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"X-£*br DIGITAL COMPUTER LABORATORY 
 
 7lO>!34> UNIVERSITY OF ILLINOIS 
 
 C ° F' 3 URBANA, ILLINOIS 
 
 REPORT NO. 136 
 
 STATIONARY DISTRIBUTION OF PARTIAL REMAINDERS 
 IN S-R-T DIGITAL DIVISION 
 
 Richard Robert Shively 
 
 May 15, 1963 
 
 (This work is being submitted in partial fulfillment of the 
 requirements for the Degree of Doctor of Philosophy in 
 Electrical Engineering, May 1963). 
 
DIGITAL COMPUTER LABORATORY 
 UNIVERSITY OF ILLINOIS 
 URBANA, ILLINOIS 
 
 REPORT NO. 136 
 
 STATIONARY DISTRIBUTION OF PARTIAL REMAINDERS 
 IN S-R-T DIGITAL DIVISION 
 
 by 
 
 Richard Robert Shively 
 
 May 15, 1963 
 
 (This work is "being submitted in partial fulfillment of the 
 requirements for the Degree of Doctor of Philosophy in 
 Electrical Engineering, May 1963). 
 
,, 
 
 ACKNOWLEDGMENT 
 
 The author wishes to express his gratitude to Professor J. E. Robertson 
 for certain important observations^ and for his encouragement and guidance in 
 pursuing this work. 
 
 The author is also indebted to Professor G. A. Metze for his 
 invaluable counsel at several points during this study. 
 
 The friendly help of Mrs. Phyllis Olson who typed the thesis in its 
 final form is sincerely appreciated. 
 
TABLE OF CONTENTS 
 
 BACKGROUND 
 
 Page 
 
 1.1 Introduction ...................... 1 
 
 1.2 S-R-T Division ..................... 2 
 
 1.3 Division as a Markov Process .............. 5 
 
 1.3-1 Existence and Uniqueness of a Stationary- 
 Distribution .................. 5 
 
 1.3-2 Formulation of the Chain ............. 10 
 
 1.3.3 Examples ..................... 12 
 
 1.3-4 Limitations of the Markov Chain Approach ..... 14 
 
 2. DERIVATION OF PARTIAL REMAINDER DENSITIES .......... l6 
 
 2.1 The Defining Equation .................. l6 
 
 2.2 End Point Relations ................... 22 
 
 2.3 Symmetry ........................ 23 
 
 2.4 Corollaries to Symmetry ................. 28 
 
 2.5 Generation of Solutions ................. 31 
 
 2.5.1 Properties of Solution Intervals ......... 44 
 
 2.5.2 Comments on the Set (D„) ............. 49 
 
 E 
 
 3. SHIFT AVERAGE ........................ 52 
 
 3.1 Equation for <S> .................... 52 
 
 3.2 Evaluation of <S(D)> for ^ < D < 1 ........... 54 
 
 3.3 <S(D)> f or I < D < I . ? . .............. . 60 
 
 4. SUMMARY AND CONCLUSIONS ................... 68 
 
 4.1 Summary ......................... 68 
 
 4.2 Conclusions and Areas for Future Investigation ..... 70 
 
 BIBLIOGRAPHY ........................... 73 
 
 ■IV- 
 
STATIONARY DISTRIBUTION OF PARTIAL REMAINDERS IN S-R-T DIGITAL DIVISION 
 
 Richard Robert Shively 
 Department of Electrical Engineering 
 University of Illinois, 19&3 
 
 The execution time required for digital division in electronic 
 computers can be reduced if a) more than a minimum number of divisor multiples 
 are available, and b) provisions exist to either by-pass or shorten steps which 
 generate zero quotient digits. A division algorithm which uses this redundancy 
 to increase the likelihood of zero digits (i.e., when the quotient is expressed 
 in the recoded form implied by the divisor selections) then is desirable. 
 
 To provide a basis for comparing various division algorithms in this 
 respect, C. V. Freiman observed, using a Markov chain model for the division 
 process, that for any given divisor, D, and a randomly distributed set of 
 dividends, the distribution of partial remainders at successive steps approaches 
 a steady-state or stationary distribution, which is independent of the distribu- 
 tion of dividends. In attempting to determine this stationary distribution as 
 a function of D for an algorithm known as S-R-T division, Freiman found that 
 the pointwise solutions provided by a Markov chain are inadequate to analyze the 
 function for certain intervals of D. 
 
 The purposes in further pursuing this work started by Freiman are: 
 
 a) to obtain a more complete understanding of S-R-T division, 
 
 b ) to derive methods which might be found applicable in 
 analyzing other division algorithms, and 
 
 c) to provide information which could prove useful in 
 formulating new division algorithms. 
 
 If partial remainders, R., are defined to be in one-to-one corres- 
 pondence with quotient digits (rather than considering only those associated 
 with nonzero quotient digits), the stationary probability density, F(x), which 
 
describes R., can be shown to be symmetric with respect to the point D, F(D), 
 for all D in the allowed range, — < D < 1. This symmetry can be used to derive 
 a straightforward method for evaluating the stationary density for intervals in 
 the divisor range, rather than for individual points „ The method essentially 
 consists of locating points at which F(x) is. discontinuous, as an ordered 
 process, starting with points of discontinuity in the relation between R. and 
 R. , and terminating when the first such discontinuity point falls in the 
 interval (— , 2D - p). Then, by observing that the jump sizes at successive 
 points in this ordered set of argument values form a geometric progression, and 
 that the sum of these jumps is — , the solution follows directly. Solution 
 intervals, i.e., neighborhoods in the range of D for which F(x) has similar 
 form, may be grouped into infinite families, based on the sequence of operations 
 used to locate discontinuity points, and when considered in the order of 
 increasing complexity, these families form a nested pattern,, 
 
 The stationary densities describing R. can be used to determine the 
 shift average, <S>, i.e., the average number of quotient digits formed per use 
 of the adder. <S> is equivalent to the mean recurrence time of the event 
 
 "R. > — " in S-R-T division, and is therefore the reciprocal of the probability 
 
 1 2D 
 that R. > p. Using properties of F(x), <S> can be shown to be — =■ for 
 
 0.75 < D < 1, reaching a maximum of 3.? over the interval 0.6 < D < 0.75 • In 
 
 the interval 0.5 < D < 0.6, <S> requires an infinite number of sub intervals to 
 
 express completely, and exhibits a nonmonotonic behavior at certain points. 
 
 Analysis of other division algorithms can use the fact that a) the 
 
 piecewise constant functions which describe the stationary density can be 
 
 formed by locating discontinuity points as an ordered process, and b) the sizes 
 
 of jumps at successive arguments in this process form a geometric progression. 
 
1 . BACKGROUND 
 
 1.1 Introduction 
 
 Numerous factors are effecting a willingness to invest in redundant 
 equipment to improve digital computer operation times. Reliability and 
 decreasing cost of components, modularization of circuits, and miniaturization 
 are but a few of these. In arithmetic processes, some commonly considered 
 applications of redundancy within the structure of a general purpose computer 
 are: a) redundant number representation to reduce or eliminate adder carry 
 propagation time, b) multiple shift paths to reduce the number of steps required 
 in iterative processes such as multiplication and division, and c) generation 
 of more than the minimum number of multiples of the operand, again to improve 
 multiplication and division. 
 
 In digital division, redundancy in the generation of divisor multiples 
 may be used to serve either or both of two general aims: a) providing overlap 
 in the partial remainder ranges to which the respective divisor multiples are 
 assigned, and b) minimizing the expectation value of the partial remainder 
 magnitude. The first of these is particularly important when signed digits or 
 stored carries are used, since distinguishing between nonoverlapping intervals 
 by inspection of leading digits of unassimilated numbers would be impossible. 
 Reduction in the partial remainder magnitude can, under certain circumstances, 
 reduce the time required for the division process. (This will be discussed in 
 Section 1.2. ) 
 
 The effectiveness of a particular digital division algorithm (and the 
 redundant equipment it may require) in reducing the average partial remainder, 
 R., can be measured if the probability distribution (or density) functions 
 
 Hereafter, R. will be used to denote the magnitude of the partial remainder 
 occurring on the ith step of a division process. 
 
 -1- 
 
-2- 
 
 which describe R. are known. In Section 1.3 it will be shown, using the theory 
 of finite Markov chains, that for a particular divisor, D, the distribution 
 describing R. converges with increasing i toward a unique steady-state dis- 
 tribution. Characterizing digital division as a discrete Markov process is due 
 to C. V. Freiman (see Ref. 4). As Freiman discovered, however, the 
 complicated behavior of the steady-state distribution of R., for certain ranges 
 of the divisor, is not amenable to investigation using Markov chain techniques 
 alone „ 
 
 The purpose of the dissertation is to derive a method for more readily 
 finding the steady-state distribution of R., assuming the S-R-T division 
 algorithm is being used, and to apply this method in examining the function's 
 behavior in those ranges of D where detailed analysis was previously impractic- 
 able. Other than satisfying a curiosity, the worth of this information has been 
 
 2 
 demonstrated by Professor Gemot Metze who, using properties indicated in 
 
 previous efforts and proved herein, has proposed a modification to S-R-T divi- 
 sion which yields, for all values of the divisor, D, the maximum shift average 
 (i.e., maximum number of quotient digits per use of adder) obtainable under the 
 constraint that the available divisor multiples are +D, 0, and -D. 
 
 1.2 S-R-T Division 
 
 Descriptions of conventional restoring and nonrestoring division can 
 
 3 
 be found in various texts on digital computer arithmetic. 
 
 By common usage, the phrase "S-R-T division" connotates what may be 
 
 regarded as nonrestoring, binary division modified to utilize the facilities of 
 
 2 
 ' See Ref. No. 7. 
 
 3 
 
 J e.g., Ref. No. 9. 
 
-3- 
 
 k 5 6 
 
 a floating-point computer. Sweeney, Robertson, and Tocher independently- 
 observed that if the absolute value of the fractional portion of the divisor, 
 
 7 
 D, is restricted to the normalized range .5 < D < 1, then any steps in which 
 
 the partial remainder magnitude is found to have leading zeros can be by-passed, 
 
 and zeros inserted as the corresponding quotient digits. (After being recoded 
 
 from a representation using +1, 0, and -1 as digits to binary, the quotient 
 
 digits just mentioned will be all zeros or all ones, depending on whether the 
 
 next lower weighted nonzero quotient digit is positive or negative.) This is 
 
 equivalent to saying we will always normalize the partial remainder between 
 
 additions, and is profitable if paths to perform multiple shifts exist, or if 
 
 the time for a given step is measurably dependent on whether or not that step 
 
 requires use of the adder. Under these circumstances the average partial 
 
 remainder magnitude is of interest, since the average number of leading zeros in 
 
 R. is equal to the average number of quotient digits formed per addition in the 
 
 divison process. 
 
 To state the S-R-T division algorithm formally, the following defini- 
 te 
 
 tions will be useful: 
 
 a) P. = the partial remainder after the ith step, where 
 i = 0, 1, . . ., m-1 
 
 b) R. = |p. I 
 
 l ' i ' 
 
 c) D = divisor 
 
 h 
 
 Ref. No. 2. 
 
 ? Ref. No. 10 
 
 6 
 
 Ref. No. 11 
 
 7 
 
 Inclusion of end points is dependent on choice of number representation. 
 
 1 8 T 
 
 In each case, the fraction portion of a floating-point number is implied, 
 
-k- 
 
 d) q = the quotient digit generated by the ith step, 
 
 with possible values of +1, 0, or -1 
 
 m-1 _• 
 e ) Q = quot ient Z 2 q . 
 i=0 x 
 
 One way of stating the recursion relation is as follows: 
 
 l) If R. < ==, then P. , = 2P. 
 l 2' l+l i 
 
 q ± = 
 
 2 ) If R . > ^ and 
 1 — 2 
 
 a) the signs of P. and D agree, then P. ., = 2(P. - D) 
 
 b) the signs of P. and D disagree, the P. , =2(P. +D) and 
 
 l i+l v l 
 
 q ± = -i 
 
 Since the ranges of — < P < 1 and — < D < 1 would imply — < Q < 2, 
 corrective initial and/or terminal steps may be required to obtain a quotient 
 for which Q is a proper fraction. It can be proved by induction that the above 
 process yields the correct quotient, since: 
 
 P. . - 2(P. - q.D) (2.1) 
 
 i+l v l ^i v ' 
 
 and therefore after m steps are completed 
 
 m-1 
 
 P_ = 2" m P + D • Z q. • 2 _1 (2.2) 
 
 m . _ i 
 i=0 
 
 or 
 
 P = QD + 2 _m P (2.3) 
 
 m 
 
-5- 
 
 1.3 Division as a Markov Process 
 
 The partial remainder in a division process becomes a random variable 
 if either or both of the dividend and divisor are random variables. A useful 
 and somewhat surprising relation is that for any fixed divisor, the distribution 
 of R. is independent of the distribution of dividends, for i suff icientely large, 
 provided variation in the dividend density is bounded. 
 
