By R. Sedgewick
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Extra info for Algorithms
Then, splitting q(x) in the same way, we have: P(x) = Pi(x) + zN’2ph(x), q(x) = 41(x) + “N’2qh(x). h(x) + d+h(x))xN’2 + Ph(x)qh(x)xN. -h(x)xN. Polynomial addition requires a linear algorithm, and the straightforward polynomial multiplication algorithm of Chapter 2 is quadratic, so it’s worthwhile to do a few (easy) additions to save one (difficult) multiplication. Below we’ll look more closely at the savings achieved by this method. For the example given above, with p(x) = 1 +x +3x2 -4x3 and q(x) = 1 + 2x - 5x2 - 3x3, we have Q(X) = (1+ x)(1 + 2x) = I + 3x + 2x2, Q(X) = (3 -4x)(-5 - 3x) = -15 + 11x + 12x2, T,(X) = (4 - 3x)(-4 - x) =: -16 +8x + 3x2.
The interpolation problem is to find the polynomial, given a set of points and values. The evaluation problem is to find the values, given the polynomial and the points. ) The classic solution to the interpolation problem is given by Lagrange’s interpolation formula, which is often used as a proof that a polynomial of degree N - 1 is completely determined by N points: This formula seems formidable at first but is actually quite simple. s+13s5=j which simplifies to x2 +a:+ 1. For x from xl, x2, .
Random number generators can be very, very good, but when they are bad they are horrid. Many tests have been developed for determining whether a sequence shares various properties with a truly random sequence. Most of these tests have a substantial basis in mathematics, and it would definitely be beyond the scope of this book to examine them in detail. However, one statistical test, the x2 (chi-square) test, is fundamental in nature, quite easy to implement, and useful in several applications, so we’ll examine it more carefully.