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 UIUCDCS-R-72-5^2 
 
 THE STATUS OF INVESTIGATIONS INTO THE USE OF 
 CONTINUED FPACTIONS FOR COMPUTER HARDWARE 
 
 by 
 
 August, 1972 
 
 James E. Robertson* 
 
 and Kishor Trivedi* ME LiSRARY q £ tjje 
 
 SEP 12 1972 
 
 UNIVERSITY OF ILLINOIS 
 ATU r ^ '"-CHAMPAIGN 
 
UIUCDCS-R-72-5^2 
 
 THE STATUS OF INVESTIGATIONS INTO THE USE OF 
 CONTINUED FRACTIONS FOR COMPUTER HARDWARE 
 
 by 
 
 James E. Robertson* 
 and Kishor Trivedi* 
 
 August, 1972 
 
 The research on which this paper is based was supported in part 
 by the National Science Foundation under Contract GJ81J, by the Applied 
 Mathematics Department of Brookhaven National Laboratory, and by the 
 Department of Computer Science of the University of Illinois. 
 
 ^Department of Computer Science, University of Illinois, Urbana, Illinois, 
 
Abstract 
 
 The purpose of this paper is to demonstrate that representations 
 of numbers other than positional notation may lead to practical hardware 
 realizations for the digital calculation of classes of algorithms. It is 
 the authors' opinion that practicality of the use of continued products 
 has been demonstrated. This paper describes current research in the use 
 of continued fractions. Although practicality has not been demonstrated, 
 theoretical results are promising, and the results thus far are presented 
 as a case study of the difficulties which arise when use of a new 
 representation is attempted. 
 
 There appear to be three fundamental requirements for a proposed 
 representation to be useful. Existence of sets of digital values for 
 coefficients (e.g., for partial denominators and partial numerators for 
 continued fractions) and a concomitant simple procedure for conversion 
 to positional notation are necessary. Secondly, a compatible set of 
 algorithms must be found, with the set of sufficient scope to justify 
 its use. Thirdly, simple rules for selection of the coefficients must 
 be found. 
 
 For continued fractions, the first requirement necessitates the 
 use of simple binary coefficients (e.g., 1/2, l), which primarily 
 distinguishes the approach taken here from results of the past several 
 centuries of research. For the second requirement, emphasis is given to 
 the first algorithm to be found, namely, solution of a quadratic equation. 
 Finding a set of algorithms of sufficiently wide scope seems to be the 
 major problem, and at this time, only a few techniques which may lead to 
 new algorithms can be described. An example of selection rules for 
 solution of the quadratic is presented in detail. 
 
1. History and Motivation 
 
 This paper is essentially a report on research in progress. The 
 fundamental observation is that, currently, virtually all digital hardware 
 calculations are based on the use of positional notation; equivalently, on 
 weighted sums of series. Other representations of numbers exist; the 
 concern here will be with continued products and continued fractions. 
 
 The use of positional notation has been limited to addition, 
 subtraction, multiplication, division, and, to a lesser extent, square 
 and higher roots. It has been shown [1] that use of continued products 
 extends the list of implementable algorithms to the logarithm, the 
 exponential, the trigonometric and inverse trigonometric functions, as 
 well as multiply, divide, and square root. Both time of execution and 
 cost of hardware are reasonable with current technology; in comparison 
 with a conventional arithmetic unit, factors of 2 to 3 for both time and 
 cost are typical. A small read-only memory fast enough to match accumulator 
 rates is also needed. The investigation of the use of continued products 
 was originally limited to the binary case. Higher radix techniques appear 
 promising, and are being investigated [2]. Otherwise, emphasis will be 
 given here to investigations into the use of continued fractions. Results 
 to date are theoretically promising, but not yet practical in the sense of 
 hardware implementation. 
 
 There appear to be three fundamental requirements for a proposed 
 representation of numbers to be useful for implementation in hardware. 
 
These are: 
 
 1) Conversion to conventional series form (positional notation) 
 must be "both possible and simple. Implicit here is the requirement 
 that the set of possible results spans continuously (in the limit 
 of infinite precision) some permissible range of values. For 
 floating point arithmetic, it seems sufficient to require that 
 
 the ratio of the upper limit to the lower limit of the range be 
 at least two. 
 
 2) The set of algorithms should include algorithms which are 
 easily soluble for the representation of numbers employed. 
 Compatability among algorithms, in the sense of hardware sharing, 
 is also a desirable goal. 
 
 3) Since most algorithms, other than multiplication, appear to 
 require trial and error procedures in the absence of redundancy, 
 it must be possible to devise techniques such that the selection 
 of each of the successive coefficients is practical (cf., quotient 
 digit selection in division). 
 
 It should be pointed out that the use of the coefficients of a 
 representation is ephemeral, since conversion to positional notation 
 occurs in parallel with the successive steps of the algorithm. For 
 example, for a continued product, 
 
 fr (1+2-%) = (l+S-\) k -if 1 (l^- i 6 ,). 
 
 i=l 1 k 1=1 X 
 
 At any one step the (k-l)st continued product has been determined, the 
 coefficient e, is determined by the selection rules appropriate to the 
 algorithm, and the kth value of the continued product (in positional 
 notation) is calculated by adding the (k-l)st value to a shifted version 
 
of itself. The entire set of coefficients e. (i = 1, 2, . .., m) is never 
 simultaneously available. 
 
