System.Random.MWC

mwc-random-0.13.3.2: Fast, high quality pseudo random number generation

Copyright | (c) 2009-2012 Bryan O'Sullivan
License | BSD3
Maintainer | bos@serpentine.com
Stability | experimental
Portability | portable
Safe Haskell | None
Language | Haskell98

System.Random.MWC

Contents

- Gen: Pseudo-Random Number Generators
  - Type helpers
- Variates: uniformly distributed values
- Seed: state management
- References

Description

Pseudo-random number generation. This module contains code for generating high quality random numbers that follow a uniform distribution.

For non-uniform distributions, see the `Distributions` module.

The uniform PRNG uses Marsaglia's MWC256 (also known as MWC8222) multiply-with-carry generator, which has a period of 2^8222 and fares well in tests of randomness. It is also extremely fast, between 2 and 3 times faster than the Mersenne Twister.

The generator state is stored in the `Gen` data type. It can be created in several ways:

1. Using the `withSystemRandom` call, which creates a random state.
2. Supply your own seed to `initialize` function.
3. Finally, `create` makes a generator from a fixed seed. Generators created in this way aren't really random.

For repeatability, the state of the generator can be snapshotted and replayed using the `save` and `restore` functions.

The simplest use is to generate a vector of uniformly distributed values:

      vs <- withSystemRandom . asGenST $ \gen -> uniformVector gen 100

These values can be of any type which is an instance of the class `Variate`.

To generate random values on demand, first `create` a random number generator.

      gen <- create

Hold onto this generator and use it wherever random values are required (creating a new generator is expensive compared to generating a random number, so you don't want to throw them away). Get a random value using `uniform` or `uniformR`:

      v <- uniform gen

      v <- uniformR (1, 52) gen

Synopsis

- data Gen s
- create :: PrimMonad m => m (Gen (PrimState m))
- initialize :: (PrimMonad m, Vector v Word32) => v Word32 -> m (Gen (PrimState m))
- withSystemRandom :: PrimBase m => (Gen (PrimState m) -> m a) -> IO a
- createSystemRandom :: IO GenIO
- type GenIO = Gen (PrimState IO)
- type GenST s = Gen (PrimState (ST s))
- asGenIO :: (GenIO -> IO a) -> GenIO -> IO a
- asGenST :: (GenST s -> ST s a) -> GenST s -> ST s a
- class Variate a where
  - uniform :: PrimMonad m => Gen (PrimState m) -> m a
  - uniformR :: PrimMonad m => (a, a) -> Gen (PrimState m) -> m a
- uniformVector :: (PrimMonad m, Variate a, Vector v a) => Gen (PrimState m) -> Int -> m (v a)
- data Seed
- fromSeed :: Seed -> Vector Word32
- toSeed :: Vector v Word32 => v Word32 -> Seed
- save :: PrimMonad m => Gen (PrimState m) -> m Seed
- restore :: PrimMonad m => Seed -> m (Gen (PrimState m))

# Gen: Pseudo-Random Number Generators

data Gen s

State of the pseudo-random number generator.

create :: PrimMonad m => m (Gen (PrimState m))

Create a generator for variates using a fixed seed.

initialize :: (PrimMonad m, Vector v Word32) => v Word32 -> m (Gen (PrimState m))

Create a generator for variates using the given seed, of which up to 256 elements will be used. For arrays of less than 256 elements, part of the default seed will be used to finish initializing the generator's state.

Examples:

    initialize (singleton 42)

    initialize (fromList [4, 8, 15, 16, 23, 42])

If a seed contains fewer than 256 elements, it is first used verbatim, then its elements are `xor`ed against elements of the default seed until 256 elements are reached.

If a seed contains exactly 258 elements, then the last two elements are used to set the generator's initial state. This allows for complete generator reproducibility, so that e.g. `gen' == gen` in the following example:

    gen' <- initialize . fromSeed =<< save

withSystemRandom :: PrimBase m => (Gen (PrimState m) -> m a) -> IO a

Seed a PRNG with data from the system's fast source of pseudo-random numbers ("`/dev/urandom`" on Unix-like systems), then run the given action.

This is a somewhat expensive function, and is intended to be called only occasionally (e.g. once per thread). You should use the `Gen` it creates to generate many random numbers.

Note: on Windows, this code does not yet use the native Cryptographic API as a source of random numbers (it uses the system clock instead). As a result, the sequences it generates may not be highly independent.

createSystemRandom :: IO GenIO

Seed a PRNG with data from the system's fast source of pseudo-random numbers. All the caveats of `withSystemRandom` apply here as well.

