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of code Pseudo-random numbers: a line at a time mostly Nelson H. F. Beebe Research Professor University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA

  1. of code � �� � Pseudo-random numbers: a line at a time � �� � mostly Nelson H. F. Beebe Research Professor University of Utah Department of Mathematics, 110 LCB 155 S 1400 E RM 233 Salt Lake City, UT 84112-0090 USA Email: beebe@math.utah.edu , beebe@acm.org , beebe@computer.org (Internet) WWW URL: http://www.math.utah.edu/~beebe Telephone: +1 801 581 5254 FAX: +1 801 581 4148 11 October 2005 Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 1 / 82

  2. What are random numbers good for? ❏ Decision making (e.g., coin flip). Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 2 / 82

  3. What are random numbers good for? ❏ Decision making (e.g., coin flip). ❏ Generation of numerical test data. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 2 / 82

  4. What are random numbers good for? ❏ Decision making (e.g., coin flip). ❏ Generation of numerical test data. ❏ Generation of unique cryptographic keys. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 2 / 82

  5. What are random numbers good for? ❏ Decision making (e.g., coin flip). ❏ Generation of numerical test data. ❏ Generation of unique cryptographic keys. ❏ Search and optimization via random walks. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 2 / 82

  6. What are random numbers good for? ❏ Decision making (e.g., coin flip). ❏ Generation of numerical test data. ❏ Generation of unique cryptographic keys. ❏ Search and optimization via random walks. ❏ Selection: quicksort (C. A. R. Hoare, ACM Algorithm 64: Quicksort , Comm. ACM. 4 (7), 321, July 1961) was the first widely-used divide-and-conquer algorithm to reduce an O ( N 2 ) problem to (on average) O ( N lg ( N )) . Cf. Fast Fourier Transform (Gauss (1866) (Latin), Runge (1906), Danielson and Lanczos (crystallography) (1942), Cooley and Tukey (1965)). Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 2 / 82

  7. Historical note: al-Khwarizmi Abu ’Abd Allah Muhammad ibn Musa al-Khwarizmi (ca. 780–850) is the father of algorithm and of algebra , from his book Hisab Al-Jabr wal Mugabalah (Book of Calculations, Restoration and Reduction) . He is celebrated in a 1200-year anniversary Soviet Union stamp: Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 3 / 82

  8. What are random numbers good for? . . . ❏ Simulation. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 4 / 82

  9. What are random numbers good for? . . . ❏ Simulation. ❏ Sampling: unbiased selection of random data in statistical computations (opinion polls, experimental measurements, voting, Monte Carlo integration, . . . ). The latter is done like this ( x k is random in ( a , b ) ): � � � b √ N ( b − a ) ∑ a f ( x ) dx ≈ f ( x k ) + O ( 1 / N ) N k = 1 Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 4 / 82

  10. Monte Carlo integration Here is an example of a simple, smooth, and exactly integrable function, and the relative error of its Monte Carlo integration: f(x) = 1/sqrt(x 2 + c 2 ) [c = 5] Convergence of Monte Carlo integration Convergence of Monte Carlo integration 0.200 0 0 -1 -1 0.150 -2 -2 log(RelErr) log(RelErr) -3 -3 f(x) 0.100 -4 -4 -5 -5 0.050 -6 -6 0.000 -7 -7 -10 -8 -6 -4 -2 0 2 4 6 8 10 0 20 40 60 80 100 0 1 2 3 4 5 N N log(N) Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 5 / 82

  11. When is a sequence of numbers random? ❏ Computer numbers are rational, with limited precision and range. Irrational and transcendental numbers are not represented. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 6 / 82

  12. When is a sequence of numbers random? ❏ Computer numbers are rational, with limited precision and range. Irrational and transcendental numbers are not represented. ❏ Truly random integers would have occasional repetitions, but most pseudo-random number generators produce a long sequence, called the period , of distinct integers: these cannot be random. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 6 / 82