 It is difficult to generalize about the procedures necessary to 
 determine whether or not a proposed representation satisfies the requirements 
 previously discussed. For continued products and continued fractions, 
 determination of the set of coefficient values and the associated conversion 
 procedure has been relatively simple. The identification of suitable 
 algorithms appears to be by far the most difficult requirement to satisfy. 
 In retrospect, for continued products, the observation that the logarithm 
 of a continued product is the sum of the logarithms of the individual 
 terms leads to the identification of the logarithm and its inverse, the 
 exponential, as suitable algorithms. Similarly, the properties of the 
 complex exponential indicate that the trigonometric functions and their 
 inverses are identifiable as algorithms for continued product representations. 
 No such general observation is as yet apparent to the authors 1 for continued 
 fractions. Formulating selection rules appears to be very much a function 
 of the individual algorithm. 
 
2. Examples: Division Algorithms 
 
 For illustrative purposes, algorithms for division are developed 
 for positional notation, for continued products, and for continued 
 fractions. In each case, the initial assumption is that 
 
 Era 
 D " X - ° 
 
 or some variant thereof, where N is the dividend, D is the divisor, and X 
 
 is the quotient. The same procedures are then used for developing the 
 
 algorithms, except for the representation of X. For positional notation 
 
 the division algorithm in common use is developed. For continued products, 
 
 a new algorithm with many useful properties is found. The continued 
 
 fraction algorithm is obviously an exercise in futility, since the 
 
 conversion procedure requires of itself a division as its terminal step. 
 
 2.1 Positional Notation 
 
 We define the remainder at the ith step by 
 
 N - DX. s Y.j and also N - DX. , = Y. . 
 l i 7 l-l l-l 
 
 For positional notation 
 
 X. = X. . + 2~ 1 x. = i 2~ J 'x. 
 i i-l i j=1 J 
 
 Y. = N - DX. = N - DX. . - 2 -1 Dx. = Y. _ - 2 _1 Dx. 
 l l l-l l i-l l 
 
 Since the allowed range of Y. decreases by a factor of two at each 
 step, it is convenient to define a shifted remainder r. 
 
 r. =2 I., and also r. - = 2 " i. . 
 i i' i-l i-l 
 
 r. = 2 X Y. . - Dx. = 2r. . - Dx. 2.1.1 
 
 l i-l l i-l l 
 
Equation 2.1.1 is the familiar recursion for most "binary division 
 
 procedures in common use. The initial remainder Y = r = N, the dividend, 
 
 and X = 0. 
 o 
 
 2.2 Continued Products 
 
 As in the previous example, the remainder is 
 
 N - DX. = Y., and N - DX. - = Y. - 
 l i l-l l-l 
 
 For a continued product 
 
 X. = X. n (l+2 _1 x. ) = tt (l+2" J "x.) 2.2.1 
 
 l i-l v i y . - j 
 
 Y n = N - DX. = N - DX. _ - 2" X DX. -X. 
 1 l l-l l-l l 
 
 = Y. . - 2 _1 (N-Y. _)x. 
 l-l l-l l 
 
 = Y. n (l+2" 1 x. ) - 2 _1 Nx. 
 l-l l i 
 
 It is again convenient to define a shifted remainder r. . 
 
 r. = 2 X Y., and r. _ = 2 1 ~ 1 Y. , 
 l i' l-l l-l 
 
 r. = 2 X Y. n (l+2 _1 x.) - Nx. 
 l i-l v l i 
 
 = 2r. n (l+2" 1 x. ) - Nx. 2.2.2 
 
 l-l l l 
 
 The conversion procedure is 
 
 X. = X. n + 2 _1 x.X. -, with X * 1 
 1 1-1 1 i-l' o 
 
 and x. = 1, 0, 1, or x. = 0, 1. The initial remainder is Y = r ■ = N - D. 
 l 7 ' ' l oo 
 
 An alternative which simplifies the selection procedure is to let 
 I = 1 in equation 2.2.2, and compensate by letting X q = N as the initial 
 condition for the conversion procedure of equation 2.2.1. 
 
7 
 
 2.3 Continued Fractions 
 
 P. 
 
 For continued fractions, let X. = — , and define the remainder Y. as 
 
 ' i Q. i 
 
 y. = m. - dp. 
 