 1.3=1 Existence and Uniqueness of a Stationary Distribution 
 
 In order to demonstrate that R. has a unique stationary distribution 
 
 Q 
 in the limit, Freiman observed that the division process may be characterized 
 
 as a finite, regular (or aperiodic and irreducible) Markov chain. The states 
 
 of this Markov model are intervals in the range of R. , each of which satisfies 
 
 i' 
 
 the following requirements: 
 
 a) The probability density function describing R. is uniform 
 within the interval for all i. That is, if x n and x are 
 the end points of such an interval, 
 
 Pr[x < R < x + dx] - K • dx, for x < x < x (3.l) 
 
 where K is a constant for any particular i, but may vary 
 with io 
 
 b) One such interval in the distribution of R. will map 
 
 into an integral number of intervals in the distribution 
 
 of R. . . for all i. This means no new sub intervals are 
 l+l' 
 
 required in order to insure that (a) is satisfied at 
 each step. 
 
 g 
 
 Reference k 
 
 10 
 
 Proof that such states can be generated is deferred to Section 1.3-2 
 
-6- 
 
 Denote these states (S.), j = 1, 2, . .., N, and let X(i), 
 
 J 
 
 i = 0, 1, 2, . .., represent the stochastic process which ranges over this state 
 
 space, where i corresponds to the step number, and X(p) = q denotes the event 
 
 "R is in state S ." The Markovian character of X(i) is a result of the two 
 P q . 
 
 properties of all states listed above. If X(i) = m is know to be true, the 
 
 probability that X(i + l) = n, for example, is not altered by information 
 
 regarding which state X(i - l) assumed, because, due to the uniformity of the 
 
 density function within an interval, this information would tell us nothing more 
 
 about the probability that R. is in the subset of S which maps into S . By 
 
 l m n 
 
 induction, the random variable X(i + l) is also independent of all X(p), 
 p < (i - l), as well, or: 
 
 Pr[X(i + l) = n|x(i) = m, X(i - l) = k, ..., X(o) - a] 
 
 (3.2) 
 
 = Pr[X(i + l) = n|x(i) = m] 
 
 The rather trivial observation that the division algorithm is invariant 
 from one step to the next enables further classifying the stochastic process as 
 a stationary Markov chain, i.e., conditional probabilities Pr[x(i + l) = n|X(i) = m] 
 are transition probabilities which are independent of i. The procedure for 
 finding the distribution of R. , given that of R. (or equivalently, finding the 
 distribution of the discrete random variable X(i + l), given the distribution 
 of X(i)) can be formally expressed by use of a transition matrix, P, with dimen- 
 sions NXN for N states, and which has as its row m, column n entry the transition 
 probability Pr[X(i + l) = n|x(i) = m], i.e., rows are identified with sources, 
 columns with destinations. Then if P is premultiplied by an N-entry row vector, 
 Jt(i), which describes the distribution of X(i), i.e., element Jt. (i) 
 = Pr[X(i) = m], the product is Jt(i + l). Since P is independent of i, this 
 result can be generalized to: 
 
ir(i + k) = *(i) • P k ; k = 1, 2, ... (3-3) 
 
 which expresses a k-step transition. 
 
 The desired result is that for i sufficiently large, fl(i) approaches 
 a unique final distribution which is independent of u(o). If X denotes this 
 limit distribution, then 
 
 X = lim(*(o) ■ P k ) (3.^) 
 
 k-»o 
 
 would be true for any 1 x N probability vector, rt(0). Equation (3-^) would 
 obviously be true if each row of P approached X as a limit, since the sum of 
 the elements in rt(0) equals 1. We can prove that is precisely what the limiting 
 behavior of P is, and can state as a 
 
 Theorem: If the entire interval < R. < 1 can be covered with a finite number 
 1 
 
 of intervals, each of which has the two requisite properties of a state listed 
 at the start of I.3.I, then there exists a unique vector X such that equation 
 (3»*0 is true, for any tt(0)} each row of P approaches X as a limit. 
 Proof: 
 
 1. The conclusion holds if the chain is irreducible and aperiodic (e.g., see 
 Feller, p. 356, or Kemeny, et . al., Chapt . 6). 
 
 2. Both properties are defined to apply if there exists a finite integer B, 
 
 such that for all k > B and all state pairs S , S , there is a nonzero 
 
 n 7 nr 
 
 probability of the system being in state S after k steps, having started 
 in state S . (This applies to the case n = m as well.) Equivalently, the 
 properties apply if all elements of P are positive for k > B. 
 
3. To show that B exists, we refer to equation (2.l), which, if attention is 
 confined to absolute values, is represented in Figure 1. The proof is 
 completed with four observations : 
 
 a) In the range of R., any interval of length < — which 
 does not include the points R. = .5 or R. = D ; projects 
 onto an interval twice its length in the range of R. . 
 The same comment applies to descendents, until one or 
 both of .5 and D are covered. 
 
 b) This doubling of the image with each step applies also 
 to an interval which includes R. = .5, provided 
 
 R. = (D - .5) is not also included (see Figure l). 
 When both of these points are included, clearly D is 
 included in the projected image on the range of R. , 
 to which observation (c) applies . 
 
 c) An interval in the range of R. containing D has a 
 projected image expressible as < R. < 5, with 
 5 > 0; on successive iterations the image will be 
 (0, 2 • 5), n - 1, 2, . .. until the entire range of 
 the partial remainder magnitude, (0,l), is covered. 
 
 d) The first three observations established that, having 
 started in any state (nonzero interval), the system can 
 reach any other state, i.e., irreducibility. The fact 
 that the last, single image of the sequence listed in 
 (c) covers all of (0, l) proves the existence of the 
 integer B, since this is equivalent to all elements 
 
 11 
 
 Figure 1 derives from equation (2.l), together with R. - |P. | 
 
i+l 
 
 1/2 
 
 Ri 
 
 FIGURE 1' Rj +1 vs. Rj 
 
-10- 
 
 of P being positive for some k, Clearly all higher 
 powers of P must also contain only positive elements., 
 and hence the chain is aperiodic. 
 The last observation leads to an immediate 
 
 Corollary : All elements of A, are positive. 
 
 The fact that the stationary density function is nowhere zero in the 
 unit interval will be used in Chapter 2. 
 
 1.3.2 Formulation of the Chain 
 
 First it will be shown that it is possible to generate a set of 
 intervals covering (0, l) which meet the two requirements of states listed at 
 the start of 1.3.1; for a large class of dividend distributions. With this 
 established, the conditional hypothesis of the theorem in 1„3°1 can be removed, 
 and for computational simplicity, the states may be selected on the assumption 
 that the distribution of dividends is uniform, since the final result is 
 independent of the actual distribution of dividends. 
 
 The states of the Markov chain for a given rational divisor, D, can 
 be selected as follows: 
 
 a) While otherwise arbitrary, the density function describing 
 dividend magnitudes is assumed to be sufficiently well 
 behaved that it can be approximated in measure to any 
 specified accuracy by a piecewise uniform density, with all 
 discontinuities in the approximation occurring at rational 
 numbers. (in particular, the density is everywhere finite; 
 there is no single point in the range R n to which a non- 
 zero probability is assigned. ) Denoting the point set 
 consisting of the divisor D and the arguments at which 
 
■ 11- 
 
 the discontinuities occur as -s — r (each point 
 represented in reduced fractional form), we find the 
 least common denominator. Denote this number as L. 
 b) The intervals, each L in length, defined by the 
 
 integral multiples of L , are then the states . Due 
 to the definition of L in (a), the density function for 
 dividends is uniform within any such interval. Discon- 
 tinuity arguments for the density functions of R., 
 i = 1, 2 t . .., all are either images of these initial 
 discontinuities, or of the points R = .5 or R = 1. 
 (See Figure 1; the latter two are the points in the 
 
 range of R. at which the R. n versus R. relation is 
 to l l+l l 
 
 discontinuous.) Since each such image is expressible 
 in terms of integral multiples, and sums and differences, 
 of elements in the set -{- — r s it is clear that density 
 function uniformity within the specified intervals is 
 preserved, and that the set of interval end points 
 maps only onto itself. 
 Other than showing that the hypothesis in the theorem in 1.3-1 is 
 satisfied, the above formulation reveals another interesting property of the 
 stationary density function, viz., all discontinuities are images of the points 
 .5 or 1, since the uniqueness property implies none of the discontinuities in 
 the R density and hence none of their images can influence the limiting 
 density. 
 
-12- 
 
 1.3-3 Examples 
 
 At this point two examples may be found helpful. We know that 
 successive distributions of X(i) are related by the equation: 
 
 *(i + 1) = K(i) • P (3.5) 
 
 for all i, and that for i sufficiently large: 
 
 «(i) = *(i + 1) = X (3.6) 
 
 Hence ^ the solution for a given D requires finding P, then solving the equation 
 set 
 
 \ = XP (3-7) 
 
 To determine state boundaries we generate the images of the points x = .5 and 
 
 x = 1 by application of equation (2.l) or Figure 1, stopping when repetition 
 
 begins. To determine the n,m element of the transition matrix, P, we answer 
 
 the question: Given an ensemble of partial remainder uniformly distributed in 
 
 state S , what fraction of them will be in S after one step? 
 n m 
 
 7 
 
 Example 1: D = ^ 
 
 a) Images of the point R. - — are: 
 (i) 2(D- |) -| 
 
 (ii) 2[D - (2D - l)] = 2 - 2D = - 
 
 o 
 (iii) 2[2 - 2D] = k - ^D = 2 
 
 (iv) 2[k - k-D - D] = 8 - 10D = | 
 
 Since the next application of equation (2.l) yields 
 
 k 
 
 q, which is also the first image point of 1, all 
 
 7 
 
-13- 
 
 potential locations of discontinuities have been 
 
 determined. 
 
 The states are therefore : 
 
 2 
 
 ? 
 
 D 
 
 b) The transition matrix then is 
 
 P = 
 
 .5 
 
 .5 
 
 
 
 
 
 
 
 
 
 
 
 • 25 
 
 • 75 
 
 
 
 
 
 
 
 • 5 
 
 
 
 
 .66... 
 
 .33... 
 
 
 
 
 
 
 
 
 
 l 
 
 
 
 
 
 
 
 c) Solving the equation set X = XP, we obtain: 
 
 X lA2 X l3*lA X 15 : 
 
 [£ g 1 _3 _1 1 
 L 7 7 7 lU lV 
 
 Example 2: D = ^ 
 
 a) Images of »5 are: 
 
 (i) 2(D -|) -i 
 (ii) 2(i') = | 
 
 (in) M§) = | 
 
 (iv) 2(| - D) = | 
 
 It 
 
 The first image of 1 is also — , 
 
 5 
 
 The states are 
 
 therefore: 
 
-14- 
 
 o 
 
 D 
 
 b) 
 
 P = 
 
 • 25 
 
 
 
 
 
 1 
 
 
 
 
 
 
 1 
 
 
 
 .25 
 
 U ll K 12 \l \k ] = [ 3 I 3 12 ] 
 
 1.3.4 Limitations of the Markov Chain Approach 
 
 The Markov model has been very useful both conceptually and as a 
 vehicle for proving the existence and uniqueness of a stationary probability 
 distribution for partial remainders. However, when attempting to use the 
 Markov chain to solve for this distribution, two significant obstacles are 
 encountered : 
 
 a) The size of the linear equation set to be solved is 
 
 governed by the value of the denominator of the divisor, 
 D, rather than by the complexity of the density function 
 being sought. This is a result of the definition of 
 states which requires the set of end points to be closed 
 under the recursion relation. Thus, in example 1 above, 
 we find by normalizing the probability of each interval 
 with respect to the length of the interval, the prob- 
 ability density function has only one discontinuity, 
 although five equations were required (see Figure 2). 
 
-15- 
 
 PR[x<R.< x + dx] 
 
 FIGURE 2: R. STATIONARY DENSITY FOR D = jr 
 i 9 
 
 b) Much more fundamental is the fact that a solution for a 
 given value of D may reveal little about the behavior 
 of the density function in that neighborhood of the 
 divisor range. Thus it happens that in example 1 
 if we change D to any other value in the interval 
 (j- f l), the stationary density still has only one 
 discontinuity, as will be shown in Chapter 2. However, 
 
 if D is varied even an infinitesimal distance from 
 
 3 
 D = — in example 2, the number of discontinuities in 
 
 the resulting density function may become arbitrarily 
 
 large. 
 In Chapter 2 } the equation which defines the steady-state distribution 
 is examined. By deriving from it certain properties which must characterize 
 the R. stationary density functions for all values of D, a method to systemat- 
 ically generate solutions for intervals of D, without the use of simultaneous 
 linear equations is developed. 
 
2. DERIVATION OF PARTIAL REMAINDER DENSITIES 
 
 2,1 The Defining Equation 
 
 The distribution of partial remainders at the i+1 step of an S-R-T 
 division process may be expressed in terms of the distribution at the preceding 
 step as follows : 
 
 x 
 
 f.(|) + f . (D + |) - f*(D - £), < x < 1 
 
 f i+1 (x) =-< 1, > 1 
 
 0, x < 
 
 (1.1) 
 
 where 
 
 ) f.(y) = Pr[R. < y J( for a given divisor D; 
 
 b) fj(y) =< 
 
 t ± (j), T>\ 
 
 f t (f), J<\ 
 
 (1.2) 
 
 c) | < D < 1 
 
 This relation can be demonstrated graphically as in Figure 3> where the intervals 
 
 of R. which yield an R. , less than a specified number, x, are indicated. In 
 
 l J l+l * ' ' 
 
 the limit, if distributions of successive partial remainders are equal, let 
 
 f(x) ■ lim[f .(x)] 
 i— »oo 
 
 (1.3) 
 
 Then we have equation (l.l) with subscripts deleted 
 
 -16- 
 
■17- 
 
 Ri+i 
 
 X/2 
 
 I /2 (D-X/2) (O+X/2) 
 
 Ri 
 
 1 
 
 FIGURE 3= .«x)= f(x/2)+f(D+x) 
 
 -f(D-X/2) 
 
-18- 
 
 f(|) + f (D + §) - f*(D - §), < x < 1 
 
 f(x) = "S 1, x > 
 
 0, x < 
 
 (1A) 
 
 to which (1.2) applies, again with subscripts deleted. 
 
 Because attention is restricted to piecewise uniform distributions, 
 i.e., piecewise linear distribution functions, f(x) is differentiable at all 
 but a finite number of points. Therefore, to find the density function, we can 
 take the derivative of both sides of equation (l.h) (to simplify notation, let 
 F(x) = f(x)): 
 
 where 
 
 r 
 
 F(x) = 
 
 |[F(|) + F(D +§) + F*(D - §)], 0<x <1 
 
 = < 
 
 0, otherwise 
 
 (1.5) 
 
 ^F(y), y>i 
 
 F*(y) = < 
 
 0, 
 
 y 2 
 
 Since F(x)dx has the interpretation: Pr[x < R. < x + dx], equation (1.5) is 
 expressing the probability of a partial remainder being in any differential 
 interval, say dx n , as the sum of probabilities that its predecessor was in one 
 of the (at most three) mutually exclusive differential intervals, each half as 
 
 long as dx , which map into dx». Hence we have, in terms of a function whose 
 
-19- 
 
 domain is a continuum, the relation that was expressed in terms of discrete 
 state probabilities by equation 1:(3.T)» The theory of Markov chains can be 
 applied to demonstrate that a piecewise constant density function which 
 satisfies (1.5) for a given D is unique. As in Chapter 1, generality will be 
 slightly compromised by assuming D is rational. The first theorem below is 
 instrumental in the proof of the second. 
 
