Business web site - 642 ANSWERS TO EXERCISES 4.6.4 These calculations assume
642 ANSWERS TO EXERCISES 4.6.4 These calculations assume that the given numbers F(t) are complex. If the F(t) are real, then f(s) is the complex conjugate of f(2 -s), so we can avoid the redundancy by computing only the 2 independent real numbers f(O), Xif(l), . . . , Bf(2 - -l), fP-1), Sf(l), . . . , Sf(2 -l -1). The entire calculation in this case can be done by working with 2n real values, using the fact that flkl(&-k+l, .. . , sn, ti, . . . , i&-k) will be the Complex COnjUgate Of flkl(sk-k+i,. . . , SL, ti, . . . , tn.-k) when (S1 . . . S~)Z + (si . . sk)z E 0 (modulo 2 ). About half as many multiplications and additions are needed as in the complex case. [The fast Fourier transform algorithm is essentially due to C. Runge and H. Kijnig in 1924, and it was generalized by J. W. Cooley and J. W. Tukey, Math. Comp. 1s (i965), 297-301 . Its interesting history has been traced by J. W. Cooley, P. A. W. Lewis, and P. D. Welch, Proc. IEEE 55 (1967), 1675-1677. Details concerning its use have been discussed by R. C. Singleton, CACM 10 (1967), 647-654; M. C. Pease, JACM 15 (1968), 252-264; G. D. Berglund, Math. Comp. 22 (1968), 275-276, CACM11(1968), 703-710; A. M. Macnaghten and C. A. R. Hoare, Comp. J. 20 (1977), 78-83. See also exercises 53, 57, and 59.1 15. (a) The hint follows by integration and induction. Let f( )(e) take on all values be- tween A and B inclusive, as 19 varies from min(zs, . . , xn) to max(zc, . . . , zn). Replacing f( ) by each of these bounds, in the stated integral, yields A/n! 2 f(zo, . . . , z,) 5 B/n!. (b) It suffices to prove this for j = 7~. Let f be Newton s interpolation polyno- mial, then f( ) is the constant n!a,. 16. Carry out the multiplications and additions of (43) as operations on polynomials. (The special case ~0 = ~1 = . . . = zn is considered in exercise 2. We have used this method in step C8 of Algorithm 4.3.3C.) 17. T. M. Vari has shown that n -1 multiplications are necessary, by proving that 12 multiplications are necessary to compute ~2 + . . . + 2: [Cornell Computer Science Report 120 (Jan. 1972)]. 18. cio = +(us/uq+l), p = uz/uq-oo(cyo-1), ~1 = crop-uul/ur, CYZ= p-2~~1, a3 = uo/u4 -W(crl + (Yz), a4 = u4. 19. Since cy5 is the leading coefficient, we may assume without loss of generality that U(Z) is manic (i.e., us = 1). Then QO is a root of the cubic equation 40z3 - 24.2~4~~+ (4U: + 2Us)Z + (Uz - usu,) = 0; this equation always has at least one real root, and it may have three. Once (~0 is determined, we have 01s = u4 -4o0, CY~ = us - 4acos - 6a;, ff2 = 1~1-QO(CXOQ~+ 4c&3 + 2oi(rs + a;), (~4 = uo -CQ(~; + ala; + (~2). For the given polynomial we are to solve the cubic equation 40z3 -120~~ + 802 = 0; this leads to three solutions (CYO, cyi, cy2, cys, CY~, as) = (0, -10,13,5, -5, l), (1, -20,68,1,11, l), (2, -10,13, -3,27,1). 20. LDA X FADD =a~= FADD =cY~= FMUL TEMP2 STA TEMPI FADD =a~= FADD =a,~-a3= FMUL TEMPl STA TEMPZ FADD =a4= FMUL TEMP2 FMUL =cx5= I STA TEMP2 21. z = (z+l)s-2, w = (x+5)z+9, U(Z) = (w+z-8)w-8; or z = (%+9)x+26, w = (5 -3)~ + 73, u(x) = (w + z - 24)~ -12.