 Ill 
 The conversion procedure is given by the recursions 
 
 P l " «i P i-l + P l P i-2 P o " ° P l " *1 
 
 \ " «A-1 + P i\-2 Q o " X 9 1 " *1 
 
 which must he followed by a terminal division (i=m) as indicated by 
 
 P 
 
 X = 7T— . Otherwise, the conversion consists of additions and shifts if 
 m Q, ' 
 
 m 
 
 p. and q. are simple binary coefficients; e.g., 1/4, 1/2, 1, and 2. 
 We note that 
 
 2.3.1 
 
 Y. = HQ. n - DP. „ 
 i-2 i-2 i-2 
 
 Y. _ = NQ. _ - DP. . 
 l-l i-l l-l 
 
 therefore 
 
 Y. = Kfq.Q.^ + p.Q._ 2 ) 
 
 -D( q .P._ 1+ p.P._ 2 ) 
 
 Y i " Vi-1 + P i Y i-2 2 - 5 - 2 
 
 Equation 2.3.2 is derived here for illustrative purposes only. Due 
 to the obvious impracticality of the process, neither the rate of 
 convergence (i.e., decrease in range of Y. ) nor the selection procedure 
 (i.e., method of choosing q. and p.) have been studied. 
 
3. The First Quadratic 
 
 Consider the finite continued fraction with k partial numerators 
 p. and k partial denominators q. (i = 1, 2, ..., k), whose value is 
 
 W i - e *' 
 
 p p 
 k 1 
 
 \ \ + V 
 
 2 
 
 *2 + !l 
 
 q 3 + 
 
 q k 
 
 P and Q- are determined from the recursions: 
 
 p i ■ h p i-i - p ± p ±-2 l .- a » 5 > ••' k 
 
 % ■ % Q i-1 + % Q i-2 P = ° P l = ?l S' 1 \=1l 
 
 It is clear that P, and Q. can be separately and simultaneously 
 determined in two binary arithmetic units in k-1 addition times if the p. 
 and q. are chosen to be simple in the binary sense. It is convenient to 
 make the choice p. = 1 for all i; it can be shown (Section 6) that other 
 values of p. are admissable. The digit set for q. is initially assumed to 
 be two-valued, and after some investigation it was found that choice of 
 the digit set q. = [1/2, 1} yields continued fractions whose values — 
 are continuous in the limit over the interval as defined by the following 
 
 equation: 
 
 P k 
 1/2 < lim gp < 1 
 
 k-> oo K 
 
These properties indicate that a suitable continued fraction representation 
 exists, such that conversion to conventional binary can be achieved by 
 repetitive use of two binary adders in parallel, followed by a division 
 to determine the quotient P /Q . 
 
 Determination of an algorithm and the appropriate corresponding 
 computational procedure is much more difficult. The particular algorithm 
 chosen for investigation was the solution of the limited class of quadratics 
 
 x + b, x - c = (x-u)(x+v) = 3-1 
 
 such that l/2 < u < 1. The problem, specifically, is, given b and c , 
 find u (and hence v = b, + u). This algorithm was selected because of the 
 following property of infinite periodic continued fractions, of period k. 
 If the sum of the first k-1 terms is P, -i/Q^ -\ an( ^ the sum of the first 
 k terms is P,/Q, , then the quadratic coefficients b and c are b = 
 (Q, - P )/Q» , and c = P./Q. -> • The value of the infinite periodic 
 continued fraction is then u, the positive root of the quadratic. The 
 problem is then resolved specifically to the following one. Given 
 (0, - P -1 )/ < \. -1 a *id P,/Qn -, (Note that k is unknown.), find the sequence 
 of partial denominators q. (i = 1, 2, ..., k). 
 
 The first two approaches to a computational procedure were similar. 
 Given the limited information about the values of continued fractions of 
 order k-1 and order k implicit in b and c , determine the sequence of 
 partial denominators 
 
 a) q k+1 = q a , q k+2 = q^ ..., q^ = q k , or 
 
 'b) q k , q k-1 , ..-, q-L 
 
10 
 
 After extensive investigation, it appears possible to prove that either 
 of these approaches requires knowledge of the solution in order to 
 determine the q., hence these approaches were abandoned. 
 
 The third and fourth approaches were based on the observation that 
 the value u of the infinite periodic continued fraction of period k with 
 p. = 1, q. = 11/2, 1}, (i = X f 2, . .., k) is also the value u of the 
 infinite periodic continued fraction of period one with each p. = c and 
 
 each 
 
 q. = b (i = 1, 2, . .., oo). That is, 
 
 C k 
 
 u = 
 
 b. + u 
 k 
 
 Approaches three and four may therefore be considered as methods of 
 conversion from one form of infinite continued fraction to that form which 
 is easily converted to a conventional binary representation. 
 
 Computational approach number three, in successive steps, generates 
 q , q^, etc., such that 
 
 u = c. =1 =1 
 
 k 
 
 K + "v ^ + 7 k-l q l + ^-T 
 
 — 8+7 ^ 7 k-2 
 
 b k + °k k_1 k - 1 B +y 
 
 ZT-Z 6. , + P k-2 7 k-2 
 b + H k-1 s 
 
 K . \ P k-2 + 
 
 For this approach, the recursion relations are 
 
 2 ^^and 1 + qjl ^~ n+1 2 
 
 k " n "' _Qn " Vn+1 ailC Vn " 7 k-n+l ^ 
 
 for n = 1, 2, ..., with 8 = b and y = c for the first step. Clearly, 
 
 this procedure requires two divisions as well as other operations at each 
 
 step, and is unsuitable for mechanization. 
 