  13. When is a sequence of numbers random? ❏ Computer numbers are rational, with limited precision and range. Irrational and transcendental numbers are not represented. ❏ Truly random integers would have occasional repetitions, but most pseudo-random number generators produce a long sequence, called the period , of distinct integers: these cannot be random. ❏ It isn’t enough to conform to an expected distribution: the order that values appear in must be haphazard. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 6 / 82

  14. When is a sequence of numbers random? ❏ Computer numbers are rational, with limited precision and range. Irrational and transcendental numbers are not represented. ❏ Truly random integers would have occasional repetitions, but most pseudo-random number generators produce a long sequence, called the period , of distinct integers: these cannot be random. ❏ It isn’t enough to conform to an expected distribution: the order that values appear in must be haphazard. ❏ Mathematical characterization of randomness is possible, but difficult. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 6 / 82

  15. When is a sequence of numbers random? ❏ Computer numbers are rational, with limited precision and range. Irrational and transcendental numbers are not represented. ❏ Truly random integers would have occasional repetitions, but most pseudo-random number generators produce a long sequence, called the period , of distinct integers: these cannot be random. ❏ It isn’t enough to conform to an expected distribution: the order that values appear in must be haphazard. ❏ Mathematical characterization of randomness is possible, but difficult. ❏ The best that we can usually do is compute statistical measures of closeness to particular expected distributions. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 6 / 82

  16. Distributions of pseudo-random numbers ❏ Uniform (most common). Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 7 / 82

  17. Distributions of pseudo-random numbers ❏ Uniform (most common). ❏ Exponential. Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 7 / 82

  18. Distributions of pseudo-random numbers ❏ Uniform (most common). ❏ Exponential. ❏ Normal (bell-shaped curve). Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 7 / 82

  19. Distributions of pseudo-random numbers ❏ Uniform (most common). ❏ Exponential. ❏ Normal (bell-shaped curve). ❏ Logarithmic: if ran () is uniformly-distributed in ( a , b ) , define randl ( x ) = exp ( x ran ()) . Then a randl ( ln ( b / a )) is logarithmically distributed in ( a , b ) . [Important use: sampling in floating-point number intervals.] Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 7 / 82

  20. Distributions of pseudo-random numbers . . . Sample logarithmic distribution: % hoc a = 1 b = 1000000 for (k = 1; k <= 10; ++k) printf "%16.8f\n", a*randl(ln(b/a)) 664.28612484 199327.86997895 562773.43156449 91652.89169494 34.18748767 472.74816777 12.34092778 2.03900107 44426.83813202 28.79498121 Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 9 / 82

  21. Uniform distribution Here are three ways to visualize a pseudo-random number distribution, using the Dyadkin-Hamilton generator function rn01() , which produces results uniformly distributed on ( 0 , 1 ] : Uniform Distribution Uniform Distribution Uniform Distribution Histogram 1.0 1.0 150 0.8 0.8 100 rn01() 0.6 rn01() 0.6 count 0.4 0.4 50 0.2 0.2 0.0 0.0 0 0 2500 5000 7500 10000 0 2500 5000 7500 10000 0.0 0.2 0.4 0.6 0.8 1.0 output n sorted n x Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 11 / 82

  22. Exponential distribution Here are visualizations of computations with the Dyadkin-Hamilton generator rnexp() , which produces results exponentially distributed on [ 0 , ∞ ) : Exponential Distribution Exponential Distribution Exponential Distribution Histogram 10 10 1000 8 8 800 rnexp() rnexp() 6 6 600 count 4 4 400 2 2 200 0 0 0 0 2500 5000 7500 10000 0 2500 5000 7500 10000 0 1 2 3 4 5 6 output n sorted n x Even though the theoretical range is [ 0 , ∞ ) , the results are practically always modest: the probability of a result as big as 50 is smaller than 2 × 10 − 22 . At one result per microsecond, it could take 164 million years of computing to encounter such a value! Nelson H. F. Beebe (University of Utah) Pseudo-random numbers 11 October 2005 13 / 82

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