 Theorem 2 . la ) : For D rational, and F(x) a piecewise constant density function 
 satisfying (l.5)> all discontinuities in F(x) occur at rational values of x. 
 Proof : 
 
 1= To construct a contradiction, assume there exists an irrational number, x~, 
 at which the conclusion is violated. 
 
 2, Since the number of discontinuities occurring at irrational arguments must 
 be the same for both sides of (1.5 )j, jumps in two different terms on the 
 right of that equation cannot cancel each other for x irrational. (As 
 demonstrated later, such mutual cancellation can occur for x rational.) 
 Hence, (1.5) implies that a second irrational discontinuity point exists 
 
 at one of 2x , 2D - 2x n , or 2x„ - 2D, depending on which of (0, — ), (— , D), 
 or (D, l) includes x~. 
 
 3. By induction, the second such discontinuity implies the existence of a 
 third, etc. The n'th term in the sequence is obviously expressible as: 
 
 x =+2 n • x^ + C (1.6) 
 
 n - 
 
 where C is some rational number (because D is rational). 
 k, Since a piecewise constant function is defined to have at most a finite 
 number of discontinuities, the sequence described by (1.6) must cycle. 
 
 In the mathematical model, representation of any partial remainder, rational 
 or irrational, is assumed possible. 
 
-20- 
 
 If x is a point in this cycle, and recurs after k steps, we could infer 
 from (1.6): 
 
 C 
 x^ = 
 
 l + 2 k 
 
 which is impossible with x irrational. 
 
 Theorem 2.1h ) : There is one, and only one, function which satisfies equation 
 (1.5) for a given rational divisor, D, under the implied restrictions of a piece- 
 
 wise uniform probability distribution: (i) / F(x)dx =1, (ii) F(x) is piece- 
 wise constant, and (iii) F(x) > for all x. 
 
 Proof : 
 
 1. At least one solution exists, since the vector representing state 
 probabilities of a Markov chain (i.e., the solution to equation 1:(3.2)), 
 can alternatively be expressed as a density which must satisfy (1.5) at 
 each point. 
 
 2. Conversely, a solution to (1.5 ) for a given divisor must be unique if the 
 
 function can alternatively be expressed in terms of state probabilities of 
 
 2 
 a finite Markov chain, i.e., if there exists a partitioning of the 
 
 remainder range such that the distribution is uniform within each interval, 
 
 and the finite partition-point set is closed under the S-R-T recursion 
 
 relation. Then, for two alleged solutions to (1.5 ) with D fixed, we could 
 
 select a single partitioning (viz., the union of two partition point sets 
 
 which apply to the solutions individually) for both functions, and thereby 
 
 construct a contradiction based on the uniqueness of regular Markov chain 
 
 state probabilities. 
 
 2 
 
 While this uniqueness extends to regular Markov chains with a denumerably 
 infinite number of states, the a priori knowledge that the number of discon- 
 tinuities is finite, gained by restricting D to the set of rational numbers, 
 is useful in the subsequent development (see Ref. 1, Chapt. 1 for properties 
 of chains with an infinite number of states). 
 
-21- 
 
 3. To select the partition point set hypothesized in 2, we observe that the 
 number of discontinuities is finite, and all occur at rational numbers 
 due to theorem 2.1a. Hence, the set \t\} n = 0, 1, . .., L, where L is the 
 least common denominator of the divisor and discontinuity arguments, is 
 always sufficient for the partition point set described in step 2 above, 
 Q.E.D. 
 
 The problem of finding the stationary density of partial remainders 
 is now reduced to solving (1.5) for F(x), as a function of D. Since no single 
 number in the range of partial remainders has a nonzero probability, the defini- 
 tion of F(x) for x = 0, 1, or any of the finite number of discontinuity arguments 
 will not be a primary concern. Moreover, D will be restricted to the open 
 interval, — < D < 1, both for expediency in derivations, and because F(x) for 
 D = — is known; to wit: F(x)=l, < x < 1. 
 
 The definition of a single step in the S-R-T division process used 
 
 3 
 here does not agree with that used by Freiman. Instead of defining a step to 
 
 be either one binary shift (i.e., if q. = 0) or one addition and shift, Freiman 
 
 defines a step to be one addition and partial remainder normalization. If g(x) 
 
 denotes the steady-state distribution of the normalized partial remainder 
 
 magnitudes, it is apparent from what computer normalization entails that: 
 
 ' 00 
 
 E [f(— ) - f(— )]; i< x < 1 
 n n 2 
 n=0 2 2 
 
 g(x) = < 
 
 (1.7) 
 
 
 x < I 
 
 from which we obtain: 
 
 3 Ref . 4. 
 
-22- 
 
 *(x) = < 
 
 io? F( ^ ^ <x<1 
 
 0, otherwise 
 
 (1.8) 
 
 Hence, information obtained using one definition can be readily compared with 
 results for which the other was used However, two advantages were found in 
 using the definition adopted here. One is the very useful symmetry which 
 characterizes all solutions to (l 5)^ as proved in Section 2=3» The other is 
 a simplification in evaluating the expectation value for the number of shifts 
 occurring between additions „ The latter point will be discussed in Chapter 3° 
 
 2o2 End Point Relations 
 
 Two properties of all solutions can be deduced by direct substitution 
 into (l„5): 
 
 F(0+) = 2F(D) 
 
 Ft! -) = 2F(1~) 
 
 (2.1) 
 (2.2) 
 
 where;, for example, 
 
 F(0+) = lim[F(0 + e)] 
 
 In (2 l), F(D) has the interpretation 
 
 | [F(D+) + F(D-)1 
 
 if D is itself the argument of a discontinuity . Both (2.l) and (2=2) depend 
 on the fact that the number of discontinuities is finite, since the limits 
 
mi 
 
 -23- 
 
 ight not exist if 0, -, D or 1 were accumlation points of discontinuities 
 
 (2.2) also relies on the fact that D = — is not being considered, 
 
 2„3 Symmetry 
 
 The orem s All solutions to equation (1.5 ) a ^e symmetric with respect to the point 
 defined by coordinates D, F(D), the symmetry extending over the interval 
 < x < 2D; i.e., 
 
 F(D _ Q) _ F(D) = F(D) - F(D + 9), < 9 < D 
 
 /l > 
 for all D in the (— , 1) interval. 
 
 Proof ; 
 
 To summarize what follows, F is extended to a function with period 
 
 2D, and it is shown that the even portion (excluding an additive constant) of 
 
 the convergent Fourier series representation of this extension must sum to zero 
 
 for all x, which, due to Fourier series uniqueness, implies each cosine term 
 
 is identically zero. The conclusion is an immediate sequel. 
 
 1. Define 
 
 r F(x), < x < 2D 
 
 H(x) = < (3.1) 
 
 v 
 
 F(X M0D.2D^ othe ™ ise 
 
 Due to the fact that F(x) is a piecewise constant function for all D and finite 
 for all x, F(x) has bounded variation. Therefore, the Dirichlet -Jordan theorem 
 applies when formulating a Fourier series representation of H(x). If we define: 
 
 h 
 
 e.g., see Zygmund; Theorem 8.1, Chapt . II < 
 
■2k- 
 
 r2D 
 b n = ij sin(^) - F(x)dx (3.2) 
 
 n =±J cos(^) • F(x)dx (3.3) 
 
 such that the series corresponding to H(x) is: 
 
 S[H(x)] = -| + S K sin O + a n cos Of (3 ° U) 
 
 n=l ^ -* 
 
 then the Dirichlet- Jordan theorem states that S[H(x)] converges uniformly 
 to H(x) within any closed interval in which H(x) has no discontinuities, 
 
 and, moreover, if x is the argument of a discontinuity 
 
 
 
 S[H(x Q )] = ||h(x +) + H(x -)| (3»5) 
 
 Therefore the notational distinction between H(x) and its series 
 representation is unnecessary if we define H(x) (and F(x)) as the average 
 of left- and right-hand limits at each jump,, 
 2. Since the period is 2D, and there is unit area under F(x), we know the 
 constant of the series : 
 
 ? - | B (3-6) 
 
 To evaluate a , n > 0, the expression for F(x) in (l„5) may be substituted 
 
 into (3.3). 
 
■25- 
 
 a = 
 
 2D, 
 
 cos(— ) 
 
 1 f2D 
 F(|)dx + ±J cos(^)F(D + |)dx 
 
 (3 = 7) 
 
 2D-1 
 
 2D, 
 
 ,nnx 
 
 
 
 cos(^)F*(D - f)dx 
 
 The limits of integration in (3° 7) are determined by the intervals through 
 ■which x may vary in the respective terms on the right of (l„5)° The upper 
 limit on the second integral may he any value between 2 - 2D and 2D 
 inclusive ^ all other limits are fixed „ The flexibility of this one limit 
 is due to H(D + — ) being zero for that interval of x ; together with the 
 fact that it was unnecessary to impose any nonlinearity on the argument of 
 the F(D + — ) term in (l„5). (The equation defining F(x) could alternatively 
 be written as ° 
 
 F(x) - |[F A (|) + F(D + §) + F B (D - |)], | < D < 1 
 
 (3.8) 
 
 where 
 
 F(y), y < | 
 
 F A (y) = i 
 
 0^ otherwise 
 
 F B (y) = { 
 
 F(y), y>§ 
 
 0y otherwise 
 
 no other restrictions on the range of x are necessary, ) 
 
 Substituting the series representation for F in the integrands of 
 (3°7);> we obtain the following expressions: 
 
-26- 
 
 .) First term on the right of (3-7 ): 
 
 p 1 °° r 
 
 1 / 1 /iwxv 1 v r / V\ flx / 
 
 2D J 2D COs( — } + 2 5, l a v [c ° s(n + 2 } ~ + C03(n 
 
 
 
 2 ; D ■ 
 
 + b v Ls-in(n + -) — + sin(n - -) -^-J f dx 
 
 (3.9) 
 
 1 . /nit. 1 1 vi 
 
 tt — sin— +y-^a r , + i — Lia 
 4Dnit v D ' 4D 2n 4it 
 
 V=l 
 
 sm(n + -)^ sin(n - ^ - 
 
 V 
 
 n + 2 
 
 V 
 
 n " 2 
 
 ri - 
 
 + b 
 
 / Vxlt 
 
 cos(n + -)- 
 
 V 
 
 n + - 
 
 1 - 
 
 / VnIT 
 
 cos(n - -)-• 
 
 V 
 
 n - g 
 
 where a prime on a summation symbol means the index, v, assumes all 
 
 positive integer values excluding 2n for terms in which (n - 
 is a factor. The same notation is used below, 
 b) Second term on the right of (3. 7) is: 
 
 Vn-1 
 
 r 2D oo r 
 
 1 / 1 /nitxv 1 v / -, \VJ 
 
 2dJ 2D COs( — > + 2 ^"^r 
 
 / VvItX / 
 
 cos(n + ^)-^r + cos(n 
 
 VvItX 
 
 2' D 
 
 . / V s-KX . , V » Jtx 
 
 b v | sin(n + -y~ + sin(n - -)— 
 
 dx 
 
 (3-10) 
 
 1 a 1 v. b fi - (-i) v . i - (-D 
 
 2 a 2n " ^ t| % D v 1 V V 
 
 V=l ^ n + - n - - 
 
 n is constant, V is the index of the series. 
 
 6 1 
 
 Trigonometric identities used are: cosA cosB = —[cos (A + B) + cos (A - B)] 
 
 cosA sinB = -[sin(A + B) + sin(A - B)] 
 
-27- 
 
 c) Third term on the right of (3. 7) 
 
 r2D-l . 00 
 
 1 / ,nrtx N 1 v / . O 
 
 2D J cos(— ) + - Z(-l) 
 
 
 
 cos(n + -)-g- + cos(n - -)— 
 
 - b 
 
 sin(n + -)-g- + sin(n - gj-jj- 
 
 dx 
 
 (3.11) 
 
 sin(^) a gn (2D - l) 1 . 
 kDnx + 4D + Iwt ~ 
 
 fsin(n + g-J^ sin(n - ^"l 
 
 . + . 
 
 n + 
 
 L J 
 
 + b 
 
 cos(n + -)- 
 
 V 
 
 n + 2 
 
 (-1) cos(n - -)- 
 
 + ■ 
 
 \V 
 
 - (-l)l 
 
 V 
 
 n - g 
 
 Summing the expressions developed in (3 = 9)^ (3<10)<, an d (3«ll)j we 
 find that (3-7) reduces to: 
 
 a = a^ 
 n 2n 
 
 (3.12) 
 
 3° As mentioned earlier, the variation per period of H(x) is known to be 
 bounded. For variation V per period of length 2D,, it can be shown that 
 
 d. v. 
 
 n ' — nD 
 
 (3.13) 
 
 Now, -by induction, (3.12 ) implies that a can be equated to any term of 
 the form: a, , k = 2- ° n, where j is an arbitrarily large integer . Hence , 
 
 (3«12) and (3.13) together imply that every |a j, n - 1, 2, 
 
 is less 
 
 e.g., see Zygmund, Theorem ^„12, Chapt. 2„ He derives the bound on a for a 
 function with period 2ft, but the modifications to his derivation are 
 straightforward . 
 
-28- 
 
 than any specified positive number. The question of summing an infinite 
 number of arbitrarily small numbers is resolved by applying Cauchy's con- 
 dition for convergence, i.e., given e > 0, there exists an integer N such 
 that all partial sums of the series after the Nth are within e of the limit 
 
 00 00 
 
 sum. Because E a cosf-7— ) < E a , and a I can be shown to be less 
 n x D — ' n" ' n 1 
 
 than e/N by use of (3.12) and (3.13), it follows that: 
 
 |Ea n cos(^)| < E (3.14) 
 
 Since e can be arbitrarily small, the cosine portion of the series must be 
 zero for all x. 
 
