What is the maximum number of random real number generated by Mersenne Twister?

We know that the period of Mersenne Twister is 2^19937 -1. I want to know what is the max random real number that can be generated in Matlab 32 bits and 64 bits?

Answer 1

[Walter Roberson]

The possible output values are (1 : 2^53 - 1) / 2^53 which is 2^53 - 2 distinct values. Most of the random number generators are the same (but a few are different.)

[Jay Liu]

How did you get this possible value? Is there any different between 32bits and 64 bits version? Thanks.

[Walter Roberson]

http://www.mathworks.com/help/matlab/math/creating-and-controlling-a-random-number-stream.html

"mt19937ar 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 is the same in all versions of MATLAB that support the twister.

[Roger Stafford]

I think there are actually more than that, Walter. For values between 1/2 and 1 there are 2^52 possible. For values between 1/4 and 1/2, there are another 2^52. For 1/8 to 1/4 yet another 2^52, and so forth down to very small (and therefore improbable) ranges each having 2^52 possibilities in it. This variation is made possible by the eleven bits in the exponential portion of double precision floating point numbers as well as the fifty-two bits of the significand.

[Walter Roberson]

"The possible values are multiples of 2^−53 in the interval (0,1)" is a quote: that is how it is implemented, integer divided 2^53 (or equivalent, such as by taking a fixed exponent and stuffing the mantisa with 52 bits.)

This contrasts with swb2712, for which

"The generator works natively in double precision to create U(0,1) values, and all values in the open interval (0,1) are possible"

swb2712 is the only one of the generators that has that property; the other ones all generate as-if an integer divided by another integer.

For mcg16807 the divisor is (2^31 - 1); for mlfg6331_64, mrg32k3a, and mt19937ar, the divisor is 2^53; for dsfmt19937 the divisor is 2^52; for shr3cong the divisor is 2^32 . swb2712 is the only algorithm that uses the full density.

By the way, I analyzed the median of swb2712 in http://www.mathworks.com/matlabcentral/answers/246999-probability-1-2-with-matlab-random-number-generators; it is on the order of 1E-154

[Jay Liu]

Thanks for all your comment. What if I need a double precision real number? I need a random number at least 9 decimals. Could I use mt19937ar as a random number generator? OR I should use the 64-bits Multiplicative Lagged Fibonacci random number generator?

[Walter Roberson]

Accuracy to 5E-10 requires only 31 bits, so use any of the generators.

[Jay Liu]

Thanks again. If my accuracy is > 5E-10, 32 bits generator is no longer available. Then, The only way in Matlab is to choose 64 bits generator such as Multiplicative Lagged Fibonacci. Is it correct?

[LeChat]

I would like to compare some results I have from a code in c that uses the Mesenne twister on 64 bits (mt19937-64.c) with the Matlab code that I wrote (the idea is to use the same rng seed and the same rng and check if the results are exactly the same, at least to numerical precision). Unfortunately the Mersenne RNG is on 32 bits on Matlab. What would be the syntax for a workaround? (should I put this as an independent question? I guess it pretty much the same as Jay's though...)