# The Art of Computer Programming, Volume 2: Seminumerical Algorithms (3rd Edition)

## Donald E. Knuth

Language: English

Pages: 1

ISBN: 0201038226

Format: PDF / Kindle (mobi) / ePub

The second book in a three volume set, this book provides a complete introduction to this topic. It presents a readable and coherent summary of the major paradigms and basic theory of semi numerical algorithms, providing a comprehensive interface between computer programming and numerical analysis.

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३2k蜢 2(sk 舑 2sk 蜢 1)2 A[1](sk 蜢 1, 1, tk 舑 3, . . . , t0). . . . Pass k. Set A[k](sk 蜢 1, . . . , s1, s0) 薒 A[k蜢1](sk 蜢 1, . . . , s1, 0) + ३(s0s1...sk舑1)2 A[k蜢1](sk 蜢 1, . . . , s1, 1). It is fairly easy to prove by induction that we have where t = (tk 蜢 1 . . . t1t0)2, so that (It is important to notice that the binary digits of s are reversed in the final result (39). Section 4.6.4 contains further discussion of transforms such as this.) To get the inverse Fourier transform (u0, . . . ,

[Comp. J. 37 (1994), 219舑222] has observed that Algorithm P can be speeded up significantly for large n by using Montgomery舗s fast method for remainders (exercise 4.3.1舑41). A completely rigorous and deterministic way to test for primality in polynomial time was finally discovered in 2002 by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, who proved the following result: Theorem A. Let r be an integer such that n 衩 r and the order of n modulo r exceeds (lg n)2. Then n is prime if and only if

mod p(u) are easy to evaluate. In the second place, to evaluate the coefficients of x(u)y(u) mod p(u), when the polynomial p(u) can be factored into q(u)r(u) where gcd(q(u), r(u)) = 1, we can use the identity where a(u)r(u)+b(u)q(u) = 1; this is essentially the Chinese remainder theorem applied to polynomials. In the third place, we can always evaluate the coefficients of the polynomial x(u)y(u) mod p(u) by using the trivial identity Repeated application of (58), (59), and (60) tends to

general application of the chi-square method. In general, suppose that every observation can fall into one of k categories. We take n independent observations; this means that the outcome of one observation has absolutely no effect on the outcome of any of the others. Let ps be the probability that each observation falls into category s, and let Ys be the number of observations that actually do fall into category s. We form the statistic In our example above, there are eleven possible outcomes

sequence states that this result is true in the special case that for some constants u1, v1, . . . , uk, vk. Therefore Eq. (8) is true whenever f = a1f1 + a2f2 + ... + amfm and when each fj is a function of type (9); in other words, Eq. (8) holds whenever f is a 舠step-function舡 obtained by partitioning the unit k-dimensional cube into subcells whose faces are parallel to the coordinate axes, and assigning a constant value to f on each subcell. Now let f be any Riemann-integrable function. If ॉ