The Mersenne Twister, as described in [11], has period $$2^{19937}-1$$ and each U(0,1) value is created using two 32-bit integers. The possible values are multiples of $$2^{-53}$$ in the interval (0, 1). This generator does not support multiple streams or substreams. The randn algorithm used by default for mt19937ar streams is the ziggurat algorithm [7], but with the mt19937ar generator underneath.

The double precision SIMD-oriented Fast Mersenne Twister, as described in [12], is a faster implementation of the Mersenne Twister algorithm. The period is $$2^{19937}-1$$ and the possible values are multiples of $$2^{-52}$$ in the interval (0, 1). The generator produces double precision values in [1, 2) natively, which are transformed to create U(0,1) values. This generator does not support multiple streams or substreams.

A 32-bit multiplicative congruential generator, as described in [14], with multiplier $$a=7^5$$, modulo $$m=2^{31}-1$$. This generator has a period of $$2^{31}-2$$ and does not support multiple streams or substreams. Each U(0,1) value is created using a single 32-bit integer from the generator; the possible values are all multiples of $${(2^{31}-1)}^{-1}$$ strictly within the interval (0, 1). For mcg16807 streams, the default algorithm used by randn is the polar algorithm (described in [1]).

A 64-bit multiplicative lagged Fibonacci generator, as described in [10], with lags $$l=63$$, $$k=31$$. This generator is similar to the MLFG implemented in the SPRNG library. It has a period of approximately $$2^{124}$$. It supports up to $$2^{61}$$ parallel streams, via parameterization, and $$2^{51}$$ substreams each of length $$2^{72}$$. Each U(0,1) value is created using one 64-bit integer from the generator; the possible values are all multiples of $$2^{-64}$$ strictly within the interval (0, 1). The randn algorithm used by default for mlfg6331_64 streams is the ziggurat algorithm [7], but with the mlfg6331_64 generator underneath.

A 32-bit combined multiple recursive generator, as described in [2]. This generator is similar to the CMRG implemented in the RngStreams package in C. It has a period of $$2^{191}$$ and supports up to $$2^{63}$$ parallel streams through sequence splitting, each of length $$2^{127}$$. It also supports $$2^{51}$$ substreams, each of length $$2^{76}$$. Each U(0,1) value is created using two 32-bit integers from the generator; the possible values are multiples of $$2^{-53}$$ strictly within the interval (0, 1). The randn algorithm used by default for mrg32k3a streams is the ziggurat algorithm [7], but with the mrg32k3a generator underneath.

A 4x32 generator with 10 rounds as described in [15]. This generator uses a Feistel network and integer multiplication. The generator is specifically designed for high performance in highly parallel systems such as GPUs. It has a period of 2¹⁹³ (2⁶⁴ streams of length 2¹²⁹).

A 4x64 generator with 20 rounds as described in [15]. This generator is a non-cryptographic adaptation of the Threefish block cipher from the Skein Hash Function. It has a period of 2⁵¹⁴ (2²⁵⁶ streams of length 2²⁵⁸).

Marsaglia's SHR3 shift-register generator summed with a linear congruential generator with multiplier $$a=69069$$, addend $$b=1234567$$, and modulus $$2^{-32}$$. SHR3 is a 3-shift-register generator defined as $$u=u(I+L^{13})(I+R^{17})(I+L^5)$$, where $$I$$ is the identity operator, $$L$$ is the left shift operator, and R is the right shift operator. The combined generator (the SHR3 part is described in [7]) has a period of approximately $$2^{64}$$. This generator does not support multiple streams or substreams. Each U(0,1) value is created using one 32-bit integer from the generator; the possible values are all multiples of $$2^{-32}$$ strictly within the interval (0, 1). The randn algorithm used by default for shr3cong streams is the earlier form of the ziggurat algorithm [9], but with the shr3cong generator underneath. This generator is identical to the one used by the randn function beginning in MATLAB Version 5, activated using randn('state',s).

A modified Subtract-with-Borrow generator, as described in [8]. This generator is similar to an additive lagged Fibonacci generator with lags 27 and 12, but it is modified to have a much longer period of approximately $$2^{1492}$$. The generator works natively in double precision to create U(0,1) values, and all values in the open interval (0, 1) are possible. The randn algorithm used by default for swb2712 streams is the ziggurat algorithm [7], but with the swb2712 generator underneath.

Computes a normal random variate by applying the standard normal inverse cumulative distribution function to a uniform random variate. Exactly one uniform value is consumed per normal value.

The polar rejection algorithm, as described in [1]. Approximately 1.27 uniform values are consumed per normal value, on average.

The ziggurat algorithm, as described in [7]. Approximately 2.02 uniform values are consumed per normal value, on average.