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4.6.4 EVALUATION OF POLYNOMIALS 475 largest real part is real, but the root with second smallest (or second largest) real part is nonreal, the same transformation applies. If the two roots with smallest (or largest) real parts are both real, they can be expressed in the form a -b and a + b, respectively; let c = -a and CY, = b2. Again x2 -QI, is a factor of U(Z -c). (Still other values of c are often possible; see exercise 24.) The coefficient a,-1 will be nonzero for at least one of these alternatives, unless q(x) is identically zero. 1 Note that this method of proof usually gives at least two values of c, and we also have the chance to permute al, . . . , am-i in (m -l)! ways. Some of these alternatives may give more desirable numerical accuracy than others. *Polynomial chains. Now let us consider questions of optimality. What are the best possible schemes for evaluating polynomials of various degrees, in terms of the minimum possible number of arithmetic operations? This question was first analyzed by A. M. Ostrowski in the case that no preliminary adaptation of coefficients is allowed [Studies in Mathematics and Mechanics presented to R. von Mises (New York: Academic Press, 1954) 40-481, and by T. S. Motzkin in the case of adapted coefficients [cf. Bull. Amer. Math. Sot. 61 (955), 1631. In order to investigate this question, we can extend Section 4.6.3 s concept of addition chains to the notion of polynomial chains. A polynomial chain is a sequence of the form where U(X) is some polynomial in x, and for 1 2 i 5 r either Xi = (*Aj) o Xk, 0 I j, k < i, or Xi = ffj 0 Xk, ONote: If you are looking for high quality webhost to host and run your jsp application check Vision jsp web hosting services

474 ARITHMETIC 4.6.4 Theorem E. Every nth degree polynomial (1) with real coefficients, n 2 3, can be evaluated by the scheme = y=x+c, w y2; z = (unv + ~O)Y + PO (n even), z = u,y + PO (n odd); U(X) = (. . . ((z(w -al) + Pl)(W -a2) + P2). . .>(w -GL) + Pm; (20) for suitable real parameters c, ok and Pk, where m = [n/2] -1. In fact, it iS possible to select these parameters so that ,& = 0. Proof. Let us first examine the circumstances under which the Q S and p s can be chosen in (20), if c is fixed. Let p(x) = u(x-cc) = u,xn +a,-lxn-l +~~~+alx+uo. (21) We want to show that p(x) has the form pi(x)(x2–cw,)+& for some polynomial pi(z) and some constants a,, ,&. If we divide p(z) by x2 - cr,, we can see that the remainder pm is a constant only if the auxiliary polynomial q(x) = u2m+lxm + a2m–15m-1 + . . . + al, (22) formed from every odd-numbered coefficient of p(x), is a multiple of x -Q,. Conversely, if q(x) has x -CY, as a factor, then p(x) = pi(x)(x2 -a,) + pm, for some constant & that may be determined by division. Similarly, we want PI(X) to have the form ps(z)(x2 -a,_~) + ,&-I, and this is the same as saying that g(x)/(x -a,) is a multiple of x -CZ,-1; for if Q(X) is the polynomial corresponding to pi(x) as q(x) corresponds to p(x), we have Q(X) = 4(x)/(x -am). C on t inuing in the same way, we find that the parameters ai, pi, . . . , cy,, Pm will exist if and only if q(x) = U2mfl(X -al). . .(x -%n). (23) In other words, either q(x) is identically zero (and this can happen only when n is even), or else q(x) is an mth degree polynomial having all real roots. Now we have a surprising fact discovered by J. Eve [Numer. Math. 6 (1964), 17-211: If p(x) has at least n -1 complex roots whose real parts are all nonnega- tive, or all nonpositive, then the corresponding polynomial q(x) is identically zero or has all real roots. (See exercise 23.) Since U(X) = 0 if and only if p(x+c) = 0, we need merely choose the parameter c large enough that at least n -1 of the roots of U(X) = 0 have a real part 2 -c, and (20) will apply whenever u,-~ = u,-1 -ncu, # 0. We can also determine c so that these conditions are fulfilled and also that Pm = 0. First the n roots of U(X) = 0 are determined. If a + bi is a root having the largest or the smallest real part, and if b # 0, let c = –a and Qm= -b2; then x2 -oym is a factor of U(X -c). If the root with smallest or
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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.