11 
 
 The fourth approach, in successive steps, generates partial 
 quotients q,, q„, etc., by increasing the periodicity of periodic continued 
 fractions, as follows : 
 
 u - c. =1 = 1 
 
 k 
 
 rmmm w mn 
 
 h k ¥ c k q l + C k-1 q l + X 
 
 b~TZ Vi + i % + Vg 
 
 BT+ q i + °k-l \-2 + 
 
 k 
 
 Vi + 
 
 After a considerable amount of algebra, the recursion relations can be 
 shown to be; 
 
 b k-n = %. C k-n+l " V-l C k-n+2 + b k-n+2 
 
 3.2 
 k-n ti k-n ti k-n+1 k-n+2 
 
 It should be noted that the relative simplicity of these recursions is 
 
 dependent on the fact that P Q, , - P , Q, = (-1) , which requires that 
 * n n-1 n-1 h ' 
 
 the partial numerators p. be 1. For n = 1, the recursions require that 
 
 3 
 
 b 
 
 k+1 
 
 m 0, c k+1 = 1, and a^ = 
 
12 
 
 k. Extension of the Range and Domain of Quadratic Solutions 
 
 In the previous section, the generality of the solution of the 
 quadratic of equation 3»1 is limited by the requirement that the root u 
 is representable. For the choice q.e{l/2, 1} and p. = 1, the range of u 
 is 1/2 < u < 1. Replacing x in equation 3»1 by u . = 1/2 and u max = 1, 
 the solutions are limited to the triangular "wedge in the c b plane 
 
 1/2 b n + 1 /k < c < b +1 (k-.l) 
 
 ' k ' — k — k 
 
 It will be shown in section 5 that selection procedures impose the further 
 requirement b, > 0. (^.2) The purpose of this section is to show that any 
 point in the upper half of the c , b plane (i.e., c > 0) can be mapped 
 onto a point in the region defined by conditions A-.l and k.£. 
 
 At this point, it is convenient to delineate four areas in the 
 c , b plane and relate each area to properties of the root magnitudes u and 
 
 V. 
 
 b 2 
 
 1) c, < - k. Both roots are imaginary. 
 
 h 2 T 
 
 2) - k < c < 0. Both roots are real and of the same sign. 
 
 3) c > 0, b, < (second quadrant). The roots are real and of 
 opposite sign, with u > v. 
 
 k) c > 0> b > (first quadrant). The roots are real and of 
 
 opposite sign, with v > u. 
 It is first shown that any point in the first quadrant of the c, , b 
 
 plane may be scaled to lie within a triangular wedge such that 1/2 < u < 1. 
 
 2 
 Since v = u + b, and c, = uv, it follows that c, = ub, + u , and the range 
 k k ' k k 
 
 l/2 < u < 1 is equivalent to 
 
 1/2 b k + l/k < c^ < b k + 1 (k.l) 
 
13 
 
 21 
 Multiplying equations 3-1 and k.l by 2 J (j an integer) yields 
 
 (2 J x) 2 + (2 J b k )(2 J 'x) - 2 2J c k = (If. 3) 
 
 2 3 '" 1 (2 d b. ) + 2 2(;j " l) < 2 2j a < 2 J (2 d b. ) + 2 25 (1^) 
 
 Let 2 J x = x', 2^b k = b', and 2 2j c fc = c' Then 
 
 (x' ) 2 + b. 1 x' - C = (lj-5) 
 
 2 '" 1 b.' + 2 2 ^'" 1 ^ -: C' < 2 D ' b; + 2 2 ^ 0.6) 
 
 Given cl* and b', the scaling procedure is then 
 
 a) Determine the value of j, such that equation k.6 is satisfied. 
 
 -2 i - i 
 
 b) Multiply c' and b' by 2 d and 2 , respectively, to obtain 
 
 K. K. 
 
 c and b, , which satisfy equation k.l. 
 
 c) When the root u is determined, find the positive root u' of 
 equation k.5> by scaling u in accordance with u 1 = 2 u. 
 
 Note that the scaling procedure reduces to that normally employed for 
 square roots in floating point computers, when b, = 0. 
 
 For any point c' b' in the first quadrant, an integer value of j 
 can be found such that equation k.6 is satisfied. It is therefore 
 sufficient, for the first quadrant, to solve equation 3*1 subject to the 
 constraints of equation k-.l, with b, > 0. 
 
 For the second quadrant, with b, < it is sufficient to replace 
 b, = v - u by b" = -b, = u - v. Equation 3*1 becomes 
 
 x 2 + b k x - c k = (x-v)(x+u) = 0. (k.7) 
 
 Solution of ('+.7) yields the magnitude v of the negative root. The value 
 
 of u is then u = b" + v. 
 k 
 
Ik 
 
 The solution for the case of two imaginary roots has not been 
 considered. Attempts to find a method of solution for two real roots of 
 the same sign have thus far been unsuccessful. The preceding observations, 
 however, indicate that a continued fraction solution of the quadratic can 
 be found if c > 0; i.e., if the two roots are real and are of opposite 
 sign. 
 