 For any integer n, sin(— — ) is symmetric with respect to D, and there- 
 for any sum of" terms of this form is also symmetric. Q.E.D. 
 
 The significance of being able to select 2D as the upper limit on the 
 second integral in (3«7) is now apparent. If any other limit in the allowed 
 range of [2 - 2D, 2D] were used, the expression for the unknown, a , would be 
 in terms of an infinite number of unknowns, the a and b , which, though correct, 
 would have presented a more formidable problem than equation (3. 12). 
 
 2.4 Corollaries to Symmetry 
 
 The symmetry of F(x) with respect to D will be used frequently in the 
 remainder of this chapter, as well as the next. The properties listed in this 
 section are some of the more immediate consequences of symmetry. 
 
 Theorem 2.4a : 
 
 (i) F(D) -i (4.1) 
 
 (ii) F(x) =|, 0<x<2D-l (4.2) 
 
-29- 
 (iii) F[(2D - l)-] - F[(2D - l)+] = F(l-) (4.3) 
 
 Proof 
 
 1. The first assertion follows from the fact that all terms except the 
 
 constant are zero on the right of (3°4) at x = Do The constant is given 
 by (3«6). (Again, the average of left- and right-hand limits is implied 
 if D is itself a discontinuity point . ) 
 
 2» Since, for < 9 < D, symmetry implies; 
 
 F(D - 0) - F(D) = F(D) - F(D + 9) (4.4) 
 
 and therefore, due to (4.l): 
 
 F(D - 6) + F(D + 0) = |, < < D (4.5) 
 
 equation (4.2) follows if we note that F(D + ) = if is in the 
 interval: 1 - D < 9 < D. 
 3o Equation (4.3) is a particular, but useful, example of symmetry. 
 
 From properties of solutions established up to this point, we can 
 
 o 
 
 deduce the following about F(x) for all D in the interval r < D < 1. 
 
 a) F(|) - |, due to (4.2); 
 
 b) F(l-) - i=r, due to (2.2) and (4.6a),- 
 
 dD (4.6) 
 
 c) F(D) - —, due to (4.1) 
 
 d) F[(2D- l)+] = ~, due to (4.3) and (4.6t). 
 
 8 
 
 The upper limit on 9 is merely to confine attention to the interval to which 
 symmetry applies. 
 
-30- 
 
 Because F(x) is equal to — at each end and the midpoint of the 
 interval 2D - 1 < x < 1 (for r- < D < l), the question naturally arises as to 
 whether or not solutions for all D in (r-, l) are constant over this intervals 
 This question is one of many answered by the 
 
 Theorem 2„Mj ; F(x) is monotonic for x > 0, i.e.^ there is no x > such that 
 
 F(x Q +) - F(x Q -} > (if .7) 
 
 P roof ' 
 
 (A discontinuity described by (4.7) will be termed a positive jump> 
 if the inequality is reA/ersed,, it is a negative jump, ) 
 
 1. Considered individually^ the respective terms on the right of equation (l.5| 
 would tend to yield a positive jump in F(x)., for x = x 3 if there were: 
 
 a) a positive jump at x /2 
 
 b) a positive jump at D + X n /2 3 for x < 2 - 2D, or 
 
 c) a negatiAre jump at D - x r /2 } for x_< 2D - 1„ 
 
 Due to {h,2) s the last of these is impossible „ Therefore, any positive 
 jump implies the existence of another, 
 
 2. Both la) and lb) above cannot be true for the same value of x oj) since the 
 number of positive jumps on the right in (1.5) would not then be the sum 
 of the numbers of such jumps in the individual terms , which would lead to 
 an inconsistency, since (1,5) is expressing F(x) in terms of itself. 
 
 3. Assume x. > is the argument of the largest positive jump (i.e., the 
 difference in (4.7) is maximum). The contradiction is immediate., since 
 (1.5) together with the above comments require the existence of a positive 
 jump twice as large. Hence,, there are none . 
 
•31- 
 
 2„5 Generation of Solutions 
 
 Due to equations (^.2) and (k.6), together with the theorem just 
 
 proved, we can confirm that the stationary partial remainder density functions 
 
 3 
 for all divisors in the range r- < D < 1 are : 
 
 fl 
 
 D ' 
 
 < x < 2D - 1 
 
 F(x) = -J | D, 2D - 1 < x < 1 (5.1) 
 
 0,, otherwise 
 
 For D < j- 3 we know that x = 2D - 1 is again the argument of a dis- 
 continuity,, nonzero in size,, since this jump is equal to that at x - 1, and the 
 
 latter is nonzero since F(x) is nowhere zero inside the unit interval. In 
 
 x 3 
 
 (l,3)} this discontinuity appears in the range of the F(— ) term,, for D < r-, and 
 
 cannot be cancelled Toy an equal hut opposite jump in one of the other terms in 
 that equation because F(x) is monotonic Therefore, there will be a discon- 
 tinuity at x = k-D - 2} if 2D - 1 < y-, then discontinuities exist at ^+D - 2 and 
 8D - k; etc. 
 
 Similar reasoning applies to discontinuities at points in the 
 (2D - — , l) interval. Due to symmetry, a discontinuity in the range of either 
 F(D + — ) or F*(D - — ) is cancelled by a compensating change in the other of 
 
 these terms in (1,5), provided both terms are nonzero, However, for 
 
 x 1 
 
 x > 2D - 1, i.e., T) + — > 2D - — , the F* term is zero; in this case discon- 
 tinuities in the range of the F(D + — ) term cannot be cancelled, again because 
 F(x) is monotonic. The following summarizes these observations; 
 
 Theorem 2 , 5a : A discontinuity in F(x), for x = y n , < y < - } implies the 
 existence of a discontinuity for x = 2y n , Similarly, a discontinuity for 
 
-32- 
 
 x = y , 2D - — < y < 1, implies the existence of a discontinuity at 
 x - 2y - 2D. 
 
 Another useful conclusion which follows almost directly from the 
 above observations is: 
 
 Theorem 2.^b : There is at most one pair of discontinuities in F(x) for the 
 interval — < x < 2D - — „ 
 Proof : 
 
 This restriction is simply a consequence of the fact that the number 
 of jumps on both sides of (1.5 ) must be equal for x > 0. A pair of jumps m 
 the — < x < 2D - — interval cancel each other on the right of (1.5), due to 
 symmetry. The two discontinuities in the defining equation for F(x) (see equa- 
 tion (3°8)) are each the source of one negative jump,, and thereby compensate 
 for the mutual cancellation, or "loss,," of two other jumps on the right side, 
 but no more. 
 
 Since the intersection of the domains of the three terms on the right 
 of equation (3=8) is null, while the union covers the unit interval (excluding 
 the points x = D, x - —) f each discontinuity in F(x) appears in precisely one 
 of these individual terms. Due to the two additional discontinuities intro- 
 duced by the definitions of F„ and F^ in (3.8), the number of discontinuities 
 in F'(x), x > is, in general,, two less than the sum of the numbers of discon- 
 tinuities in these individual terms. This difference can be accounted for 
 either by the mutual cancellation of jumps in the (— , 2D - — .) interval, 
 described in theorem 2.5b, or by two pairs of jumps in the individual terms 
 
 r ) 
 
 viz. : jumps at x - 2D - 1 and x = 1 due to the zero-going of F(D - — ) and 
 F(-), respectively. 
 
-33- 
 
 summing to a single pair in F(x). The stationary density function for D = -jr 
 is an example of the latter (see Figure k). Since discontinuity arguments are 
 all linear expressions in D (e.g.., 2D - 1, h - GV) } etc.), this effective coin- 
 cidence of two pairs of discontinuity arguments in general occurs only for 
 isolated values of D, which will be called the "exceptional divisors," denoted 
 
 [D„}„ To be precise, the set fD^l is defined to include all divisors to which 
 v E E 
 
 (5. 2d) below applies. 
 
 We are now in a position to find the arguments of all discontinuities 
 in a very straightforward way, A sequence {x.} is formed., using theorems 2.5a 
 and 2.5b, starting with x - 2D - 1: 
 
 a) If x. < — , then x. , = 2x. . 
 l 2. 3 l+l l 
 
 (5.2) 
 
 b ) If x . > 2D - \, then x . . - 2D - 2x . . 
 
 l 2 J l+l l 
 
 c) If — < x. < 2D - — , the sequence is complete, 
 
 2—i 2 J 
 
 d) The sequence is otherwise terminated at the ith step 
 
 if there exists an x . , j < i„ such that x„ = x. . 
 
 J J i 
 
 Elements of a set fx!}, where x! = 2D - x. , are then also discon- 
 
 ± JJ l i^ 
 
 tinuity arguments, because of symmetry. The fact that [x.) and {x!} include all 
 discontinuity arguments is evident from considerations similar to those used 
 to prove no jumps are positive: in general, any discontinuity other than the 
 pair resulting from discontinuities in the defining equation (at x = 1; 
 x = 2D - l) implies the existence of a predecessor (in the sense of the sequence 
 defined by (5.2)) at which the jump is twice as large, because of the coefficient, 
 
 10 1 
 
 x i ~ p anc * x i ~ ^ are limiting cases where two otherwise distinct 
 discontinuities become coincident. 
 
-34- 
 
 F(X) 
 
 5/3 
 5/4 
 5/6 
 5/12 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 2 
 
 r 
 
 4 .8 1 
 
 
 FIGURE 4q: SOLUTION FOR D = 3/5 
 
 1/2 F(X/2) 
 (X<1) 
 
 1/2 F(D+X/2) 
 
 FIGURE 4b 
 
 FIGURE 4c 
 
 1/2 F*(D-X/2) 
 
 FIGURE 4d 
 
-35- 
 
 -, on the right in (1.5). Since the two just cited are the only x. which do 
 not require such a predecessor, all discontinuities must he descendent from 
 these, in the sense of (5-2) 
 
 With the number of discontinuities known, the size of the respective 
 jumps can also be easily evaluated if we ignore the set of divisors, (-D^}. 
 Because F(x) is monotonic, the sum of the jumps for x > must be — (see (k.2j). 
 Moreover, the sizes of the jumps in the sequence {x.} form a geometric progres- 
 sion, i.e., the jump at x = x . -.is twice as large as the jump at x = jc. because 
 of the coefficient of — in equation (1.5 ) - Denote 
 
 a. = F(x.-) - F(x.+) (5.3) 
 
 then for T discontinuity pairs: 
 
 £ = 2 L f 2 Z a^) 1 = 1^(1 - (|) T ) (5J0 
 
 i=l i=0 
 
 or 
 
 T-2 
 2 
 
 1 (2 T - 1) 
 
 (5.5) 
 
 (The coefficient, 2, in (5-^) accounts for the fact that the {a.} are half of 
 the jumps.) Therefore, the jump at x . is: 
 
 a, = (I) 1 ' 1 ■ a = 2T " X " (5=6) 
 
 1 2 1 D(2 T - 1) 
 
 11 
 
 If the divisor is in {D }, or either of the cases x. = D or x- = p occur, the 
 predecessor requirement applies to all except one "generation of discon- 
 tinuities, the exception resulting from the fact that one jump is half the 
 sum of two others. 
 
-36- 
 
 jumps at corresponding symmetric discontinuities are,, of course, 
 equal. 
 
 The solutions derived using (5.2) and (5° 6) apply to intervals of the 
 divisor range., rather man individual values of the divisor (excluding solutions 
 for [B } ) } since there are T pairs of discontinuities in F(x) for all divisors 
 which result in x being included in the interval {— s 2D - — ) As demonstrated 
 below., there are an infinite number of such intervals-, eacn of which involves 
 a distinct sequence of operations when applying (^.,2) y though several non- 
 adjacent intervals may result In the same number of discontinuities „ Equation 
 (5«l) gives the solution for one of these intervals,, 
 
 To formulate systematically the R_. distributions for — < D < f^ 
 
 1 ^ T .3 
 
 * 2 < D < Tp 
 
 consider the intervals of D for which (— ) < (2D - l) < (— ) 9 n = 1 ; 2 9 . ..., 
 
 x> ts i - 1 /lvn+2 . ^ - 1 ,l\n+l 
 is true; denote tnese intervals of D (viz. : — + {—} < D < — + {^) 3 
 
 n = 1, 2 f . . . ) as I , n = I, 2 } . ...j respectively. To express a sequence of 
 
 discontinuities generated by use of (5-2)^ the following definitions will be 
 
 useful : 
 
 N("k.) = k consecutive normalizing operations (i.e., (5.2a)) 
 
 S = one application of (5 -2b) 
 
 (Listing tne operations required to form, the discontinuity arguments^ 
 rather than listing the arguments themselves s will enable observing similarities 
 in different intervals of D which can then be used as a basis to form infinite 
 families of solution intervals.) 
 
 Then every 1 includes a sublnterval for which the sequence of opera- 
 
 tions in (5.2) is N(n)j, i.e., the first x. > - also satisfies (5. 2c). Tn 
 
 general, I will also include a subinterval for which the discontinuity locations 
 n 
 
-37- 
 
 in F(x) are determined in (5.2) by the sequence N(n)S; another for which the 
 sequence is N(n)SN; etc. Table 1 is a list of all subintervals of I , n > 2 } 
 for which the divisor results in n + 5 discontinuity pairs or less (excluding 
 
 {D ]). The following two examples are provided to show how Table 1 was 
 
 E 
 
 derived; the solution for I appears in the first example 
 
 Example 1, Consider the sub interval of I for which the sequence in (5°2) is 
 N(n); this means the only discontinuities in F(x), for all values of D in this 
 subinterval are at: x = 2D - 1; x = 2 (2D - l); ...; x = 2 n (2D - l), 
 together with the respective 2D complements. For this portion of I , the fol- 
 lowing is true : 
 
 | < 2 n (2D - 1) < 2D - \ (5.7) 
 
 or 
 
 n+1 n+1 
 
 J-± < D < " , n = 1, 2, ... (5.8) 
 
 n+2 - n+2 ^ > > 
 
 (See Figure 5«) Note that I , .625 < D < .75> is covered completely in this 
 family of solutions. Since the number of discontinuity pairs, T, is two for 
 I (in general, T exceeds the number of steps in (5°2) by one) ? we find by use 
 of (5.6); 
 
-38- 
 
 TABLE 1. SOLUTION INTERVALS IN THE DIVISOR RANGE 
 
 2r(5o2) 
 
 Interval 
 Bounds 
 
 Examples 
 
 (to eight si 
 
 gnificant fi 
 
 gures ) 
 
 Sequence 
 
 n = 2 
 
 n = 3 
 
 n = 4 
 
 n = 5 
 
 1) N(n) 
 
 2 n+1 - 1 
 2 n+1 - 4 
 
 .5833.... 
 
 o.53571^8 
 
 .516..... 
 