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472 ARITHMETIC 4.6.4 polynomials of higher degrees. This idea is due to C. T. Fike [CACM 10 (1967), 175-1781, who has presented several interesting examples. Any polynomial of the fifth degree may be evaluated using four muliiplica- tions, six additions, and one storing, by using the rule U(X) = U(Z)Z+UO, where U(Z) = u5×4 + ~42~ + u3z2 + zlzz + Q is evaluated as in (9). Alternatively, we can do the evaluation with four multiplications, five additions, and three storings, if the calculations take the form Y = (x + ao12, 4×1 = (NY + W)Y + Q2)(5 + a31 + a4)a5. (11) The determination of the cr s this time requires the solution of a cubic equation (see exercise 19). On many computers the number of storing operations required by (11) is less than 3; for example, we may be able to compute (X + ~0)~ without storing z + a~. In fact, many computers have more than one arithmetic register for floating point calculations, so we can avoid storing altogether. Because of the wide variety of features available for arithmetic on different computers, we shall henceforth in this section count only the arithmetic operations, not the operations of storing and loading an accumulator. The computation schemes can usually be adapted to any particular computer in a straightforward manner, so that very few of these auxiliary operations are necessary; on the other hand, it must be remembered that this extra overhead might well overshadow the fact that we are saving a multiplication or two, especially if the machine code is being produced by a compiler that does not optimize. A polynomial u(z) = ugz6 + . . . + ulx + ug of degree six can always be evaluated using four multiplications and seven additions, with the scheme 2 = (x + Qo)X + al, w = (x + a,)2 + Q3, U(x) = ((w + 2 + a4)w + a5)a6. (12) [See D. E. Knuth, CACM 5 (1962), 595-599.1 This saves two of the six multi- plications required by Horner s rule. Here again we must solve a cubic equation: Since a6 = L16,we may assume that 216 = 1. Under this assumption, let PI = gu5-Q P2 = u4-Pl(Pl+l), P3 = u3-p1@2, p4 = pl-p2, p5 = u2-PlP3. Let ,& be a real root of the cubic equation 2Y3 + (% -P2 + l)Y2 + (SD5 -P2P4 -i33)Y + (UI -h/35) = 0. (13) (This equation always has a real root, since the polynomial on the left approaches +oo for large positive y, and it approaches -oo for large negative y; it must assume the value zero somewhere in between.) Now if we define h=&+p4p6+P5, ~8=~3—~6-b, we have finally 00 =P2-2/36, a2 = Pl -Qo, & = /36 -aOa2, a3 = P7 -%Q2, &4=/38-P7-m, a5 = UO -@7p8* (14)
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4.6.4 EVALUATION OF POLYNOMIALS 471 Adaptation of coefficients. Let us now return to our original problem of eval- uating a given polynomial U(Z) as rapidly as possible, for random values of 2. The importance of this problem is due partly to the fact that standard functions such as sinx, cos Z, eZ, etc., are usually computed by subroutines that rel.y on the evaluation of certain polynomials; such polynomials are evaluated so often, it is desirable to find the fastest possible way to do the computation. Arbitrary polynomials of degree five and higher can be evaluated with fewer operations than Horner s rule ,requires, if we first adapt or precondition the coefficients ~0, ~1, . . . , u,. This adaptation process might involve a lot of work, as explained below; but the preliminary calculation is not wasted, since it must be done only once while the polynomial will be evaluated many times. For examples of adapted polynomials for standard functions, see V. I^a. Pan, USSR Computational Math. and Math. Physics 2 (1963), 137-146. The simplest case for which adaptation of coefficients is helpful occurs for a fourth degree polynomial: u(x) = %$x4 + z&x3 + u2×2 + UlX + uo, u4 # 0. (8) This equation can be rewritten in a form originally suggested by T. S. Motzkin, Y = (x + ao)x + al, u(x) = ((Y + x + Q2)Y + &3)&4, (9) for suitably adapted coefficients a~, (~1, ax, as, ~4. The computation in (9) involves three multiplications, five additions, and (on a one-accumulator machine like MIX) one instruction to store the partial result y into temporary storage. By comparison with Horner s rule, we have traded a multiplication for an addition and a possible storage command. Even this comparatively small savings is worthwhile if the polynomial is to be evaluated often. (Of course, if the time for multiplication is comparable to the time for addition, (9) gives no improvement; we will see that a general fourth-degree polynomial always requires at least eight arithmetic operations for its evaluation.) By equating coefficients in (8) and (9), we obtain formulas for computing the aj s in terms of the Q S: Qo= 4@3/9J4 -l), P = u2/u4 -~O(~O + I), Ql = w/u4 -aoPt et2 =p-2cq, a3 = uo/u4 -Ql(Q + Q2), a4 = u4. (10) A similar scheme, which evaluates a fourth-degree polynomial in the same num- ber of steps as (9), appears in exercise 18; this alternative method will give greater numerical accuracy than (9) in certain cases, although it yields poorer accuracy in others. Polynomials that arise in practice often have a rather small leading coeffi- cient, so that the division by 214 in (10) leads to instability. In such a case it is usually preferable to replace x by Iu4 11j4 x as the first step, reducing (8) to a polynomial whose leading coefficient is f 1. A similar transformation applies to