15 
 
 5. Selection Procedures for the First Quadratic 
 
 This section develops a selection procedure for p. and q. of the 
 algorithm of equations 3»2 for solving quadratics using continued fractions. 
 We decide to have p. = 1 for all i. Thus the problem reduces to the 
 selection of q. . 
 
 First, we must choose the set from which to pick q. ; we call this 
 a digit set of q. . We put five requirements on this digit set. 
 
 a) All elements must be of the form 2 J where j is an integer. 
 
 b) Let the range of numbers representable as infinite continued 
 fractions using this digit set be [a,b]. We require that this 
 range form a continuum between a and b. 
 
 c) The range [1/2, 1] should be a subset of the range [a, b]. 
 
 d) The cardinality of the digit set should be as low as possible. 
 
 e) It should be possible to develop a selection procedure for our 
 algorithm, with this digit set. 
 
 The set [1, 2} does not satisfy the requirement (b). The set [1, 1/2} 
 satisfies all requirements except (e). The reason for this is that, with 
 this set, every number representable as an infinite continued fraction, has 
 a unique representation. We will see later in this paper, that a certain 
 amount of redundancy in representation is necessary to satisfy the requirement 
 (e). The set [1, 1/2, 1/k) satisfies all five requirements; so now we focus 
 our attention on this digit set. 
 
 The requirement (a) is clearly satisfied. It is easily shown that 
 the range [a, b], is approximately [0.39* 1«56] with this digit set. Thus 
 the requirement (c) is satisfied. The requirement (d) is also satisfied. 
 To show that the requirement (b) is satisfied we can proceed as follows. 
 
16 
 
 First any number f e[a, b] can be expanded as a continued fraction 
 
 as follows . Let 
 
 1 
 
 , = , ■ „ and m general, let 
 
 1 q, + f to ' 
 
 1 2 
 
 1 % + f i + i 
 
 If a < f . < 1/2 then choose q. = 1. 
 
 If 1/2 < f . < 1 then choose q. = 1/2. 
 
 If 1 < f. < b then choose q. = 1/k. 
 — 1 — 1 ' 
 
 It can easily be verified that with f e[a, b] and using the above 
 
 rules, f.e[a, b] for all i. Therefore the above rules can be used for all 
 
 i > 1. We call such a method of expansion a consistent method of expansion. 
 
 By an expansion of f., to k terms is meant the fraction — , — ...,—. 
 
 1 q i q 2 q k 
 
 Next we use the following theorem, which we state without proof. 
 
 Theorem 1 : For a number f e[a, b], if there is a consistent method 
 of expansion of f.. in the form of a continued fraction, then such an 
 expansion converges to the value f as the number of terms in the expansion 
 increases, provided that the smallest element in the digit set is greater 
 than [ 5] • 
 
 Thus every number in [a, b] has an infinite continued fraction 
 expansion with the digit set {1, 1/2, 1/^} and hence the requirement (b) is 
 satisfied. 
 
 We devote the rest of this section to show that the requirement (e) 
 is satisfied. 
 
 We restrict the problem to b > 0. 
 
 C k-i 1 
 
 ; be expanded to f . = ■ — - — . 
 
 -i + u 1 Vi + f i+i 
 
 Let f . = - 
 1 \ 
 
17 
 
 Given that 0.39 < f . < 1*56, we have to find q. e [1, 1/2, 1/4} such that 
 0.39 < f. .-i < 1.56. From these, we get, 
 
 for 0.39 fa .+u) < c, . < 0.72 fa .+u); choose q. ,_ = 1. 
 k-i — k-i — k-i 1+1 
 
 for 0.485 fa .+u) < c, . < 1.124- fa .+u); choose q. ,_ = 1/2. 
 k-x — k-i — k-i 7 l+l ' 
 
 and 
 
 for 0„553 fa .+u) < c n . < 1*56 fa .+u); choose q. ,_ = 1/4. 
 k-i — k-i — k-i l+l ' 
 
 The regions where two choices are allowed are, 
 
 0.485 fa .+11) < c, . ■: 0oT2 fa .+u) then q. ,. = 1/2 or 1, 
 
 k-i — k-i — k-i l+l ' 
 
 and 
 
 0.553 fa . +u) < c. . C 1.124 (b ? .+u) then a. = l/k or 1/2. 
 
 iC — 1 j£-"l K.-1 1+1 
 
 Both these are triangular wedges in the (c, . , b, .) plane. We 
 
 k-i' k-i * 
 
 will call these the (l/2 & l) and the (1/4 & 1/2) overlap regions, 
 respectively. Clearly these wedges vary with u. To get a selection line 
 which decides between q = 1/2 or 1 and which is u-independent, (since 
 u is unknown) we should first take the intersection of all (l/2 & l) regions 
 as u varies over the range [1/2, 1] and then take a line which is completely 
 within this intersection. A similar statement can be made about the (l/4 & 1/2) 
 region but unfortunately the resulting triangular wedges are not yet wide 
 enough for cur problem. It is clear that if we let u vary over a smaller 
 range, we shall have wider overlap regions. Thus partitioning the u-range 
 into three subranges, namely, I. = [1/2, 5/8), I = [5/8, 3/4) and I = [3/4, 1] 
 
 works well. It is clear that from the given values of c and b it is simple 
 
 k k * 
 
 to determine the subrange for root u with shift, add and comparison operations 
 
18 
 
 only. For example, 
 
 c k - 1/2 b k > 1/1+ and c - 5/8 b fc < 25/61)- => u e I . 
 
