7, 13, 19, 5, 5, 185, 0, 0, 0, 0},
{1, 3, 1, 3, 13, 57, 97, 131, 0, 0, 0, 0},
{1, 1, 5, 5, 21, 25, 125, 197, 0, 0, 0, 0},
{1, 3, 3, 9, 31, 11, 103, 201, 0, 0, 0, 0},
{1, 1, 5, 3, 7, 25, 51, 121, 0, 0, 0, 0},
{1, 3, 7, 15, 19, 53, 73, 189, 0, 0, 0, 0},
{1, 1, 1, 15, 19, 55, 27, 183, 0, 0, 0, 0},
{1, 1, 7, 13, 3, 29, 109, 69, 0, 0, 0, 0},
{1, 1, 5, 15, 15, 23, 15, 1, 57, 0, 0, 0},
{1, 3, 1, 3, 23, 55, 43, 143, 397, 0, 0, 0},
{1, 1, 3, 11, 29, 9, 35, 131, 411, 0, 0, 0},
{1, 3, 1, 7, 27, 39, 103, 199, 277, 0, 0, 0},
{1, 3, 7, 3, 19, 55, 127, 67, 449, 0, 0, 0},
{1, 3, 7, 3, 5, 29, 45, 85, 3, 0, 0, 0},
{1, 3, 5, 5, 13, 23, 75, 245, 453, 0, 0, 0},
{1, 3, 1, 15, 21, 47, 3, 77, 165, 0, 0, 0},
{1, 1, 7, 9, 15, 5, 117, 73, 473, 0, 0, 0},
{1, 3, 1, 9, 1, 21, 13, 173, 313, 0, 0, 0},
{1, 1, 7, 3, 11, 45, 63, 77, 49, 0, 0, 0},
{1, 1, 1, 1, 1, 25, 123, 39, 259, 0, 0, 0},
{1, 1, 1, 5, 23, 11, 59, 11, 203, 0, 0, 0},
{1, 3, 3, 15, 21, 1, 73, 71, 421, 0, 0, 0},
{1, 1, 5, 11, 15, 31, 115, 95, 217, 0, 0, 0},
{1, 1, 3, 3, 7, 53, 37, 43, 439, 0, 0, 0},
{1, 1, 1, 1, 27, 53, 69, 159, 321, 0, 0, 0},
{1, 1, 5, 15, 29, 17, 19, 43, 449, 0, 0, 0},
{1, 1, 3, 9, 1, 55, 121, 205, 255, 0, 0, 0},
{1, 1, 3, 11, 9, 47, 107, 11, 417, 0, 0, 0},
{1, 1, 1, 5, 17, 25, 21, 83, 95, 0, 0, 0},
{1, 3, 5, 13, 31, 25, 61, 157, 407, 0, 0, 0},
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}