 . 50806452 
 
 
 2 n+1 + 1 
 n+2 
 
 »5625 
 
 .53125 
 
 .515625 
 
 .5078125 
 
 2) N(n)S 
 
 2 n+2 - 1 
 2 n+ 3 _ 8 
 
 0625 
 
 o5535 7 l43 
 
 .525 
 
 .51209677 
 
 
 2 n+2 + 1 
 2 n+3 - 4 
 
 060714286 
 
 °55 
 
 .52419355 
 
 . 51190476 
 
 3a) N(n)SN 
 
 2 n+3 - 1 
 2 n+k - 12 
 
 59613846 
 
 .54310345 
 
 .52049180 
 
 .51 
 
 
 2 n+3 - 1 
 2 n+k - 8 
 
 . 58928571 
 
 o54l6.o.o 
 
 . 52016129 
 
 .50992603 
 
 3b) N(n)SS 
 
 2 n+3 - 1 
 2 n+k - 16 
 
 
 .5625 
 
 .52916.00 
 
 51411290 
 
 
 2 n+3 + 1 
 2 n+k - 12 
 
 ......... 
 
 . 56034483 
 
 52868852 
 
 .514 
 
 4a) N(n)SNN 
 
 2 U+k - 1 
 2 n+5 - 20 
 
 ......... 
 
 .52813559 
 
 .51829263 
 
 50896414 
 
 
 2 n+k + 1 
 2 n+5 - 16 
 
 
 .5375 
 
 51814516 
 
 .50892857 
 
 4b) NfnjSNS 
 
 2 n+k - 1 
 2 n+5 - 2k 
 
 060576923 
 
 o5474l379 
 
 .52254098 
 
 .511 
 
 
 2 n+k + 1 
 2 n+5 - 20 
 
 060185185 
 
 54661017 
 
 .52235772 
 
 51095618 
 
 4c) N(n)SSN 
 
 2 U+k - 1 
 2 n+5 - 28 
 
 
 .55701754 
 
 . 52685950 
 
 . 51305221 
 
 
 2 n+k + 1 
 2 n+ 5 . 24 
 
 000000.00 
 
 .55603448 
 
 .52663934 
 
 .513 
 
 4d) N(n)SSS 
 
 2 n+4 - 1 
 
 2 n+5 - 32 
 
 .... . 
 
 ......... 
 
 .53125 
 
 .51512097 
 
 
 2 n+4 + 1 
 2 n+5 - 28 
 
 
 ......... 
 
 .53099174 
 
 .51506024 
 
•39- 
 
 TABLE 1. SOLUTION INTERVALS IN THE DIVISOR RANGE (CONT'D) 
 
 2:(5»2) 
 
 Interval 
 Bounds 
 
 Examples 
 
 (to eight si 
 
 gnificant fj 
 
 gures ) 
 
 Sequence 
 
 n = 2 
 
 n = 3 
 
 n = 4 
 
 n = 5 
 
 5a) N(n)SNNN 
 
 2 n+5 - 1 
 
 2 n+6 - 36 
 
 
 
 .51720648 
 
 1 . 50844930 
 
 
 2 n+5 + 1 
 2 n+6 - 32 
 
 ......... 
 
 
 .51713710 
 
 .50843254 
 i 
 
 5b) N(n)SNNS 
 
 2 n+5 - 1 
 2 n+6 - ^0 
 
 . 58796296 
 
 . 54025424 
 
 . 51930894 
 
 .50946215 
 
 
 2 n+5 + 1 
 
 2 n+6 - 36 
 
 . 58636364 
 
 o53991597 
 
 .51923077 
 
 o 5094433^ 
 
 5c) N(n)SNSN 
 
 2 n+5 - 1 
 2 n+6 - 44 
 
 059905660 
 
 . 54487179 
 
 .52142857 
 
 ; o 51047904 
 1 
 
 
 2 n+5 + 1 
 2 n+6 - 40 
 
 .5972,. ,o 
 
 „ 54449153 
 
 .52134146 
 
 j .51045817 
 
 t 
 
 5d) N(n)SNSS 
 
 2 n+5 - 1 
 2 n+6 - U8 
 
 
 . 5495689 T 
 
 o52356557 
 
 | ^5115 
 
 
 2 n+5 + 1 
 2 n+6 - 44 
 
 
 .5491^-530 
 
 . 52346939 
 
 .51147705 
 
 5e) N(n)SSNN 
 
 2 n+5 - 1 
 2 n+6 - 52 
 
 
 „ 55434783 
 
 „525720l6 
 
 1 .51252505 
 
 1 
 
 
 2 n+5 + 1 
 2 n+6 - 48 
 
 ......... 
 
 o55387931 
 
 .52561475 
 
 .5125 
 
 5f) N(n)SSNS 
 
 2 n+5 - 1 
 2 n+6 - 56 
 
 
 .55921053 
 
 .52789256 
 
 ,51355422 
 
 
 2 n+5 + 1 
 2 n+6 - 52 
 
 
 o55869565 
 
 .5277.o,. 
 
 . 51352705 
 
 5g) N(n)SSSN 
 
 2 n+5 - 1 
 2 n+6 - 60 
 
 
 
 .53008299 
 
 .51^58752 
 
 
 2 n+5 + 1 
 2 n+6 - 56 
 
 
 1 
 
 .52995868 
 
 .51^55823 
 
 5h) N(n)SSSS 
 
 Q n+5 
 2_-_l_ 
 
 2 n+6 - 64 
 
 
 — -■■ -\ 
 
 
 ,515625 
 
 
 2 n+5 + 1 
 2 n+6 - 60 
 
 
 
 ......... 
 
 ,51559356 
 

 
 -1+0- 
 
 
 
 lO 
 
 
 
 
 
 
 • 
 
 
 
 
 
 
 o 
 
 CM 
 1 
 
 q 
 
 cvl 
 
 
 V 
 
 
 
 N. 
 
 Q 
 
 
 
 
 
 
 V > 
 
 
 
 
 
 lO 
 
 Q 
 CVJ 
 
 
 
 
 
 ID 
 
 • 
 
 — 
 
 
 
 
 
 
 c 
 CVJ 
 
 
 
 
 
 lO 
 CM 
 
 
 
 
 
 
 (& 
 
 
 
 \ 
 
 
 
 
 
 
 ^ 
 
 
 
 CM 
 
 
 
 ^ 
 
 
 
 
 
 
 (0 
 
 
 
 ^ogf 
 
 
 \ 
 
 CO 
 CM 
 
 to 
 
 
 
 9T=gg^ 
 
 
 "" - ■X-cz: 
 
 ro 
 lO 
 
 
 CM 
 
 ro 
 
 V, 
 
 ro 
 ro 
 
 0> 
 
 CO 
 
 CD 
 
 to 
 
-la- 
 
 r . 
 
 1 
 
 D > 
 
 < x < 2D 
 
 — , 2D - 1 < x < min(lj-D - 2 ; 2 - 2D) 
 
 F(x) == ^ — ', min(^D - 2 } 2 - 2D) < x < max(^D - 2, 2 - 2D) (5.9) 
 
 7==: , max(>D - 2, 2 - 2D) < x < 1 
 
 \_ 
 
 0,, otherwise 
 
 For D in any given solution subinterval,, x and x' = 2D - x (and only these two 
 discontinuity arguments) may be related by any of: x > x', x' > x _, or 
 x' = x = Do This is the reason for the "max" and "min" in (5°9)° 
 
 Example 2 „ If the sequence in (5-2) is N(n)SS., the following relation is 
 satisfied by D° 
 
 ~ < 2{2 n (2D - l) - D] - D) < 2D - | 
 
 (5«io) 
 
 or 
 
 2 n+3 + 1 2 n+3 - 1 
 - — , — < D - 
 
 2 U+k - 12 
 
 2 n+k - 16 
 
 (5=11) 
 
 There are two reasons why any given sequence of operations in (5-2) 
 may not correspond to a solution interval in the range of D. 
 
 a) Some intermediate term may necessarily satisfy 
 
 — < x. < 2D - — } if the implied final term also does, 
 
 b) The sequence may not be self-consistent; e.g.,, if a 
 sequence begins with N(k)S . .,, then (2D - l) > (— ) 
 
is true, and therefore an inconsistency would exist 
 if k + 1 (or more) successive normalizations occurred 
 at any point subsequently, because 2D - 1 is the 
 smallest discontinuity argument „ 
 In the example being considered,, N(l)SS cannot correspond to a solution 
 because all values of D for which only one step is required to normalize 2D - 1 
 are part of the interval in which N(l) is the complete sequence „ N(2)SS cannot 
 yield a solution since, if the first three steps are N(2)S, it can easily be 
 shown that x, < 2D - — 3 and therefore the sequence must either terminate after 
 N(2)S, or be followed by a normalization as the next step. 
 
 For all n > 3^ equation (5.1l) correctly defines intervals of D for 
 which the discontinuities in F(x) are generated by N(n)SS in (5»2), since 
 neither of the two reasons for exclusion are violated in these areas , 
 
 To facilitate deciding whether a given sequence of operations when 
 applying (5-2) corresponds to some interval of D, there are four simple rules 
 to test for consistency. 
 
 1) In a sequence starting with N(k)S, the maximum number of successive 
 normalizations which can occur anywhere in the sequence is k. This is 
 because 2D - 1 is the smallest discontinuity argument 
 
 2) A sequence starting with N(k)S cannot also terminate with N(k). Consider 
 
 Figure 5 for a given value of D. If 2 (2D - l) > 2D - -, then any other 
 
 x. occurring in the (-) < x < (— ) interval will, after normalization, 
 
 also exceed 2D - — . again because all x. > 2D - 1„ 
 2' 1 — 
 
 The other two restrictions are simply duals of the above „ Recalling 
 that (5-2) is generating half of the total set of discontinuity arguments, it 
 may be noted the sequence of operations to generate the 2D complement set, 
 
-43- 
 
 {x! ), starting with x! = 1, is the dual (i.e., interchange N's and S's) of the 
 
 sequence to generate (x.}. To determine the bound on the number of successive 
 
 S's which may occur for D in a given interval, I , we assume D is such that 
 
 some x. is almost equal to one. Rather than examining bounds on x . ., . etc.. 
 $ J+l ' 
 
 attention may, at that point, be shifted to the 2D complement sequence x! , 
 
 J 
 
 x'. , i etc., and the limits on the number of successive normalizations in the 
 J+l 
 
 complement sequence determined. From these considerations one obtains: 
 3) A sequence starting with N(k)S can have at most k successive S's. 
 k) A sequence starting with N(k)S cannot terminate with a subsequence of k 
 successive S's. 
 
 After rejecting by inspection those sequences which are not consistent, 
 the final test for the validity of a proposed sequence of operations in (5»2) 
 is comparison of the interval of D implied with intervals corresponding to all 
 subsequences (formed by truncation) to determine whether an intermediate x. 
 was in the (— , 2D - — ) interval as well as the implied final one. 
 
 It is therefore only necessary to find the smallest value of n, say 
 n , for which a given sequence in Table 1 correctly describes a solution 
 interval, and the expression then applies for all n > n . Solutions in Table 1 
 are arranged in the order of increasing complexity, such that all subintervals 
 of I for which F(x) has k discontinuity pairs are listed before those with 
 k + 1, Moreover, by ordering those which have the same number of discontinuities 
 for I according to a binary weighting, such that S corresponds to 1 and N to 
 zero, it happens that constants in the denominators of successive upper bounds 
 (as well as successive lower bounds) form a readily discernible arithmetic 
 progression. If desired, this progression may be used to extend the table 
 indefinitely. 
 
-hk- 
 
 2.5.1 Properties of Solution Intervals 
 
 Though by no means unique, the ordering of solution intervals given 
 by Table 1 displays one of several interesting characteristics of the solutions 
 Aside from the finite number of I for which a given form of sequence does not 
 apply,, all I include : 
 
 a) one subinterval which yields n + 1 discontinuity pairs 
 in F(x),° 
 
 b) one for which there are n + 2 pairs.; 
 
 c) two for which there are n + 3 pairs ; 
 
 d) four for which there are n + k pairs; 
 
 and. in general,, 
 
 e ) 2 for which there are n + j + 2 pairs . 
 
 I, represented by the column for n = 5j is an example of this,, up to ten 
 discontinuity pairs „ 
 
 A second property which is apparent from Table 1 is that solution 
 intervals are of the form 
 
 <D<— T ^-i— - (5.12) 
 
 2 m+1 - B 2 m+1 - (B + k) 
 
 where B and m are integers „ This can be proved by observing that for an 
 arbitrary sequence of operations in (5*2),, the inequalities which determine 
 the solution interval (e.g., (5-07), (5.10)) are of the form: 
 
 | < 2' 3 (2D - 1) - kD < 2D - | (5-13) 
 
 where j and k are integers. (5.13) then yields an expression of the form of 
 (5.12). 
 