 Now we ask for three selection procedures for these three subranges 
 of u. First we discuss the case of subrange I, = [l/2, 5/8). The 
 (l/2 & l) overlap region is given by, 
 
 O.i+85 (b. .+5/8) < c. . < 0.72 (b. .+1/2) 
 
 k-i ' — k-i — k-i ' 
 
 Similarly, the (l/k & l/2) overlap region is given by, 
 
 0.553 (b k-i +5/8) < c k _. < 1.12 (b k _.+l/2). 
 
 We show these regions on the (c. . , b, .) plane, in figure 5.1. 
 The upper and the lower bounds of the (l/k & l/2) region are labelled A 
 and B respectively, and those for the (l/2 & l) region are labelled C and 
 D. We also show the greatest upper bound c . =1.56 (b .+5/8) as line 
 
 H and the least lower bound c. . = 0.39 (b, .+1/2) as line L. We also 
 
 k-i k-i ' 
 
 draw two selection lines SI and S2, which are c . = h. . + 1/2 and 
 
 ' k-i k-i ' 
 
 c „ = 1/2 b . + 5/l6 respectively. Notice that the coefficients in 
 
 these lines are chosen to be "simple" binary numbers. For any point in 
 
 the region enclosed by line H and SI we choose q.. , = l/k. For any point 
 
 between S2 and L, we choose q. _ =1 and otherwise we choose q = l/2. 
 
 Notice that with these rules our choice could be erroneous in certain 
 
 regions. Where this happens, we call these regions the forbidden regions. 
 
 The quadrilateral enclosed by lines H, B, SI and L is the (l/k & 1/2) 
 
 forbidden region and the quadrilateral enclosed by lines H, S2, C and L 
 
 is the (l/2 & l) forbidden region. We have to make sure that for no 
 
 value of i, the point (c, . , b, .) lies in one of these regions. This 
 ' k-1 7 k-i 
 
o 
 
 en 
 
 i 
 
 19 
 
 l 
 
 H 
 in 
 
 •H 
 
20 
 
 we do in appendix I in the proof of convergence. A similar treatment can 
 
 be given to the other two subranges I and I . For the subrange I , the 
 
 selection lines SI and S2 are, c . = b. . + 5/8 and c. . = 1/2 b, . + 3/8 
 
 ' k-i k-i ' k-i ' k-i ' 
 
 respectively. For the subrange I,, the selection lines SI and S2 are 
 c . = b . + 3/4 and c . = l/2 b . + l/2 respectively. 
 
 Although this general selection procedure is valid for all i > 0, 
 we want to use a special procedure for i = so that when we make tests 
 for the subrange determination, we also find q on the basis of the same 
 tests. 
 
 For 1 = } f. = u. Then from our previous analysis, we have, 
 
 0.485 < u < 1.124 then a = l/2 
 
 and 
 
 0.39 < u < 0.72 then q f 1. 
 
 Thus we can choose q = 1 for all uel and q = 1/2 for all uel or 1^. 
 
 ¥e now give the complete algorithm A. 
 
 A_0: [Check] 
 
 If b, < o then exit, no solution; otherwise 
 k 
 
 If (c k -l/2b k ) < 1/4 or If (c k -b k >l) 
 
 then exit, no solution; 
 
 A_l: [Subrange] 
 
 If c - 5/8 b < 25/64 then set q <- 1, 
 
 Kl 4- 1/2, K2 *-5/l6 and go to 
 
 step A_2; otherwise set q <- 1/2; 
 
 If c - 3/4 b < 9/16 then set 
 
 Kl *- 5/8, K2 *- 3/8 and go to step A_2; 
 otherwise set 
 Kl - 3/4, K2 - 1/2; 
 
21 
 
 A_2: [Initialize] 
 
 Set P <- 0. Q *- P. <- 1, Q n <- q_ ; 
 o o 1 1 1 
 
 Set b^ - q lV c k _ x «_ 1 + q x (b k -b k _ ] _); 
 
 Set i «- 2; 
 A_5: [Selection] 
 
 If Wi > (b k- i+ i +K1) then 
 
 set q. *- 1/k and go to step Ajj-; otherwise 
 
 If c. . , < 1/2 b n . . + K2 then 
 k-i+1 — ' k-i+1 
 
 set cl «- 1 and go to step A_^; otherwise 
 
 set q^ *- 1/2; 
 A_k: [Advance] Set 
 
 Vi - Vfc-i+1 " Vl C k-i + 2 + Vi + 2> 
 
 Vi-% (b k-i + l" Vi) +C k-i + 2> 
 
 P i - % P i-l + P i-2 
 
 i «- i + 1; 
 
 A_5: [Loop Test] If i < i then go to step A_3; 
 
 A 6: [Finall u (=R00T_ ) «- P./Q., 
 
 _ L 1 l' l' 
 
 v <- b. + u; 
 k 
 
 Note: The value of i will be decided by the machine precision, 
 in case, this algorithm is implemented in hardware. If this 
 
 algorithm is implemented in software, however, the value of i 
 
 max 
 
 will be decided by the allowable error. 
 
