(5) Suppose our computing resources are limited to performing only a few million ele? mentary operations such as addition and multiplication of two numbers. What is roughly the largest size N for a vector (f1, . . . , fN) whose ?nite Fourier transform we can compute using such limited resources? Why? How about using the Fast Fourier Transform method, what can the largest N be? Explain. (6) Let the points 0 (0,0),A (1,0),B (0,1),0 (1,1) be the corners of the unit square in the my plane. Find the continuous piecewise linear polynomial in two variables P(x, y) such that P(0,0) 1,P(1,0) 0,P(0, 1) ?1,P(1, 1) 1. Show all your work. 1 / ecos(7mc)dx. 0 (7) Consider the integral (a) Compute approximate values for this integral using the Composite Mid?Point Rule M N, Composite Trapezoid Rule TN and Composite Simpson?s Rule SN, for N 2. (b) Let f be a function de?ned on the interval [?1, 1] with the property that | f? | g 60 and | f (4)| S 3340 on this interval. Find what is the smallest integer N such that the error in the approximation of the integral LII f (x)dx by M N is less than 10?4. Answer the same question for TN and for SN as well.
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