By R. Sedgewick

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**Extra info for Algorithms**

**Sample text**

For example, to evaluate p(x) = x4 + 3x3 - 6x2 + 2x + 1 for any given x, one could compute x4, then compute and add 3x3, etc. This method requires recomputation of the powers of x; an alternate method, which requires extra storage, would save the powers of x as they are computed. A simple method which avoids recomputation and uses no extra space is known as Homer’s rule: by alternat:ing the multiplication and addition operations appropriately, a degree-N polynomial can be evaluated using only 45 CHAPTER 4 46 N - 1 multiplications and N additions.

Thus, the global variable a has been replaced by a full table plus a pointer (j) into it. This large amount of “global state” is a disadvantage of this generator in some applications, but it is also an advantage because it leads to an extremely long cycle even if the modulus m is small. The function randomint returns a random integer between 0 and r-l. Of course, it can easily be changed, just as above, to a function which returns a random real number between 0 and 1 (a b]/m). Testing Randomness One can easily detect numbers that are not random, but certifying that a sequence of numbers is random is a difficult task indeed.

Once the program is written, the numbers that it will produce can be deduced, so how could they be random? The best we can hope to do is to write programs which produce isequences of numbers having many of the same properties as random numbers. Such numbers are commonly called pseudo-random numbers: they’re not really random, but they can be useful 33 CHAF’TER 3 as approximations to random numbers, in much the same way that floatingpoint numbers are useful as approximations to real numbers. (Sometimes it’s convenient to make a further distinction: in some situations, a few properties of random numbers are of crucial interest while others are irrelevant.