22 
 
 6. Recent Related Work 
 
 In the preceding sections, the discovery of the first continued 
 fraction algorithm and its method of application have been described. 
 In this regard, the exposition is historically ordered. The purpose of 
 this section is to describe briefly the results of more recent research. 
 
 A study of the derivation of the quadratic algorithm of 
 equations 5*2 has indicated that the requirement that p. = 1 for all 
 i is unnecessary [3]» Equations 3*2 then become 
 
 % Vi 
 
 'k-n " p "k-n+1 p ., "k-n+2 ! k-n+2 
 ^n ^n-l 
 
 'k-n ti k-n ' ti k-n+1 p n c, 
 
 n-1 k-n+2 
 
 6.1 
 
 Selection rules for the digit sets p.e{l/2, 1} and q. e[l/2, 1} have been 
 determined. 
 
 In a companion paper[^-], it is shown that the Ricatti equation 
 
 o 
 y' + ay + by + c = 6.2 
 
 leads to relatively simple recursions if the partial numerators p. and 
 partial denominators q. of the associated continued fraction are simple in 
 the binary sense. Since both tan x and e satisfy the Ricatti equation 
 for particular choices of a, b, and c, there is some hope that useful 
 continued fraction algorithms for these functions can be found. Attempts 
 to find selection rules for tan x have thus far been unsuccessful, and 
 have not been attempted for the exponential. 
 
 The derivation of the recursion relations for the Ricatti equation 
 suggested a similar derivation for the quadratic equation, and led to a 
 second set of recursion relations for the quadratic. The special selection 
 
23 
 
 procedure for i = o described in section 5 suffices for selection rules 
 for this second quadratic algorithm. Recursion relations for higher 
 order polynomials can also be found by this method. 
 
2k 
 
 7« Conclusions 
 
 It should be emphasized that the primary purpose of this paper 
 is to point out that hardware construction can be based on representations 
 of numbers other than positional notation. It seems quite clear that 
 the use of continued products yields a useful set of algorithms which 
 can share the same hardware in a feasible and practical manner using 
 current technology. 
 
 The discussion of continued fractions presented here is a case 
 study of the problems which arise when a different representation of 
 numbers is proposed. The research on the use of continued fractions is 
 incomplete; the results obtained thus far do not justify hardware 
 construction based on continued fractions. 
 
 It seems appropriate, therefore, to conclude with a list of 
 questions for future research. These include: 
 
 1) Can the set of algorithms soluble with continued 
 products be extended? 
 
 2) How can the set of algorithms based on the use of 
 continued fractions be extended? Can feasible 
 selection rules for each algorithm be found? 
 
 3) What additional representations of numbers exist? 
 What is their potential usefulness? 
 
25 
 APPENDIX I 
 
 The Proof of Convergence of Algorithm A 
 
 To prove convergence, we only need to show that the selection 
 procedure in step A_]> of Algorithm A is a consistent method, and then using 
 theorem 1, we have the required result. 
 
 To show consistency, first notice that, 
 
 C k-i 
 
 f . = r — *-* — and a < f . < b 
 1 b. . + u — i - 
 
 k-i 
 
 for all i. Thus a (t>. . +u) < c. . < b (b. . +u) is always satisfied, and 
 
 k-i — k-i — k-i ' 
 
 we are well within the bounds H and L of figure 5*1. We also need to show 
 
 that for any value of i > 1, the point (c, ., b .) does not fall in any 
 
 one of the forbidden regions. It is necessary to treat each of the ranges 
 
 I , I and I, separately. We only treat the range I here, others being 
 
 similar. We use the following result [ 5] • 
 
 For all i > o; c . and b . satisfy one equation to the line 
 P. Q, x 
 
 (b . -b ). The line of closest approach to the forbidden 
 
 l l 
 
 k-i Q. . Q. -. k-i k 
 l-l i-I 
 
 region or alternatively, the left most line is given by 
 
 P. \ / Q. 
 
 11 ' x b. 
 
 'k-i Q. , . ft, J. k-i 
 
 1-1/ mm \ i-I /mm 
 
 since b, > o. We wish to show that this line is within the allowed region 
 
 in figure 5«1« 
 
 We have, l/k < c, < 'd^/Gh and b = o then with q = 1, the above line 
 
 K K 1 
 
 for i = 1 is c , = 1 - b , l/k < b < 25/6^. This is shown (labelled P) 
 lc— 1 K— A. k— 1 
 
 in figure A.l. Next consider i > 3 and i odd. We first get a lower bound on 
 
 P. 
 
 q— . We use a theorem [6], which states that odd ordered convergents approach 
 
 the value of an infinite continued fraction from above (and even ordered 
 
 convergents approach from below). 
 