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470 ARITHMETIC 4.6.4 Derivatives and changes of variable. Sometimes we want to find the coefficients of u(z + x0), given a constant x0 and the coefficients of U(X). For example, if U(X) = 3×2 + 22 -1, then u(x -2) = 3×2 -10x + 7. This is analogous to a radix conversion problem, converting from base x to base x + 2. By Taylor s theorem, the desired coefficients are given by the derivatives of U(X) at x = x0, namely u(x + x0) = u(x0) + u (xlJ)x + (U (ZO)/2!)Z2 + . . . + (dn)(xo)/n!)xn, (7) so the problem is equivalent to evaluating U(X) and all its derivatives. If we write U(X) = q(z)(z -x0) + r, then u(x + x0) = q(a: + ze)z + r; so r is the constant coefficient of u(z + ZO), and the problem reduces to finding the coefficients of ~(a: + x0), where q(x) is a known polynomial of degree n -1. Thus the following algorithm is indicated: Hl. Set vj + uj for 0 5 j 5 n. H2. For k = 0, 1, . . . , n -1 (in this order), set vj 6 Vj + xovj+l for j = n-1, . . . 7 k + 1, k (in this order). 1 At the conclusion of step H2 we have U(X + x0) = v,xn +. . . + wlx + ~0. This procedure was a principal part of Horner s root-finding method, and when Ic = 0 it is exactly rule (2) for evaluating u(z0). Horner s method requires (n2 + n)/2 multiplications and (n2 + n)/2 addi- tions; but notice that if xo = 1 we avoid all of the multiplications. Fortunately we can reduce the general problem to the case x0 = 1 by introducing compara- tively few multiplications and divisions: Sl. Compute and store the values xi, . . . , xg. S2. Set 1-j t ujx$ for 0 5 j 5 n. (Now V(X) = u(x,-,x).) S3. Perform step H2 but with z. = 1. (Now v(x) = u(x~(x+l)) = U(XOX+XO).) S4. Set vj c vj/xj, for 0 < j 2 n. (Now v(x) = u(z + 50) as desired.) 1 This idea, due to M. Shaw and J. F. Traub [JACM 21 (1974), 161-1671, has the same number of additions and the same numerical stability as Horner s method, but it needs only 2n -1 multiplications and n -1 divisions. About 3n of these multiplications can, in turn, be avoided (see exercise 6). If we want only the first few or the last few derivatives, Shaw and Traub have observed that there are further ways to save time. For example, if we just want to evaluate u(x) and u (x), we can do the job with 2n -1 additions and about n + & multiplications/divisions as follows: Dl. Compute and store the values x2, x3, . . . , xt, x~~, where t = [miz]. D2. Set 1-j t zljxf(j) for 0 2 < 1 -n,wheref(j)=t-1-((n-1-j)mod2t) for 0 2 j < n, and f(n) = t. D3. Set v~j + Vj + vj+lxg(j) for j = n -1, . . . , 1, 0; here g(j) = 2t when n-j is a multiple of 2t, otherwise g(j) = 0 and the multiplication by xg(j) need not be done. D4. Set Vj t Vj + vj+ixg(j) for j = n -1, . . . , 2, 1. NOW VO/X~( ) = U(X) and Vl/Xf(l) = u (x). 1
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4.6.4 EVALUATION OF POLYNOMIALS 469 and u(-z), this approach yields u(–5) with just one more addition operation; two values can be obtained almost as cheaply as one. Moreover, if we have a computer that allows parallel computations, the two lines of (4) may be evaluated independently, so we save about half the running time. When our computer allows parallel computation on k arithmetic units at once, a /&h-order Horner s rule (obtained in a similar manner from f(z) = xk - x; ) may be used. Another attractive method for parallel computation has been suggested by G. Estrin [Proc. Western Joint Computing Conf. 17 (1960), 33-401; for n = 7, Estrin s method is: Processor 1 Processor 2 Processor 3 Processor 4 Processor 5 al = U7x + U6 bl = U5Z + U4 cl = U3x + U2 dl = UlZ + Z&o X2 u2 = u1×2 + bl c2 = c1×2 + dl X4 a3 =a2×4+c2 Here us = u(x). However, an interesting analysis by W. S. Dorn [IRM J. Res. and Devel. 