A Practical Introduction to Data Structures and Algorithm Analysis

Sec. 16.3 Numerical Algorithms 553ii−11−11−i−iFigure 16.6 Examples of the 4th and 8th roots of unity.advantage of symmetries, such as the one we see for evaluating x and −x. But weneed to find even more symmetries between points if we want to do more than cutthe work in half. We have to find symmetries between, say, pairs of values. Andthen further symmetries between pairs of pairs, and so on.Recall that a complex number z has a real component and an imaginary component.We can consider the position of z on a number line if we use the y dimensionfor the imaginary component. Now, we will define a primitive nth root of unity if1. z n = 1 and2. z k ≠ 1 for 0 < k < n.z 0 , z 1 , ..., z n−1 are the nth roots of unity. For example, when n = 4, then z = i orz = −i. In general, we have the identities e iπ = −1, and z j = e 2πij/n = −1 2j/n .The significance is that we can find as many points on a unit circle as we wouldneed (see Figure 16.6). But these points are special in that they will allow us to dojust the right computation necessary to get the needed symmetries to speed up theoverall process of evaluating many points at once.The next step is to define how the computation is done. Define an n × n matrixA z with row i and column j asA z = (z ij ).The idea is that there is a row for each root (row i for z i ) while the columns correspondto the power of the exponent of the x value in the polynomial. For example,when n = 4 we have z = i. Thus, the A z array appears as follows.A z =1 1 1 11 i −1 −i1 −1 1 −11 −i −1 iLet a = [a 0 , a 1 , ..., a n−1 ] T be a vector that stores the coefficients for the polynomialbeing evaluated. We can then do the calculations to evaluate the polynomial