■45- 
 
 Consider now the problem of finding contiguous solution intervals, 
 rather than the successively complex but usually nonadjacent intervals of 
 Table 1. Let d < D < d represent a solution interval for which the discon- 
 tinuity arguments generated by (5.2) are y 3 y 3 „„„,, y 3 each expressed as a 
 function of D„ For D in the region immediately to the right of this interval., 
 y exceeds 2D - — 3 by definition of an interval boundary (see (5°2c))„ The only 
 difference in the first T discontinuity arguments for D = d +, as compared with 
 those for d < D < d 3 would be if some y. 3 j < T, satisfied — < y. < 2D - — 3 
 resulting in an interval of fewer discontinuitites . Otherwise stated, since 
 all y. are of the form bD - C, where b and c are positive., the only change in 
 the relations of the y, to the thresholds in (5.2a) and (5.2b) which can occur 
 by increasing D is that one (or more) of the y. cross — . If,, however, such a 
 y„, j < T does not exceed — 3 then d is an accumulation point of solution 
 intervals! To prove this 5 we note that for D = d + £ 3 and £ small but positive: 
 
 y T = (2D - |) + p • 2 T e 
 
 where — < p < 1,, and therefore 
 
 T+1 
 y T+1 = 2[y T - D] = 2D - 1 + p • 2 e 
 
 For € sufficiently small, the process defined in (5»2) can be made to approxi- 
 mate an infinite cycling for an arbitrary number of iterations since y 
 approximates y 3 and the perturbation introduced by e causes the term which 
 corresponds to y in each iteration to move further from the 2D - — threshold „ 
 To summarize, the above considerations reveal any solution interval 
 is bounded on the right by either: 
 
 a) an accumulation point of solution intervals, or 
 
 b) a solution interval yielding fewer discontinuities. 
 
-1+6- 
 
 There is never a single adjacent interval on the right which yields a greater 
 number of discontinuitites . 
 
 Next consider the relation of solution interval d < D < d to the 
 adjacent interval on the left, again denoting the terms generated by (5.2) 
 for d < D < d p as y , y , . .., y . Due to the definition of an interval 
 boundary, as D is varied from d + to d -, the y term moves out of the (— 3 
 2D - — ) interval; moreover, none of the y., j < T replace y in that interval 
 since this would imply that d < D < d is bordered on the left with an interval 
 of fewer discontinuity pairs, which contradicts what was just established above . 
 Therefore, the inequality relation of each y., j < T, with respect to the 
 (— , 2D - _-) interval is not altered as D changes from d + to d, -, which means 
 
 the first T terms generated in (5°2) for D-d - are the same as those for 
 
 1 T 
 d < D < d „ If D = d - G, with G small, then y = — - p • 2 • €, where 
 
 | < p < 1; y T+1 = 1 - P 2 T • £} y T+2 = 2 - 2D - p ' 2 T+1 £; . . . . Starting with 
 
 y , these terms may be expressed relative to the 2D complements of the first 
 
 T T+2 
 
 T terms as follows: y T+1 = y| - p • 2 e, y T+2 = y^ - p • 2 e, . .., 
 
 'prn "" </i 
 
 ?T T 11 
 
 ' - p • 2 Go Now, since y is p ° 2 G away from the (— , 2D - — ) interval 
 
 y' is as well, due to symmetry, and therefore these expressions reveal that 
 y must be inside the interval. Therefore, d < D < d , the generic solution 
 interval, is bounded on the left by an interval with twice the number of dis- 
 continuity pairs . 
 
 By induction, the results of the last three paragraphs yield the 
 
 Theorem 2.5c : Every solution interval is an element of an infinite sequence 
 adjacent intervals, each of which results in half the numbers of discontinuity 
 
 pairs in F(x) as the interval at its left. Except for the sequence beginning 
 
 3 
 with the f < D < 1 interval, the first interval in each such sequence is 
 
 bordered on the right by an accumulation point of solution intervals. 
 
-hl- 
 
 3 
 An example of such an infinite sequence begins with ■? < D < 1 ; for 
 
 which F(x) has one discontinuity pair. The next four intervals are: 
 
 a) §< D< j2 
 
 and 
 
 d) 108124 < D < ilil20^ in WhlCh there are 2j k > &J and l6 
 discontinuity pairs, respectively. 
 
 The discontinuity arguments which satisfy the — < x . < 2D - - relation for the 
 last three examples are shown in Figure 6, which illustrates the rapid con- 
 vergence of such a sequence,, It happens that all the solution intervals 
 appearing in Table 1, except those described by N(2)S i N(3)SS^ N(^)SSS. and 
 N(5)SSSS, are examples of first intervals in Infinite sequences. Due to tne 
 above theorem, it is only necessary to observe whether any of the intervals 
 resulting in fewer discontinuitites than the interval in question is adjacent, 
 to decide whether a given interval is first in such a sequence . 
 
 Parenthetically, it may be noted that equation (5ol2) provides a very 
 straightforward method for computing boundaries on successive adjacent intervals 
 in these infinite sequences . As an example, -^k = — ^o — is of the form of a 
 
 lower bound in (5.12). To express the same number in the form of an upper 
 
 17 2^ - 1 2^-1 
 
 bound. 7-5 ° — j — = — rT^r~° Then the lower bound on the same interval is 
 
 20 2^" - 1 42U 
 
 25^ + 2 
 
 Tpk — ~T} again using (5. 12). In addition, the list of operations when applying 
 
 (5«2) for each interval of D in such a sequence may be generated by application 
 
 of the following rule : 
 
-kQ- 
 
 
 
 \ 
 
 
 
 
 
 \\ 
 
 \ 
 
 
 
 
 
 
 
 
 
 CM 
 1 
 
 <D\ 
 
 
 
 
 CM 
 
 \ \ 
 
 %\ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 8i 
 
 ^t^ir 
 
 
 
 
 
 
 
 
 
 891 
 
 2£-d29( 
 
 )t>S 
 
 
 
 CM 
 
 CO 
 
 CM 
 CO 
 
 CD 
 
 ro 
 
 CO 
 
 % 
 
 or 
 
 LU 
 
 co 
 
 8 
 
 3 
 
 I- 
 
 o 
 o 
 
 u. 
 o 
 
 Q. 
 3 
 O 
 (T 
 CD 
 
 UJ 
 
 Z 
 
 o 
 
 «— 1 
 
 CO 
 
 a: 
 o 
 
 
 U- 
 
 
 Q 
 
 
 if>\ 
 
 
 >l 
 
 
 h- 
 
 
 X 
 
 CD 
 
 
 o 
 
 . . 
 
 CO 
 
 CO 
 
 
 UJ 
 
 CO 
 
 a: 
 
 ID 
 
 CM 
 
 O 
 
 IT) 
 O 
 CO 
 
 0) 
 
 00 
 
 CO 
 
-49- 
 
 If u(N,S) denotes the N-S sequence corresponding to any 
 
 given solution interval, and u(N, S) is defined to be the 
 
 same sequence with N's and S's interchanged, then the (5-l4) 
 
 solution interval to the left of the one in question is 
 
 characterized by u(N, S)Nu(N, S). 
 
 This rule is merely a restatement of an observation which led to Theorem 2.5c, 
 
 v 
 
 where it was noted that with K discontinuity pairs, the second — terms 
 generated by use of (5.2) closely approximate the 2D complements of the first 
 
 V 
 
 — . Applying it to the four intervals mentioned above, one obtains: 
 
 |<D<f : N 
 
 |J <D <|: NNS 
 
 fir < D < W NNS N SSN 
 
 65337 <Ti< 65535 
 108124 108120 
 
 As a final observation, an immediate corollary to Theorem 2.5c is 
 that all solution intervals with odd numbers of discontinuity pairs have an 
 accumulation point of discontinuitites as an upper bound, i.e., such intervals 
 are first in an infinite sequence of the type described by Theorem 2.5c 
 
 2.5.2 Comments on the Set (D_,} 
 t — 
 
 The exceptional divisors, {£>„}, defined to be those values of D for 
 which (5.2d) is satisfied, may be recognized as one of three distinctly different 
 types of limit points (or accumulation points) which would exist if Table 1 were 
 
-50- 
 
 extended indefinitely. One type of limit point is the right edge of certain 
 solution intervals; for D arbitrarily close but greater than such a boundary, 
 the sequence of operations in (5.2) appears to cycle as discussed in the 
 previous section. However, the {D } are accumulation points of solution 
 intervals on either side, unlike these interval boundary points. The third type 
 of limit point is that to which an infinite sequence of contiguous intervals 
 converges (see Theorem 2.5c). This type of accumulation point of solution 
 intervals is distinguished from the other two by the fact that the sequence of 
 operations when applying (5 = 2) does not cycle. See (5.1^-) and the examples which 
 follow it. 
 
 To generate a few subsets of [D }, consider the following examples 
 of cycling which could arise when applying (5.2): 
 
 a) x = 2D - 1; x - ^D - 2; x - 8D - k; 
 x, = ikl) - 8 = kD - 2, i.e., D = |. 
 
 b) x = 2D - 1; x = kD - 2; x = 8D - k} x^ = l6D - 8; 
 
 x - 30D - l6 = kO - 2, i.e., D = yL 
 
 The first of these is an example of the subset of {D } for which the sequence 
 
 ill 
 
 of operations is N(n)SNSN ...; n = 2, 3, k, .... The second is an example of 
 N(n)S MSM . ,.} n = 3, k, .... The values of D in the first of these subsets 
 satisfy the equation 
 
 2 n (2D - 1) = 2[2 n+1 (2D - l) - D], n = 1, 2, . . . (5.1^) 
 and therefore the subset is : 
 
 D- 3 -^— , n-1,2, ... (5.0.}) 
 
-51- 
 The values of D in the second subset are described by: 
 
 2 n (2D - 1) = 2[2 n+2 (2D - l) .- D] (5.1$) 
 
 or 
 
 D = Li^J , n = 1, 2, ... (5.17) 
 
 7 • 2 - 2 
 
 Both (5.16) and (51-7) can be shown to be proper subsets of the set in {D } 
 
 2 n - 1 
 
 where n assumes all integer values excluding powers of two. 
 
 UNIVERSITY Of 
 ILLINOIS UBR^ 
 
3. SHIFT AVERAGE 
 
 3.1 Equation for <S> 
 
 While the density of partial remainders, as a function of D, is 
 interesting in itself, its practical value lies in evaluating what fraction of 
 the total number of quotient digits produced are nonzero. This figure of 
 merit is frequently expressed in reciprocal form as the shift average, i.e., 
 the average number of binary shifts in the division process between successive 
 uses (addition or subtraction) of the divisor. The shift occurring as part of 
 a step producing a nonzero quotient digit is counted when computing the shift 
 average . Define : 
 
 S(D) = S = random variable denoting the shift count between 
 successive uses of divisor D; 
 
 <S> = expectation value of S. 
 
 Since the distribution of R is independent of the dividend (for i 
 sufficiently large), the distribution of S is also. This independence of R. 
 was based on the assumption that the probability density describing dividends 
 is piecewise constant, and finite (Section 1.3.2). Therefore, for a divisor 
 D , <S(D n )> = C means that the shift average for any ensemble of dividends 
 which satisfies the above assumption is C; it does not mean that with divisor 
 equal to D , the shift average is C for each individual dividend. 
 
 If the distribution of the shift count, S, were known, <S> could be 
 evaluated by use of the relation which defines expectation: 
 
 <S> = E m • Pr[S = m] (l.l) 
 
 m=l 
 
 -52- 
 
-53- 
 
 Pr[S = m] can be determined by converting f(x), the distribution of R. to 
 distribution of normalized partial remainders, g(x), using equation 2:(l.7)j 
 then: 
 
 Pr[S = m] = g(D + --) - g(D + — 7=-) + g(D - — ~) - g(D - — ) 
 B v m p m+l ' B v Q m+1 to v m ' 
 
 (1-2) 
 
 Alternatively, <S> can be determined without knowledge of the dis- 
 tribution of S if we make use of a rather basic property of ergodic, aperiodic 
 Markov processes: 
 
 if M. = mean recurrence time of state j, 
 
 J 
 
 and n = absolute probability of state j, (l-3) 
 
 J 
 
 then M. = (n . )" . 
 J J 
 
 Since S can be regarded as the recurrence time of the set of states 
 for which x > — (where steps of the division process correspond to units of 
 time in the terminology of a Markov process), equation (1.3) implies: 
 
 <S>" 1 = Pr[R. > |] 
 
 or 
 
 <S> - - 
 
 1 - 1<£) 
 
 Therefore, to find the shift average, it is only necessary to find 
 the area under the density function, F(x), for x > — (or x < -, whichever is 
 
 1 
 
 e.g., see Feller, Chapt. XV, Section 5. 
 
-54- 
 
 more convenient). This simplification in evaluating <S> is a by-product of 
 considering the distribution of R., rather than that of normalized partial 
 remainders . 
 
 3 • 2 Evaluation of <S(P> for j < D < 1 
 
 F(x), for r- < D < 1, is given by equation 2:(5.l), and we can easily 
 confirm Freiman's result for this region by use of (1.4): 
 
 <£> = — lj- = g^ I < S < 1 (£.1) 
 
 2D 
 
 For ■£ < D < p, F(x) is given by equation 2:(5.9)> from which we 
 obtain: 
 
 f(|) = (2D - 1)| + ^(| - 2D + 1) 
 
 = |, or (2.2) 
 
 <S> = 3, f or | < D < | 
 
 17 5 
 The next adjacent interval in the range of D is — o ^ D < tt, which appears in 
 
 Table 1 as the n = 2 entry for sequence N(n)S. Since the N(2)S sequence 
 
 involves three steps ;, there are four discontinuity pairs in F(x): 2D - 1, 
 
 4D - 2, 8D - 4, and Ikh - 8, together with the respective 2D complements. The 
 
 area under F(x), x < — } can be determined by use of equation 2: (5.6). First: 
 
 F(|) = F(D) + a T 
 
 + -L(. 1 ) = i ( 
 
 (2.3) 
 
 2D 2D V 2 T _ ' 2D V 2 T _ 1 ' 
 
-55- 
 
 where, as defined in Chapter 2, T is the number of discontinuity pairs, and 
 a T is the jump in F(x) at one of the two discontinuities in the (p, 2D - — ) 
 interval, provided D is not in the set {D ). The area in question is then 
 expressible as : 
 
 f(|) = | • P(|) + (2D - 1) &1 + (4D - 2)a 2 + (k - 6D)a 3 
 
 [1 + BD . 1 + iS^ + i4iiB] {2 .k) 
 
 D(2 T - 1) 
 
 _ 2 
 3 
 
 (Note that 2D - 1, hL - 2, and k - 6D are the only discontinuity arguments less 
 than - for D in this interval.) Therefore from (1.4): 
 
 <S> = 3, for U < D <| (2.5) 
 
 17 
 Thus far, the shift averages for all divisors in the range -~k < D < 1 
 
 have been found. In addition, it can be readily ascertained from Figure h, 
 
 using equation (1.4), that the shift average for the point D = .6 is also 3. 
 
