sec numerical algorithms - - - - figure examples of the th and th roots of unity advantage of symmetriessuch as the one we see for evaluating and - but we need to find even more symmetries between points if we want to do more than cut the work in half we have to find symmetries betweensaypairs of values and then further symmetries between pairs of pairsand so on recall that complex number has real component and an imaginary component we can consider the position of on number line if we use the dimension for the imaginary component nowwe will define primitive nth root of unity if and for - are the nth roots of unity for examplewhen then or - in generalwe have the identities eip - and pij/ - / the significance is that we can find as many points on unit circle as we would need (see figure but these points are special in that they will allow us to do just the right computation necessary to get the needed symmetries to speed up the overall process of evaluating many points at once the next step is to define how the computation is done define an matrix az with row and column as az ( ij the idea is that there is row for each root (row for while the columns correspond to the power of the exponent of the value in the polynomial for examplewhen we have thusthe az array appears as follows - - az - - - - let [ an- ] be vector that stores the coefficients for the polynomial being evaluated we can then do the calculations to evaluate the polynomial