On computing reciprocals of power series

Root-finding iterations are used to compute reciprocals of power series. We show that Sieveking's algorithm is just Newton iteration applied in the field of power series. Let L^ denote the number of non-scalar multiplications needed to compute the first rrH terms of the reciprocal. We show that We conjecture that


INTRODUCTION
We consider the problem of computing reciprocals of power series. This problem is closely related to the problems of polynomial division, evaluation and interpolation. (For example, see Borodin (1973).) Let L denote the n number of non-scalar multiplications needed to compute the first n+1 terms of the reciprocal of a power series. Recently Sieveking (1972) showed that L ^ 5n-2, n In this paper, we show: (i) Root-finding iterations can be used in the field of power series, Sieveking 1 s algorithm is just Newton iteration applied to the function f(x) = x ^-a, a ^ 0, in the field of power series.
(ii) By modifying Sieveking 1 s algorithm and analysis, Sieveking 1 s bound is improved to L n ^ 4n-log2n.
(iii) We propose a new algorithm for computing reciprocals of power series, which is competitive with Sieveking f s algorithm, and which is based on a third order iteration. The bound in (ii) can also be obtained by this new algorithm.
n In Section 2 we define some basic notation and also prove results (i) and (ii). Results (iii) and (iv) are proven in Sections 3 and 4, respectively. -2- In Section 5 we give a general family of algorithms for computing reciprocals of power series. Any algorithm in the family can compute the first n+1 terms of the reciprocal of a power series in 0(n) non-scalar multiplications (and can also compute them in 0(n log n) arithmetic operations if Fast Fourier Transform is used for polynomial multiplication). We conjecture that Newton iteration and the third order iteration are optimal among all algorithms in the family.

NEWTON ITERATION
We will use notation of Sieveking (1972) and Strassen (1973). Let k be an infinite field, a^, b^, i -0,1,...,°°, indeterminates over k, A an extension field of k, and t an indeterminate over A. Suppose that E and F are finite subsets of A and that we do computations in the field A. Let L(E mod F) denote the number of multiplications or divisions by units which are necessary to compute E starting from k U F not counting multiplications by scalars in k.
We shall prove the following theorem by using Newton iteration.
We first use a technique of Strassen (1973) to prove the following Let X., 1 £ j £ n+m -jif 1 , be any n-hn -ji+1 distinct nonzero elements in k. Observe that n-hn-ji . n m 1-1 for j = 1,...,n+m -&KI, and det(X^) £ 0. Hence c j^.^» 0 ^ i ^ n+m-j£, can be obtained by solving the linear system (2.2). Therefore, Denote S a.t by a and Jb.t by b. Suppose that (2.1) holds for all n. 0 1 0 1 Then b is the reciprocal of a with respect to the field A(t).
Define the function f: just the root of f. Applying Newton iteration to f, we obtain the iterate-;, (In this paper,derivatives of f are defined by purely algebraic methods without employing any limit concept. For example, see van der Waerden (1953, ^65).) It follows from (2.3) that L 0 ^ L + (2n+l) + (2n+l) = L + 4rrt-2, I n+1 n n We have shown (2.5). From (2.7) we also have (2.6) follows in the same as (2,5) by starting with (2.10) instead of (2.7).
One can easily check that the algorithm proposed by Sieveking (1972) is just the Newton iteration stated above. However, because of (2.8), Lemma 2.1, and careful estimation of L from (2.5) and (2.6) we are able to obtain n rather than L < 4n-log 0 n for n ^ 1 n z L £ 5n-2 for n ^ 1 n which is obtained by Sieveking (1972). One should also note that the idea of using Newton iteration to compute reciprocals has been known for a long For example, Newton iteration is used to compute matrix inverses by Schulz (1933), to compute reciprocals of real numbers by Rabinowitz (1% and to compute integer reciprocals by S. A. Cook (see Knuth (1969, ^4.33) , In this section, we have shown that Newton iteration can be used successfully in the field of power series, and hence can compute reciprocals of power series. In fact, any root-finding iteration (Traub (1964)) can be used for the problem of computing reciprocals (see Section 5). Newton iteration is a second order iteration. In the next section we propose a new algorithm for computing reciprocals of power series, which is based on a third order iteration, and which, is competitive with Sieveking 1 s algorithm.
It is easy to show that We shall now use 9 to prove Theorem 2.1 for n * 3. Let L denote n L(b Q ,...,b n mod a 0 ,...,a n ) as before. Note that L 1 £ 3 and L 2 <: 6. It is not difficult to check that it suffices to prove that for n £ 1,   (3.10), respectively instead of (3.6).

A LOWER BOUND
Under the hypotheses of Theorem 2.1, we show that  For all k define c R to be the greatest integer such that This is the third order iteration cp in (3.1).
(iii) If d = 1 and JL Q = ^ = 1 we have x (k + 1) ^x(k) (1 . ax (k-1) ) +x (k-D --ax^x^+x^ + x 00 . -1 One can check that this is the secant iteration applied to f(x) = x -In fact, (5.1) represents the algorithm which is obtained by a general Hermite interpolatory iteration (Traub (1964)) applied to the function f(x) = x ^ -a 0 A special case of (5.1) (i.e., d = 0) was pointed out before by Rabinowitz (1961) for computing reciprocals of numbers. By the same techniques used in Sections 2 and 3 one could show that L(b^,...,b^ mod ag,...,a^) is bounded by a linear function of n by using any algorithm defined by (5.1).
However, we believe that Newton iteration and the third order iteration are optimal among all algorithms defined by (5.1). More precisely, we state the following -13-
If we use the Fast Fourier Transform for polynomial multiplication, then it can be easily shown by techniques similar to those used in this paper that any algorithm defined by (5.1) is able to compute the first nH-1 terms of the reciprocal of a power series in 0(n log n) arithmetic operations, ACKNOWLEDGMENT I would like to thank Professor J. F. Traub for his comments on this paper.