 17 
 The conjecture that <S> = 3 holds for the interval .6 < D < ^ is a natural 
 
 consequence of these observations. However, the number of solution intervals 
 in this small interval of D is infinite, with the number of discontinuities in 
 F(x) becoming arbitrarily large in an infinite number of distinct neighborhoods, 
 
 The complex behavior of F(x) for ^ < D < 4 notwithstanding, the con- 
 jecture raised regarding <S> in this region can be proved by use of the 
 symmetry property, derived in Section 2.3, as demonstrated by the following 
 two theorems . 
 
-56- 
 
 Theorem 3«2a : There are no discontinuities in F(x) in the interval 
 
 h - 6D < x < 4D - 2, for | < D < |. 
 
 5 o 
 
 N N N N N N 
 
 I 1 I 2 I 3 I I 3 2 I 1 
 
 2D - 1 1-D to-2 J 2D-J 3D - 1 
 
 2 2 
 
 2 - 2D 
 
 FIGURE 7 
 
 Proof : 
 
 1. Let N, , N p , and N^ denote the numbers of discontinuity arguments appearing 
 in the intervals 2D-l<x<l-D, l-D<x<^D-2, and ^D - 2 < x < ^ 
 respectively. Due to symmetry, the same numbers of discontinuities, 
 respectively, appear in 3D - 1 < x < 1, 2 - 2D < x < 3D - 1, and 
 2D--<x<2-2D. See Figure "J. (Note that the ordering of the end 
 points of these intervals with respect to size remains as shown in Figure 7 
 for the interval of D in question. ) 
 
 2. For D < f, and 2 - 2D < x < 1, equation 2:(l.5) reduces to: 
 
 F(x)=|F(|) (2.6) 
 
 In words, F(x) for 2 - 2D < x < 1 is a copy of F(x) for 1 - D < x < .5 
 (though with horizontal scale doubled and vertical scale halved); therefore: 
 
 N 2 + N 3 - N 2 + H x 
 
 or 
 
 (2.7) 
 
 N 3 = N 1 
 
-57- 
 
 3. Due to equation (2.6), all N_ discontinuities in kD - 2 < x < — map into 
 8D - h < x < 1. This, together with equation (2.7), implies no discontinu- 
 ities inl-D<x<4D-2 map into 3D - 1 < x < 1. 
 
 k. Therefore, the N p discontinuities in 1 - D < x < ^D - 2 are related to 
 those in 2 - 2D < x < 3D - 1 in two respects: symmetry, and mapping via 
 equation (2.6). Due to the latter, if x denotes a discontinuity argument 
 such that 1 - D < x < 4D - 2, the jump at 2x is half as large. Attempting 
 to select the smallest (nonzero) jump among the N_ in 1 - D < x < kl) - 2 
 would therefore lead to a contradiction, since another jump, half as large, 
 would exist in 2 - 2D < x < 3D - 1, and also in the first interval due to 
 symmetry. Hence, N = 0. 
 
 5. In step 3, the upper limit on one of these symmetric discontinuity-free 
 intervals was extended from 3D - 1 to 8D - h. The symmetric mate to 
 (2 - 2D, 8D - k) is {k - 6D, UD - 2). Q.E.D. 
 
 Theorem 3-2b: <S> - 3 for | < D < |. 
 5 8 
 
 Proof ; 
 
 (The proof consists of relating areas under F(x) for certain intervals 
 of x. Recognizing expressions of the form: f(z ) - f(z ), as the area under 
 F(x), for z < x < z may be found helpful.) 
 
 1. Using equation 2: (1.4) for D < ■?: <, 
 
 f(^D - 2) - f(2D - l) = f(3D - l) - f(2D - |) + f(2D - l) - f(D - |) 
 
 (2.8) 
 
 2. Since F(x) = |, for < x < 2D - 1, 
 
 f(2D - 1) - f(D - |) = 1 - ^ (2.9a) 
 
 and 
 
-58- 
 f(2D - 1) - 2 - | (2.9b) 
 
 3- Substituting (2.9) into (2.8): 
 
 f(4D - 2) - f(3D - 1) - f(2D - |) + 3 - ^ (2.10) 
 
 U. Due to symmetry, and the fact that F(D) = — : 
 
 F(D + C) + F(D - C) - i, for any C < D (2.1l) 
 
 Therefore, the sum of areas under F(x) for two symmetrically located 
 
 W 
 intervals, each of width W, is — , i.e., 
 
 /Z Z "" z 
 
 2 {F(D + y) + F(D - y)}dy = 2 p X 
 
 Z l 
 
 provided all arguments are in the (0, 2D) interval. An application of this 
 relation is : 
 
 ] l D " I 1 
 
 f (±) - f(l - D) + f (3D - 1) - f(2D - i) = — ^ -I-535 (2 * 12) 
 
 Substituting (2.12) into (2.10): 
 
 fj(to - 2) = 4 - § - [f(|) - f(i - d)] 
 
 or 
 
 (2.13) 
 
 *(§) -> -|- [f(^D - 2) - f(l - D)] 
 
-59- 
 
 5. From Theorem 3«2a: 
 
 F(4D - 2-) = F(l - D) (2.l4) 
 
 Using 2: (1.5) for D < i and x = 4D - 2- : 
 
 F(4D - 2-) = |[F(2D - 1-) + F(3D - 1-)] 
 
 = ^ + |F(3D- 1-) 
 
 (2.15) 
 
 From equation (2.11):: 
 
 F(3D - 1) + F(l - D) = i (2.16) 
 
 After substitutions, the last three equations yield: 
 
 F(4D - 2-) = ~ (2.17) 
 
 6. The area under F(x), l-D<x<4D-2, can now be evaluated, using 
 (2.14) and (2.17): 
 
 f(kV - 2) - f(l - D) = ^[k-D - 2 - (1 - D)] = f - | (2.18) 
 
 Using this result in (2.13), we obtain: 
 
 f(|) = | (2.19) 
 
 which produces the desired relation in (1.4). Q.E.D. 
 
■6o- 
 
 The fact that the shift average is equal to 3 for all divisors 
 
 3 3 
 
 in — < D < r- was indicated by use of simulation studies by Freiman, though 
 
 17 3 
 proved only for the subinterval ^h < D < t-. This result has been used by 
 
 Professor G. Metze in proposing a modification to S-R-T division. Briefly, the 
 Metze form of S-R-T division uses a variable threshold value, rather than the 
 constant -^ } to determine whether a given step involves an addition (or subtrac- 
 tion) of the divisor. The threshold is selected at the start of any particular 
 division, based on an inspection of the leading digits of the divisor; if K 
 denotes the threshold, then the relation to be satisfied in order to maximize 
 the shift average is : 
 
 | K < D < | K 
 
 or 
 
 2 5 
 |D<K<7D 
 
 3 6 
 
 3.3 <S(D)> f or \ < D < | 
 
 For D > .6, <S(D)> has been found to be a relatively straightforward 
 function. However, in the interval — < D < .6, the complex behavior of the 
 partial remainder density is reflected in <S(D)> as well. 
 
 To evaluate <S(D)> for the families of solution intervals for which 
 F(x) was determined in Chapter 2, equation 2: (5.6) will be used to find f(^). 
 Due to symmetry, together with Theorem 2.5b, one of each pair of discontinuity 
 arguments in less than — , with the exception of the one pair in (— , 2D - — ). 
 
 I f x * denotes which of the pair of symmetric discontinuity arguments, x^ x ^t 
 is less than — , then: 
 
-6l- 
 
 f(|) ■ \ F(|) + Z a.x* (3.1) 
 
 i=l 
 
 where F(p) is given by equation (2.3). As an example, consider the infinite 
 set of intervals in the range of D for which the sequence of operations in 
 2: (5.2) was N(n), n = 1, 2, . ..; viz.: 
 
 n+1 n+1 _ 
 
 5-= < D < - — = = 
 
 2 n+2 2 n+2 _ k 
 
 Using (3«l) and 2: (5.6), and observing that T = n + 1 in this case, we find 
 that: 
 
 1 P n_1 
 
 f (o) " £\ [1 + n (2D - 1)], n = 1, 2, ... (3.2) 
 
 2 D(2 n+1 - 1) 
 
 for this family of intervals. As a second example, consider the family of 
 intervals corresponding to the operations N(n)S in 2: (5-2); T = n + 2, and the 
 expression is : 
 
 '<!> - n + f [1 + "< 2D " ^ + 2D ~ ^^ " ^ (3.3a) 
 2 D(2 n+2 - 1) 2 n 
 
 = —TZr [1 + (n - 1)(2D - 1) + -2-] (3.3b) 
 
 D(2 n+2 - 1) 2 n 1 
 
 In (3.3a), the term n(2D - l) results from the n multiples of 2D - 1 (i.e., 
 2 J (2D - l), j = 0, 1, ..., n - l) which are less than -r. Since the sequence 
 terminates after one subtraction, the discontinuity argument 2 (2D - l) must 
 exceed 2D - — , and therefore its 2D complement must be less than — , and is 
 represented by the last term in (3. 3a). Table 2 is a list of the expressions 
 
-62- 
 
 TABLE 2. EXPRESSIONS FOR f(|) 
 
 2: (5.2) Sequence 
 
 f(|) 
 
 1) N(n) 
 
 n = 2, 3, ... 
 
 f(|) " —|li [1 + n(2D - 1)] 
 
 2 D(2 n+1 - 1) 
 
 2) N(n)S 
 
 n = 2, 3, ... 
 
 *& = ~ nC El + (n - D(2D - 1) + -2r] 
 
 2 D(2 n+2 - 1) 2 n a 
 
 3a) N(n)SN 
 
 n - 2, 3, ... 
 
 1 P n+1 T) 
 
 f 2 = -7w5 [1 + n(2D " 1} + \ ] 
 
 2 D(2 n+3 - 1) 2 n 
 
 3b) N(n)SS 
 
 n - 3, ^ ... 
 
 t( & = — C [1 +■ (n - 2)(2D - 1) + --?-] 
 
 2 D(2 n+3 - 1) 2 n 2 
 
 4a) N(n)SNN 
 n * 3, fc, 
 
 *(i) ■ 
 
 ,n+2 
 
 >2' ^n + "5 " [1 + (n + 1)(2D " 1)] 
 
 D(2 ur " - 1) 
 
 4b) N(n)SNS 
 n - 2, 3, 
 
 n+2 
 
 f r i) = 2 
 
 — [1 + (n - 1)(2D - l)+—i-« D] 
 D(2 X ^ - 1) 2 n+1 
 
 4c) N(n)SSN 
 n - 3, fc, 
 
 '(J) ■ 
 
 ,n+2 
 
 N n+ 
 
 D(2 U " - 1) 
 
 [1 + (n - 1)(2D - 1) + -^J «D] 
 
 4d) N(n)SSS 
 n - *, 5, 
 
 n+2 - 
 
 f (|) = ha [I + (» - 3) (2D - i) + -=s -D] 
 
 2 D(2 n+4 - 1) 2 n X 
 
-63- 
 
 for f(p) for all subintervals of I , n = 2, 3; . .., in which there are n + h 
 discontinuity pairs or less; the interval of D to which each applies is given 
 by Table 1. 
 
 In Table 3> expressions derived for Table 2 are used to evaluate 
 <S(D)>, using equation (1.4), and with n = 2, 3> 4, 5- One interesting 
 property of <S(D)> in this interval is the appearance of "plateaus" for certain 
 intervals of D (e.g., N(2)S, N(3)SS, N(4)SSS, ...). The characteristic which 
 identifies plateau regions is evident from examination of expressions for f (p); 
 any sequence which involves a total of m normalizations and m subtractions 
 yields an expression of the form: 
 
 T-2 
 
 f(|) = % [1 + (m. - m )(2D - l) + Q • D] (3-4) 
 
 2 D(2 T - 1) 1 2 
 
 where 9 is a constant. (Equation (3«l) can be used to confirm readily that 
 
 (3.4) is true for the general case: 
 
 a) If the step required to generate x , given x , is a 
 
 subtraction, then x > 2D - — and the a * x* product 
 
 K 2 K K 
 
 in (3.l) is of the form 
 JC-1 
 
 2D - 2 (2D - 1) + ... 
 2 K-1 
 
 2 T-2 
 
 D(2 T - 1) 
 
 b) Analogously, if the kth step is a normalization, then 
 
 x must be less than — , and the a * x.* product 
 K 2 K K 
 
 contributes a positive multiple of 2D - 1. ) 
 Therefore, when m exceeds m by (exactly) one, (3-^) yields a constant. 
 Conversely, f (—), and therefore <S(D)>, can be constant over an interval of D 
 only when m - m = 1. Since nothing has been assumed regarding the value of 
 
■6k- 
 
 TABLE 3- EXAMPLE EXPRESSIONS FOR <S>, USING TABLE 2 AND EQUATION 3: (1.4) 
 
 Sequence 
 
 <S> 
 
 in 2:(5.2) 
 
 n = 2 
 
 n = 3 
 
 n = k 
 
 n = 5 
 
 l) N(n) 
 
 7D 
 2 - D 
 
 15D 
 8 - 9D 
 
 31D 
 2k - 33D 
 
 6 3D 
 6k - 97D 
 
 2) N(n)S 
 
 3 
 
 31D 
 
 8 - 3D 
 
 63D 
 32 - 35D 
 
 127D 
 
 96 - 131D 
 
 3a) N(n)SN 
 
 31D 
 8 - 3D 
 
 63D 
 32 - 35D 
 
 127D 
 96 - 131D 
 
 255D 
 
 256 - 387D 
 
 3b) N(n)SS 
 
 
 63 
 23 
 
 12 7D 
 32 - 9D 
 
 255D 
 
 128 - 137D 
 
 4a) N(n)SNN 
 
 
 12 7D 
 
 96 - 129D 
 
 255D 
 
 511D 
 
 256 - 385D 
 
 6UO - 1025D 
 
 4b) N(n)SNS 
 
 3 
 
 127D 
 32 - IID 
 
 255D 
 
 511D 
 
 128 - 139D 
 
 384 - 523D 
 
 kc) N(n)SSN 
 
 
 127D 
 32 - IID 
 
 255D 
 
 511D 
 
 128 - 139D 
 
 384 - 523D 
 
 4d) N(n)SSS 
 
 
 