## Type helpers

The functions in this package are deliberately written for flexibility, and will run in both the `IO` and `ST` monads.

This can defeat the compiler's ability to infer a principal type in simple (and common) cases. For instance, we would like the following to work cleanly:

    import System.Random.MWC
    import Data.Vector.Unboxed

    main = do
      v <- withSystemRandom $ \gen -> uniformVector gen 20
      print (v :: Vector Int)

Unfortunately, the compiler cannot tell what monad `uniformVector` should execute in. The "fix" of adding explicit type annotations is not pretty:

    {-# LANGUAGE ScopedTypeVariables #-}

    import Control.Monad.ST

    main = do
      vs <- withSystemRandom $
            \(gen::GenST s) -> uniformVector gen 20 :: ST s (Vector Int)
      print vs

As a more readable alternative, this library provides `asGenST` and `asGenIO` to constrain the types appropriately. We can get rid of the explicit type annotations as follows:

    main = do
      vs <- withSystemRandom . asGenST $ \gen -> uniformVector gen 20
      print (vs :: Vector Int)

This is almost as compact as the original code that the compiler rejected.

type GenIO = Gen (PrimState IO)

A shorter name for PRNG state in the `IO` monad.

type GenST s = Gen (PrimState (ST s))

A shorter name for PRNG state in the `ST` monad.

asGenIO :: (GenIO -> IO a) -> GenIO -> IO a

Constrain the type of an action to run in the `IO` monad.

asGenST :: (GenST s -> ST s a) -> GenST s -> ST s a

Constrain the type of an action to run in the `ST` monad.

# Variates: uniformly distributed values

class Variate a where

The class of types for which we can generate uniformly distributed random variates.

The uniform PRNG uses Marsaglia's MWC256 (also known as MWC8222) multiply-with-carry generator, which has a period of 2^8222 and fares well in tests of randomness. It is also extremely fast, between 2 and 3 times faster than the Mersenne Twister.

Note: Marsaglia's PRNG is not known to be cryptographically secure, so you should not use it for cryptographic operations.

Methods

uniform :: PrimMonad m => Gen (PrimState m) -> m a

Generate a single uniformly distributed random variate. The range of values produced varies by type:

- For fixed-width integral types, the type's entire range is used.
- For floating point numbers, the range (0,1] is used. Zero is explicitly excluded, to allow variates to be used in statistical calculations that require non-zero values (e.g. uses of the `log` function).

To generate a `Float` variate with a range of [0,1), subtract 2**(-33). To do the same with `Double` variates, subtract 2**(-53).

uniformR :: PrimMonad m => (a, a) -> Gen (PrimState m) -> m a

Generate single uniformly distributed random variable in a given range.

- For integral types inclusive range is used.
- For floating point numbers range (a,b] is used if one ignores rounding errors.

Instances

Variate Bool |
Variate Double |
Variate Float |
Variate Int |
Variate Int8 |
Variate Int16 |
Variate Int32 |
Variate Int64 |
Variate Word |
Variate Word8 |
Variate Word16 |
Variate Word32 |
Variate Word64 |
(Variate a, Variate b) => Variate (a, b) |
(Variate a, Variate b, Variate c) => Variate (a, b, c) |
(Variate a, Variate b, Variate c, Variate d) => Variate (a, b, c, d) |

uniformVector :: (PrimMonad m, Variate a, Vector v a) => Gen (PrimState m) -> Int -> m (v a)

Generate a vector of pseudo-random variates. This is not necessarily faster than invoking `uniform` repeatedly in a loop, but it may be more convenient to use in some situations.

# Seed: state management

data Seed

An immutable snapshot of the state of a `Gen`.

Instances

Eq Seed |
Show Seed |

fromSeed :: Seed -> Vector Word32

Convert seed into vector.

toSeed :: Vector v Word32 => v Word32 -> Seed

Convert vector to `Seed`. It acts similarily to `initialize` and will accept any vector. If you want to pass seed immediately to restore you better call initialize directly since following law holds:

    restore (toSeed v) = initialize v

save :: PrimMonad m => Gen (PrimState m) -> m Seed

Save the state of a `Gen`, for later use by `restore`.

restore :: PrimMonad m => Seed -> m (Gen (PrimState m))

Create a new `Gen` that mirrors the state of a saved `Seed`.

# References

- Marsaglia, G. (2003) Seeds for random number generators. Communications of the ACM 46(5):90–93. http://doi.acm.org/10.1145/769800.769827
- Doornik, J.A. (2007) Conversion of high-period random numbers to floating point. ACM Transactions on Modeling and Computer Simulation 17(1). http://www.doornik.com/research/randomdouble.pdf