26 
 
 H 
 
 < 
 
 CD 
 
 !h 
 
 iaJD 
 •H 
 
Thus 
 Now 
 
 
 > 
 
 i odd 
 
 min 
 uel 
 
 27 
 (u) = 1/2. 
 
 V2 + . 
 
 * 1 
 
 q l 
 
 odd 
 
 / P - \ 
 
 Thus ^ — > 9/40. 
 
 \ ' l > 3 
 
 odd 
 
 Thus the left most line (labelled OLM in figure A.l) for odd values of i, 
 
 is given by 
 
 e k _. =, 9A0 - 9/20 b k _. . 
 
 Next consider the case of even values of i > 2. It can be seen 
 
 P 2 
 (from figure A.l) that q^ = l/2 for all uel . Then ^— = 1/3- Now again 
 
 '~P. 
 
 % 
 
 using the stated theorem on convergents, | r?- 
 
 Next observe that, 
 
 > 1/3. 
 i/i > 2 
 
 even 
 
 11111 1 _. ., 
 
 > r . t . t . t . t . . T finite 
 
 L Q. . J -lt-+l + l + l + l+ +1 
 
 1-1/- T7 
 
 l even 4 
 
 1111 
 
 4+1+1+1+ infinite. 
 
 4~ 
 
 0.640388 
 
28 
 
 Thus q7 > 0.2135. 
 i > 2 
 
 even 
 
 Thus the left most line (labelled ELM in figure A.l) is given "by, 
 
 C. . = 0.2135 - 0.6k b . It is clear from figure A.l that we never 
 
 K-l k-i 
 
 transcend into the forbidden regions. Thus we have satisfied the 
 consistency requirement 
 
 As a matter of practical interest, we mention here that c, . 
 and b. . remain bounded in algorithm A. In particular bounds on c, . 
 are given by < CL . < 3«57 + 1.2 b, . 
 
 "— 1C-X ™" .K 
 
29 
 
 Bibliography 
 
 [ 1] DeLugish, Bruce G., A Class of Algorithms for Automatic 
 Evaluation of Certain Elementary Functions in a 
 Binary Computer, Ph.D. thesis, University of 
 Illinois, June 1950; also Department of Computer 
 Science Report No. 399- 
 
 [21 Ercegovac, Milos, Radix 16 Evaluation of Some Elementary 
 Functions, M.S. thesis in preparation at the 
 Department of Computer Science, University of 
 Illinois, Urbana. 
 
 [3] Bracha, Amnon, private communication. 
 
 [k] Bracha, Amnon, Application of Continued Fractions for 
 Fast Evaluation of Certain Functions on a 
 Digital Computer. 
 
 [5] Trivedi, K. S., An Algorithm for the Solution of a Quadratic 
 Equation using Continued Fractions, M.S. thesis in 
 preparation at the Department of Computer Science, 
 University of Illinois, Urbana. 
 
 [6] Wall, H., Analytic Theory of Continued Fractions, 
 Van Nostrand, New York, 1950. 
 
BLIOGRAPHIC DATA 
 IEET 
 
 Title and Subtitle 
 
 1. Report No. 
 
 UIUCDCS-R-72-51+2 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 August 1972 
 
 The Status of Investigations into the Use of 
 Continued Fractions for Computer Hardware 
 
 Date of issue 
 
 Author(s) 
 
 James E. Robertson and Kishor Trivedi 
 
 8. Performing Organization Rept. 
 No. 
 
 Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 6l801 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 NSF GJ 813 
 
 Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, D. C. 
 
 13. Type of Report & Period 
 Covered 
 
 Research 
 
 14. 
 
 Supplementary Notes 
 
 Abstracts 
 
 The purpose of this paper is to demonstrate that representations of 
 numbers other than positional notation may lead to practical hardware 
 realizations for the digital calculation of classes of algorithms. It is 
 the authors' opinion that practicality of the use of continued products 
 has been demonstrated. This paper describes current research in the use 
 of continued fractions. Although practicality has not been demonstrated, 
 theoretical results are promising, and the results thus far are presented 
 as a case study of the difficulties which arise when use of a new 
 representation is attempted. 
 
 1 Key Words and Document Analysis. 17a. Descriptors 
 
 Continued Products, Continued Fractions, Computer Arithmetic, Hardware, 
 Selection Rules, Radix, Representation of Numbers, Ricatti Equation. 
 
 3. Identifiers/Open-Ended Terms 
 
 e. COSATI Field /Group 
 
 Availability Statement 
 
 Unlimited Release 
 
 RM NTIS-3B ( 10-70) 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 21. No. of Pages 
 
 J2 
 
 22. Price 
 
 USCOMM-DC 40329-P71 
 
Ss P 
 
 4fc 
 
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