com.numericalmethod.suanshu.stats.random.pseudorandom.linear
Class Lehmer

java.lang.Object
  com.numericalmethod.suanshu.stats.random.pseudorandom.linear.Lehmer

All Implemented Interfaces:

LinearCongruentialGenerator, RandomLongGenerator, RandomNumberGenerator

public class Lehmer
extends java.lang.Object
implements LinearCongruentialGenerator

Lehmer proposed a general linear congruential generator that generates pseudo-random numbers in [0, 1]. It has this form:

 xi+1 = (a * xi + c) mod m
 ui+1 = xi+1 / m
 

We take c to be 0 because Marsaglia shows that there is little additional generality when c ≠ 0. However, there are restrictions placed on the selection of (a, m) and the seed. For example,

• the seed must be co-prime to m;
• the modulus m is a prime number or a power of a prime number;
• the multiplier a is an element of high multiplicative order modulo m

This class is essentially doing what Random.next(int) is doing (for a specific pair a and m), but it computes (ax mod m) in integer arithmetic without overflow under certain conditions. In addition, this allows customerized multiplier and modulus.

This class is the most fundamental building block for all linear random number generation algorithms in this library.

See Also:

• "Lehmer, D.H. (1951) Mathematical methods in large-scale computing units, p.141-146. Proceedings of the Second Symposium on Large Scale Digital Computing Machinery. Harvard University Press, Cambridge, Mass."
• Wikipedia: Lehmer random number generator

Field Summary

long	a the multiplier
long	m the modulus
long	q m / a
long	r m % a

Constructor Summary

Lehmer()
Construct a Lehmer (pure) linear congruential generator using default values.

Lehmer(long a, long m, long seed)
Construct a Lehmer (pure) linear congruential generator.

Lehmer(long a, long m, long k, long seed)
Construct a skipping ahead Lehmer (pure) linear congruential generator.

Method Summary

long	modulus() Get the modulus of this linear congruential generator.
double	nextDouble() Get the next random double between 0 and 1.
long	nextLong() All built-in linear random number generators in this library ultimately call this function to generate random numbers.
int	order() Get the order of recursion.
void	seed(long... seeds) Seed the random number generator to produce repeatable sequences.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

a

public final long a

the multiplier

m

public final long m

the modulus

q

public final long q

m / a

r

public final long r

m % a

Constructor Detail

Lehmer

public Lehmer(long a,
              long m,
              long seed)

Construct a Lehmer (pure) linear congruential generator.

Suggested values are:

• m = 2³¹ - 1 = 2147483647; a = 16807 (inferior to the other 3)
• m = 2³¹ - 1 = 2147483647; a = 39373
• m = 2147483399; a = 40692
• m = 2147483563; a = 40014

This implementation computes the next random number in long arithmetic without overflow. It is based on L'Ecuyer, P. (1988).

Note that a cannot be too big.

Parameters:

a - multiplier

m - modulus

seed - the seed. It should not be zero.

See Also:

• "Monte Carlo Methods in Financial Engineering. Paul Glasserman. 2004."
• "L'Ecuyer, P. (1988) Efficient and portable combined random number generators. Communications of the ACM 31:742-749, 774. Correspondence 32:1019-1024."

Lehmer

public Lehmer(long a,
              long m,
              long k,
              long seed)

Construct a skipping ahead Lehmer (pure) linear congruential generator. The pseudo-random sequence is a subset of the original Lehmer sequence, taking every k value. Equivalently, this call is the same as

Lehmer((a^k)%m, m, seed)

We compute (a^k)%m more efficiently.

Note that a cannot be too big.

Parameters:

a - multiplier

m - modulus

k - exponent

seed - the seed. It should not be zero.

Lehmer

public Lehmer()

Construct a Lehmer (pure) linear congruential generator using default values. a = 40014; m = 2147483563; seed = 8682522807148012L + System.nanoTime()

Method Detail

seed

public void seed(long... seeds)

Description copied from interface: RandomNumberGenerator

Seed the random number generator to produce repeatable sequences.

Specified by:

seed in interface RandomNumberGenerator

Parameters:

seeds - the seeds

order

public int order()

Description copied from interface: LinearCongruentialGenerator

Get the order of recursion.

Specified by:

order in interface LinearCongruentialGenerator

Returns:

the order of recursion

modulus

public long modulus()

Description copied from interface: LinearCongruentialGenerator

Get the modulus of this linear congruential generator.

Specified by:

modulus in interface LinearCongruentialGenerator

Returns:

m

nextLong

public long nextLong()

All built-in linear random number generators in this library ultimately call this function to generate random numbers. This particular function is thread safe using non-blocking synchronization. This in turn ensures thread-safety for all these rngs.

If you are to write your own rng, you should either call this function, or have your own synchronization mechanism.

Specified by:

nextLong in interface RandomLongGenerator

Returns:

a random long number

See Also:

"Java Concurrency in Practice. Brian Goetz, Tim Peierls, Joshua Bloch and Joseph Bowbeer."

nextDouble

public double nextDouble()

Get the next random double between 0 and 1.

Specified by:

nextDouble in interface RandomNumberGenerator

Returns:

the next random number between 0 and 1