6 (1962), 239-2451 shows that these methods might not actually be an improvement over the second-order rule, if each arithmetic unit must access a memory that communicates with only one processor at a time. Tabulating polynomial values. If we wish to evaluate an nth degree polynomial at many points in an arithmetic progression (i.e., if we want to calculate u(xo), 4×0 + h), u(xo + 2h), . * J, the process can be reduced to addition only, after the first few steps. For if we start with any sequence of numbers (Q,, (xl,. . . , a,) and apply the transformation ~o+~o+w, a1 +w+Qz, **a, an-1 + c&-l + Cb, (5) we find that k applications of (5) yields where ,0j denotes the initial value of aj and pj = 0 for j > n. In particular, is a polynomial of degree n in k. By properly choosing the p s, as shown in exercise 7, we can arrange things so that ar) is the desired value u(xg + kh), for all k. In other words, each execution of the n additions in (5) will produce the next value of the given polynomial. Caution: Rounding errors can accumulate after many repetitions of (5), and an error in Qj produces a corresponding error in the coefficients of x0, . . . , xj in the polynomial being computed. Therefore the values of the a s should be refreshed after a large number of iterations.
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468 ARITHMETIC 4.6.4 Complex addition and multiplication can obviously be reduced to a sequence of ordinary operations on real numbers: real + complex requires 1 addition complex + complex requires 2 additions real X complex requires 2 multiplications complex X complex requires 4 multiplications, 2 additions or 3 multiplications, 5 additions (See exercise 41. Subtraction is here considered as if it were equivalent to addition.) Therefore Horner s rule (2) uses either 472 - 2 multiplications and 3n -2 additions or 3n -1 multiplications and 6n -5 additions to evaluate U(Z) when z = z + iy is complex. Actually 2n -4 of these additions can be saved, since we are multiplying by the same number z each time. An alternative procedure for evaluating U(X + iy) is to let a1 = %, bl = w-1, r=x+x, s = x2 + y2; aj = by-1 + raj-l, bj = un–j -saj-l, lNote: In case you are looking for affordable webhost to host and run your web application check Vision cheap hosting services

4.6.4 EVALUATION OF POLYNOMIALS 467 a demonstration that certain kinds of numerical stability cannot be guaranteed for some families of high-speed algorithms.] A beginning programmer will often evaluate the polynomial (1) in a manner corresponding directly to its conventional textbook form: First u,xn, is calcu- lated, then u,-lxn- , . . . , ulx, and finally all of the terms of (1) are added together. But even if the efficient methods of Section 4.6.3 are used to evaluate powers of x in this approach, the resulting calculation is needlessly slow unless nearly all of the coefficients uk are zero. If the coefficients are all nonzero, an obvious alternative would be to evaluate (1) from right to left,, computing the values of xk and Ukxk+. . .+UO for k = 1, . . . , n. Such a process involves 2n-1 multiplications and n additions, and it might also require further instructions to store and retrieve intermediate results from memory. Homer s rule. One of the first things a novice programmer is usually taught is an elegant way to rearrange this computation, by evaluating u(x) as follows: u(x) = ((. . . (21,x + u,-1)x + . . .)x + uo. (2) Start with un, multiply by x, add ~~-1, multiply by x, . . . , multiply by x, add ~0. This form of the computation is usually called Horner s rule ; we have already seen it used in connection with radix conversion in Section 4.4. The entire process requires n multiplications and n additions, minus one addition for each coefficient that is zero. Furthermore, there is no need to store partial results, since each quantity arising during the calculation is used immediately after it has been computed. W. G. Horner gave this rule early in the nineteenth century [Philosophical Transactions, Royal Society of London 109 (1819), 308-3351 in connection with a procedure for calculating polynomial roots. The fame of the latter method [see J. L. Coolidge, Mathematics of Great Amateurs (Oxford, 1949), Chapter 151 accounts for the fact that Horner s name has been attached to (2); but actually Isaac Newton had made use of the same idea 150 years earlier. In a well-known work entitled De Analysi per Bquationes Innfinitas, originally written in 1669, Newton wrote y–4xy:+5xy:-12xy:+17 for the polynomial y4 -4y3 + 5y2 -12y + 17; this clearly uses the idea of (a), since he often denoted grouping by using horizontal lines and colons instead of parentheses. [See D. T. Whiteside, ed., The Mathematical Paperrs of Isaac Newton 2 (Cambridge Univ. Press, 1968), 222.1 Several generalizations of Horner s rule have been suggested. Let us first consider evaluating U(Z) when z is a complex number, while the coefficients uk are real. In particular, when .z = eie = cos 0 + i sin 8, the polynomial u(z) is essentially two Fourier series, (U,+U1COSe+… + un cos no) + i(ul sin 19 + . . . + un sin no).
