4.6.4 EVALUATION OF POLYNOMIALS (Web hosting company) 473 We can illustrate
4.6.4 EVALUATION OF POLYNOMIALS 473 We can illustrate this procedure with a contrived example: Suppose that we want to evaluate x6 + 13×5 +49×4 +33×3 -61×2 -37x + 3. We obtain os = 1, p1 = 6, p2 = 7, p3 = -9, p4 = -1, ,& = -7, and so we meet with the cubic equation 2y3-&Js+2y+12=0. (15) This equation has ps = 2 as a root, and we continue to find p7 = -5, ps = -6, ~0 = 3, LYZ = 3, (~1 = -7, os = 16, CY~ = 6, os = -27. The resulting scheme is therefore z = (5 + 3)x -7, w = (x + 3)~ + 16, u(x) = (w + z + 6)w -27. By sheer coincidence the quantity x + 3 appears twice here, so we have found a method that uses three multiplications and six additions. Another method for handling sixth-degree equations has been suggested by V. I^a. Pan [Problemy Kibernetiki 5 (1961), 17-291. His method requires one more addition operation, but it involves only rational operations in the preliminary steps; no cubic equation needs to be solved. We may proceed as follows: z = (x + Qo)X + Ql, w=z+x+a2, U(x) = (((2 -x + a3)w + a4)z + a5)&6. (16) To determine the (Y S, we divide the polynomial once again by us = os so that U(X) becomes manic. It can then be verified that oe = us/3 and that al = (ul -aOu2 + C&Q -c&4 + 2cuz)/(us -2ao~4 + 5&j). (17) Note that Pan s method requires that the denominator in (17) does not vanish. In other words, (16) can be used only when 27~3~~; -18 Z&?J51Lq + 5tL; # 0; (18) in fact, this quantity should not be so small that 011 becomes too large. Once 011 has been determined, the remaining o s may be determined from the equations Pl = 2ao, p2 = u4 -crop1 -Qlt P3 = u3 -aoP2 -cd%, P4 = 7J2 -aoP3 -02, a3 = f(P3 -(a0 -1)Pz + (Qo -l)(GJ -1,) -Ql, (19) cY2 = pz -(ai -1) -a3 -2cy1, ~4=04-(~2+Q1)(~3+~1), cY5 = uo -cx1p4. We have discussed the cases of degree n = 4, 5, 6 in detail because the smaller values of n arise most frequently in applications. Let us now consider a general evaluation scheme for nth degree polynomials, a method that involves at most ]n/2] + 2 multiplications and n additions.
Note: In case you are looking for affordable webhost to host and run your servlet application check Vision make web site services