 255 
 103 
 
 511D 
 
 
 128 - 25D 
 
-65- 
 
 3 3 
 D, all solution subintervals in the — < D < t- region must correspond to 
 
 sequences for which this difference is unity. A second consequence is that 
 
 F(x) must have an even number of discontinuity pairs for intervals of D in which 
 
 <S(D)> is constant, since the number of operations in 2: (5 .2) is odd if 
 
 m i " m 2 = lo 
 
 In principle, Table 3 could be extended indefinitely to obtain 
 
 algebraic expressions for <S(d)> for as great a fraction of the „5 < D < .6 
 
 interval as we desire, although the size of solution intervals decreases rapidly 
 
 with larger numbers of discontinuities and are particularly minute in the lower 
 
 portion of this interval of D„ (See Table 1 for n = 3, ^ and %) Alternatively, 
 
 by evaluating <S(D)> numerically, for a sufficiently fine grid of points in 
 
 this interval, a graphical representation can be obtained to within any desired 
 
 accuracy,, 
 
 Figure 8 was obtained with the aid of ILLIAC II, using increments of 
 
 .001 for .5 < D < 060 The discontinuity arguments, x*, were determined for 
 
 i < 30o By rewriting equation (3»l) as: 
 
 1 T-2 r T-2 r t . 
 
 2 D(2 T - 1) L i=0 L 2 1+1 
 
 we observe that the maximum error in f (^) which can occur as a result of 
 
 -30 
 T < 30, where, in fact, T may be infinite, is of the order of 2 
 
 Using increments of 10 and the trapezoidal rule, a numerical 
 
 approximation to the integral 
 
 / <S(D)> dD 
 
 (3.5) 
 
■66- 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 — 1 
 
 1 
 
 1 > 
 
 o 
 
 CO 
 
 0> 
 
 ID 
 
 in 
 
 to 
 
 to 
 in 
 
 in 
 
 ro 
 in 
 
 CM 
 
 m 
 
 in 
 
 m 
 
 o 
 
 o> 
 
 CO 
 
 h- 
 
 CD 
 
 lO 
 
 *- 
 
 ro 
 
 CM 
 
 *H 
 
 O 
 
 ro 
 
 CM 
 
 CVJ 
 
 OJ 
 
 CM 
 
 CM 
 
 CM 
 
 CM 
 
 CM 
 
 CM 
 
 CM 
 
-67- 
 
 was found to be 2.61917 + 5 x 10 . This corresponds to the average value of 
 <S> for D in this range, if all values of D are assumed equally likely. Using 
 this result, together with 3:(lA) and theorem 3-2b, the average <S> for the 
 entire divisor range is 2.6170 for S-R-T division. 
 
 Because of the short computation time required per point (less than 
 a millisecond), it was found practical to investigate in detail the behavior 
 of <S(D)> for certain portions of the divisor range. The following are listed 
 as empirical results : 
 
 a) The left limit of the plateau region in <S> which has 
 
 9 ikk 
 
 D = -rz as the right end point is D = 37^. 
 
 b) The above plateau is a local minimum in <S>. Moving 
 
 Ikk 
 to the left of D = 7^j } <S> has numerous local maxima, 
 
 each of which is a plateau region. The limiting 
 
 2k 96 
 
 maxima is a point, viz., D = r-r-, at which <S> = ^=-. 
 
 While the difference of (approximately) 0.01 in the 
 
 2k ikk 9 
 
 values of <S> at D = r-r compared to the -^p= < D < y^ 
 
 plateau may have little practical significance, this 
 
 behavior is sufficiently contrary to intuition to 
 
 warrant future investigation. Noteworthy is the much 
 
 larger peak which Freiman's simulation of the so-called 
 
 9 
 . 75-1 • 0-1. 5 method yields for D near y£. 
 
 c) The nonmonotomic behavior of <S> described in (b) 
 
 17 
 occurs again near D = — , and a plausible conjecture 
 
 is that it will be found near all D = — + (p) , 
 
 n = k, 5> ..., due to the similarities in these 
 
 neighborhoods noted in Chapter 2. 
 
h. SUMMARY AND CONCLUSIONS 
 
 4.1 Summary 
 
 Treating digital division as a Markov process in order to provide a 
 basis for evaluating and comparing division algorithms was the contribution of 
 Dr. C. V. Freiman. If the set of dividends encountered is assumed to have some 
 degree of randomness, so that it may be approximated by a piecewise uniform 
 probability distribution, then, as demonstrated in Chapter 1, the distribution 
 function describing partial remainder magnitudes approaches a limit which is 
 determined only by the divisor, D, and in particular, is independent of both the 
 dividend distribution and the step numbers in the decision process. This stationary 
 distribution is itself piecewise uniform, and the intervals of uniformity are 
 the states of the Markov chain. However, a straightforward application of the 
 theory of finite Markov chains in the case of S-R-T division was found imprac- 
 ticable, for the following reasons : 
 
 a) For S-R-T division, the number of discontinuities in 
 the partial remainder stationary density function 
 becomes indefinitely large in an infinite number of 
 distinct neighborhoods in the divisor range, as 
 demonstrated in Chapter 2. 
 
 b) When solving for the probabilities of the individual 
 states, or intervals, the number of unknowns in the 
 linear equation set must at least equal, and, in 
 general, will exceed the number of discontinuities 
 in the density function. 
 
 c ) Solving one set of linear equations yields the partial 
 remainder distribution for only one value of the 
 divisor. 
 
 -68- 
 
-6 9 - 
 
 As an alternative to the finite Markov chain approach, a functional 
 relation was derived which was shown to be a necessary and sufficient condition 
 for the stationary density function (2:(l.5)). By considering the Fourier 
 series representation of a periodic extension of the density function, and 
 applying the functional relation just mentioned together with the knowledge that 
 the total variation (per period) is bounded, it is proved that the density 
 functions for all values of D must satisfy an important symmetry relation 
 (Section 2.3). This symmetry is used to derive a number of other properties 
 which lead to a straightforward algorithm (2: (5.2)) for determining all discon- 
 tinuity arguments in the density function, and an equation for the size of the 
 respective jumps (2: (5.6)). The limiting distributions being sought may be 
 expressed in terms of these jumps. The solutions thereby obtained apply for 
 intervals in the range of D, and by grouping these intervals on the basis of 
 the sequence of steps used in 2: (5.2), infinite families of intervals are 
 formed. Properties of the array of solution intervals are derived by consid- 
 ering, among other things, the relations which must exist between adjacent 
 intervals . 
 
 While the partial remainder stationary density function is interesting 
 in itself, a practical application lies in evaluating the shift average, <S>, 
 which is the average number of quotient bits generated per iteration; this 
 problem is considered in Chapter 3. Since the shift average is equal to the 
 mean recurrence time of partial remainders which exceed 0.5 in magnitude, <S> 
 is found to be simply related to the partial remainder distribution function 
 (3:(l.*0). Properties derived in Chapter 2 enable proving that <S> = 3 for the 
 entire interval, 0.6 < D < 0.75^ a result which was previously not obtainable 
 due to the complex behavior of the density function for D near 0.6. For 
 0.5 < D < 0.6, the complicated behavior of the partial remainder density 
 
-70- 
 
 function is reflected in <S> (Table 3), and relations derived in Chapter 3 are 
 used to evaluate <S> with aid of ILLIAC II, for D in that interval (see 
 Figure 8). 
 
 These results, together with equation 3:(2.l), the expression for 
 
 3 
 <S> when r- < D < 1, satisfy a goal defined at the outset: determination of 
 
 the shift average for the entire range of divisors. 
 
 h.2 Conclusions and Areas for Future Investigation 
 
 In recent years there have been a number of efforts to better under- 
 stand the digital division process when more than a minimum number of divisor 
 multiples are available, and to apply this insight in formulating algorithms 
 which reduce the number of nonzero quotient digits . Usually the avenue of 
 approach is to consider bit patterns which can occur, then to select a set of 
 rules which provides best performance in each example. This is often an 
 iterative process. 
 
 Knowledge of stationary distributions of partial remainders for the 
 various classes of division algorithms provides a valuable theoretical 
 foundation in this search for optimum algorithms. Thus, for example, the shift 
 average obtained for S-R-T division is applicable to an entire class of what 
 might be called single-threshold division processes, which can be described as 
 follows : 
 
 a) the divisor multiples available are: mD, 0, and -mD, 
 where m > 1, 
 
 b) there is a positive number, K, such that K < |mD| < 2K 
 is true, and 
 
 c) K, which in the general case is a function of D, is 
 
 used as the threshold level with which each R. is 
 
 i 
 
■71- 
 
 compared to decide whether or not the divisor is to 
 
 be used in forming R. 
 
 & i+l 
 
 The abscissas in the <S(D)> versus D relation derived in Chapter 3 may be 
 regarded as — D rather than D. Metze's modification of S-R-T division uses 
 
 CIS. 
 
 this change of scale effected by varying K to take advantage of the maximum 
 which <S> was found to assume when 0.6 < D < 0.75 is true. 
 
 An important question then is: can any similar generalizations be 
 made to optimize multiple -threshold algorithm, perhaps using a property 
 analogous to the relative independence of m and K in the single-threshold case? 
 (Only the inequalities in (b ) above must be satisfied.) In other words, when 
 more than one (nonzero) multiple of the divisor magnitude is assumed to exist, 
 to what extent can the solution for a specific algorithm be generalized in 
 order to aid in formulating an optimum one? 
 
 Derivation of partial remainder distributions for any useful two- 
 threshold algorithms (MacSorley 1 s paper includes examples of such algorithms) 
 is complicated by the fact that a number of ranges of the divisor must be 
 analyzed individually, since the thresholds vary with D. However, the observa- 
 tions that a) the piecewise constant functions which describe the stationary 
 density can be formed by locating discontinuity arguments as an ordered process, 
 starting with discontinuities in the defining equation, and b) the size of 
 jumps at successive arguments in this process form a geometric progression, 
 should expedite these derivations. 
 
 In addition to considering binary division with further redundancy 
 added, there is, of course, the problem of considering higher radices. 
 
 A second area where further work is needed is the modification of the 
 <S(D)> versus D relation to account for the fact that, in parallel computers, 
 shifting R. an arbitrary number of places in one step is not possible. One 
 
-72- 
 
 approach to this problem would be that outlined by equations 3:(l.l), (1.2), 
 and (1.3)? with the index, m, in (l.l) confined to a finite range corresponding 
 to the shift paths available. 
 
 Finally, a more thorough understanding of some peculiarities in the 
 stationary distribution of R. for S-R-T division would be interesting if not 
 useful. A few of the questions which remain are: 
 
 a) Can the nonmonotonic behavior of <S(D)> for 0.5 < D < 0,6 
 be related to divisor bit configurations? The fact that 
 "bad examples," i.e., cases for which S-R-T type division 
 is clearly nonoptimal, often involve divisors of the 
 
 1 In 
 
 form — + (— ) , and that the local minima in <S(D)>, 
 which are plateau regions, include these points, indicates 
 an intuitive explanation for these minima and what 
 determined their end points may not be difficult. 
 
 b) What significance can be attached to the accumulation 
 points of solution intervals, i.e., neighborhoods of 
 D which lead to distributions of R. with arbitrarily 
 large numbers of discontinuities? 
 
BIBLIOGRAPHY 
 
 1, A. T. Bharuca-Reid, Elements of the Theory of Markov Processes and Their 
 Applicati ons ;, Chaptc P McGraw-Hill; i960 
 
 2. J. Cocke and D, W„ Sweeney, ''High Speed Arithmetic in a Parallel Device," 
 IBM internal report, February 1957 
 
 3„ W. Feller, An Introduction to Probability Theory and Its Applications , 
 second edition, Chapt „ Ik, 15 ? John Wiley and Sons; 196c 
 
 h. C. V. Freiman, "Statistical Analysis of Certain Binary Division 
 Algorithms," Proceedings of the IR E, vol. k$, No, 1, pp. 91-103; 
 January 1961 
 
 5. J. Go Kemeny, H, Mirkhil, J, L, Snell, G. L. Thompson, Finite Mathematical 
 Structure s, Chaptc 6, Prentice-Hall, Inc; 1959 
 
 6. 0, L. MacSorley, "High Speed Arithmetic in Binary Computers," Proceedings 
 of the IRE , vol, 49 ; No* 1, pp, 67-91,. January 1961 
 
 7. G. Ac Metze, "A Class of Binary Divisions Yielding Minimally Represented 
 Quotients," IRE Trans a ctions on Electronic Computers , vol, EC-11, 
 
 ppc j6l-hj December 1962. 
 
 8. Penhollow, J, 0,, ;i A Study of Arithmetic Recoding with Applications in 
 Multiplication and Division, : ' University of Illinois Digital Computer 
 Laboratory, (Doctoral Dissertation ;_, 1962 
 
 9c R, K. Richards, Arithmetic Operations in Digital Computers , D, Van Nostrand 
 Co.; 1955 
 
 10c J, E, Robertson, ! 'A New Class of Digital Division Methods," IRE Trans - 
 actions on Electronic Computer s.- vol, EC- 7, PP^ 218-222,; September 1958 
 
 11= T t D, Tocher, '''Techniques of Multiplication and Division for Automatic 
 Binary Computers, ,: Quart, J, Mechu Applied Math ,, vol, XI, pt, 3^ 
 pp, 364-38^ 1958 
 
 12, J, B. Wilson and P., S. Ledley, "An Algorithm for Rapid Binary Division," 
 IRE Transactions on Elect r onic Compute rs, vole EC-10, pp, 662-670/ 
 December 1961 
 
 13. A Zygmund, T rigonometric Serie s, vol., I, Chapt s. 1, 2, Cambridge j; 1959 
 
 -73-