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466 ARITHMETIC 4.6.3 37. [M,u] The binary addition chain for n = 2 O -+-. . .+2et, when eo > > et 2 0, is 1, 2, . . . , 2eo-e1, 2eo-e1 + 1, . . . , 2e0-e2 + 2e1-ez, 2 Ope2 + 2e1-ez + 1, . . . , n. This corresponds to the S-and-X method described at the beginning of this section, while Algorithm A corresponds to the addition chain obtained by sorting the two sequences (1,2,4,. . ,2 O) and (set-l + 2et, 2 t-2 + 2et-1 + 2 t,. . . , n) into increasing order. Prove or disprove: Each of these addition chains is a dual of the other. 38. [M,C?7] How many addition chains without useless steps are equivalent to each of the addition chains discussed in exercise 37, when eo > el + l? b 39. [A4.25] (J. Olivos, 1979.) Let l([nl,na,. . . , n,]) be the minimum number of mul- tiplications needed to evaluate the monomial z; z~ . . CI$ in the sense of exercise 27, where each nz is a positive integer. Prove that this problem is equivalent to the problem of exercise 32, by showing that I( [nl, n2, . . , n,]) = l(nl, n2, . , n,) + m - 1. [Hint: Generalize the directed graph construction by considering graphs with more than one source vertex.] F 40. [kC?I] (J. Olivos.) Generalizing the factor method, prove that l(mlnl + . . . + mtnt) I l(ml, . . . , mt)+l(nl,…,nt)+t-1, where l(nl, . . , n,) is defined in exercise 32. 41. [M40](P. Downey, B. Leong, R. Sethi.) Let G be a connected graph with n vertices (1,. , n} and m edges, where the edges join uj to vj for 1 5 j 5 m. Prove that 1(1,2,. . . , 2An, 2AU1 + 2A 1 + 1,. . . , 2AUm + 2Avm + 1) = An + m + k for all sufficiently large A, where k is the minimum number of vertices in a vertex cover for G (i.e., a set containing either uj or V~ for 1 5 j 5 m). 4.6.4. Evaluation of Polynomials Now that we know efficient ways to evaluate the special polynomial z%, let us consider the general problem of computing an nth degree polynomial U(X)= u,zn + un-&–l + * * * + UlZ + U-J, GL # 0, (1) for given values of Z. This problem arises frequently in practice. In the following discussion we shall concentrate on minimizing the number of operations required to evaluate polynomials by computer, blithely assuming that all arithmetic operations are exact. Polynomials are most commonly evaluated using floating point arithmetic, which is not exact, and different schemes for the evaluation will, in general, give different answers. A numerical analysis of the accuracy achieved depends on the coefficients of the particular polynomial being considered, and is beyond the scope of this book; the reader should be careful to investigate the accuracy of any calculations undertaken with floating point arithmetic. In most cases the methods we shall describe turn out to be reasonably satisfactory from a numerical standpoint, but many bad examples can also be given. [See Webb Miller, SIAM J. Computing 4 (1975), 105-10 7, for a survey of the literature on stability of fast polynomial evaluation, and for
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