Chapter 14. Random Numbers
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1 B B Chapter 14 Random Numbers
2 293 DEFINITION PboxRandom; CONST seedlimit = ; Figure 14.1 The interface of module PboxRandom. PROCEDURE Int (n: INTEGER): INTEGER; PROCEDURE Randomize; PROCEDURE Real (): REAL; PROCEDURE SetSeed (n: INTEGER); END PboxRandom.
3 PROCEDURE SetSeed (n: INTEGER); Pre 0 < n 20 n < seedlimit 21 Post The random number seed is initialized to n. 294 PROCEDURE Randomize Post The random number seed is initialized to a value derived from the system clock. Two calls to Randomize should be separated by more than one second to guarantee different values of seed. PROCEDURE Real (): REAL; Post Returns a random real between 0.0 and 1.0.
4 294 MODULE Pbox14A; IMPORT Dialog, PboxRandom, PboxStrings, StdLog; VAR d*: RECORD seed*: INTEGER; END; Figure 14.2 A procedure that prints ten random real numbers to the Log. PROCEDURE SetSeed*; IF (0 < d.seed) & (d.seed < PboxRandom.seedLimit) THEN PboxRandom.SetSeed(d.seed) ELSE StdLog.String("Seed must be greater than 0 and less than "); StdLog.Ln END END SetSeed; PROCEDURE Randomize*; PboxRandom.Randomize END Randomize;
5 PROCEDURE Display*; VAR i: INTEGER; x: REAL; realstring: ARRAY 8 OF CHAR; FOR i := 1 TO 10 DO StdLog.String("i = "); StdLog.Int(i); x := PboxRandom.Real(); PboxStrings.RealToString(x, 5, 3, realstring); StdLog.String(" END; StdLog.Ln END Display; d.seed := 1 END Pbox14A. x = "); StdLog.String(realString); StdLog.Ln 295
6 295 Figure 14.3 The output of procedure Display of Figure 14.2.
7 297 Figure 14.4 The output for the procedure of Figure 14.5.
8 297 MODULE Pbox14B; IMPORT Dialog, PboxRandom, PboxStrings, StdLog; VAR d*: RECORD seed*: INTEGER; limit*: INTEGER; END; Figure 14.5 A procedure that prints ten random integers to the Log. PROCEDURE SetSeed*; IF (0 < d.seed) & (d.seed < PboxRandom.seedLimit) THEN PboxRandom.SetSeed(d.seed) ELSE StdLog.String("Seed must be greater than 0 and less than "); StdLog.Ln END END SetSeed; PROCEDURE Randomize*; PboxRandom.Randomize END Randomize;
9 297 PROCEDURE Display*; VAR i: INTEGER; m: INTEGER; IF (0 < d.limit) & (d.limit < PboxRandom.seedLimit) THEN FOR i := 1 TO 10 DO m := PboxRandom.Int(d.limit); StdLog.Int(m) END; StdLog.Ln ELSE StdLog.String("Limit must be greater than 0 and less than "); StdLog.Ln END END Display; d.seed := 1; d.limit := 0 END Pbox14B.
10 PROCEDURE Int (n: INTEGER): INTEGER; Pre 0 < n 20 n < seedlimit 21 Post Returns a random integer in the range 0..n
11 REPEAT Statement1 UNTIL Condition1 298 S1 F C1 T Figure 14.6 The flowchart for the REPEAT statement.
12 299 Figure 14.7 The output for the procedure in Figure 14.8.
13 300 MODULE Pbox14C; IMPORT Dialog, PboxRandom, PboxStrings, StdLog; VAR d*: RECORD seed*: INTEGER; END; Figure 14.8 A procedure that simulates rolls of a pair of dice. PROCEDURE SetSeed*; IF (0 < d.seed) & (d.seed < PboxRandom.seedLimit) THEN PboxRandom.SetSeed(d.seed) ELSE StdLog.String("Seed must be greater than 0 and less than "); StdLog.Ln END END SetSeed; PROCEDURE Randomize*; PboxRandom.Randomize END Randomize;
14 PROCEDURE RollDice*; VAR die1, die2: INTEGER; sum, numrolls: INTEGER; numrolls := 0; StdLog.String("Rolls of the dice"); StdLog.Ln; REPEAT die1 := PboxRandom.Int(6) + 1; die2 := PboxRandom.Int(6) + 1; INC(numRolls); sum := die1 + die2; StdLog.Int(die1); StdLog.Int(die2); StdLog.Ln UNTIL (sum = 7) OR (sum = 11); StdLog.String("You rolled "); StdLog.Int(numRolls); StdLog.String(" times before you got 7 or 11."); StdLog.Ln END RollDice; d.seed := 1 END Pbox14C. 300
15 z n + 1 = az n MOD m 301 Example If you select values of (17, 5) for (m, a), and the current value of seed z n is 11, then the next seed is computed as z n + 1 = az n mod m = 5 11 mod 17 = 55 mod 17 = 4 The 20 successive seed values from the (17, 5) Lehmer generator are with an initial seed of 11.
16 Example The (17, 13) Lehmer generator produces the sequence starting from
17 302 MODULE PboxRandom; IMPORT Dates; CONST multiplier = 48271; modulus = ; quotient = modulus DIV multiplier; remainder = modulus MOD multiplier; seedlimit* = modulus; VAR seed: INTEGER; Figure 14.9 An implementation of the standard ( , 48271) Lehmer random number generator. PROCEDURE ComputeNextSeed; VAR low, high: INTEGER; low := seed MOD quotient; high := seed DIV quotient; seed := multiplier * low - remainder * high; IF seed <= 0 THEN seed := seed + modulus END END ComputeNextSeed;
18 PROCEDURE Int* (n: INTEGER): INTEGER; ASSERT(0 < n, 20); ASSERT(n < seedlimit, 21); ComputeNextSeed; RETURN SHORT(ENTIER(seed / modulus * n)) END Int; 302 PROCEDURE Randomize*; VAR date: Dates.Date; time: Dates.Time; i: INTEGER; Dates.GetDate(date); Dates.GetTime(time); seed := * Dates.Day(date) * time.hour + 60 * time.minute + time.second; (* Elapsed time this year in seconds *) FOR i := 0 TO 7 DO ComputeNextSeed END END Randomize;
19 PROCEDURE Real* (): REAL; ComputeNextSeed; RETURN seed / modulus END Real; PROCEDURE SetSeed* (n: INTEGER); VAR i: INTEGER; ASSERT(0 < n, 20); ASSERT(n < seedlimit, 21); seed := n MOD modulus; FOR i := 0 TO 7 DO ComputeNextSeed END END SetSeed; Randomize END PboxRandom. 302
20 Figure The transformation PboxRandom.Int makes from the real number line to the integer number line
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