LIBRARY OF THE UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAICN bi.0.%4 CENTRAL CIRCULATION BOOKSTACKS The person charging this material is re- sponsible for its renewal or its return to the library from which it was borrowed on or before the Latest Date stamped below. The Minimum Fee for each Lost Book is $50.00. Thoft, mutilation, and underlining of books are reasons for disciplinary action and may result in dismissal from the University. TO RENEW CALL TELEPHONE CENTER, 333-8400 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN APR 4 M* MAR 2 / 1994 JAN 6 When renewing by phone, write new due date below previous due date. L162 Digitized by the Internet Archive in 2013 http://archive.org/details/applicationofcon510brac r Report No. UIUCDCS-R-72-510 I) APPLICATION OF CONTINUED FRACTIONS FOR FAST EVALUATION OF CERTAIN FUNCTIONS ON A DIGITAL COMPUTER by Amnon Bracha March 1972 The person charging this material is re- sponsible for its return on or before the Latest Date stamped below. Theft mutilation, and underlining of books 2: reasons for disciplinary action and may result in dismissal from the Un.vers.ty. L161— O-1096 Report No. UIUCDCS-R-72-510 APPLICATION OF CONTINUED FRACTIONS FOR FAST EVALUATION OF CERTAIN FUNCTIONS ON A DIGITAL COMPUTER by Amnon Bracha Computer Science Department University of Illinois Urbana, Illinois 61801 March 1972 This research was supported in part by the National Science Foundation under Grant No. US NSF GJ 8I3. iii ACKNOWLEDGMENT The author is grateful to Professor J. E. Robertson and Professor K. B. Stolarsky of the University of Illinois for many helpful discussions. Thanks also to Mrs. Connie Slovak for typing the manuscript. APPLICATION OF CONTINUED FRACTIONS FOR FAST EVALUATION OF CERTAIN FUNCTIONS ON A DIGITAL COMPUTER Amnon Bracha Abstract The purpose of this paper is to develop a method for evaluation of certain elementary functions on a digital computer by the use of continued fractions. The time required for this evaluation is drastically reduced by using "short" operations like shift and add, instead of multiplications. Functional consistency is the most important factor that allows the expansion of a function into a continued fraction. Several cases are discussed and in particular the solution of the quadratic equation is discussed in more detail to demonstrate the convergence of the method. 1. INTRODUCTION The idea of using continued fraction representations for generating a solution to a limited class of quadratics was first introduced by Robertson [3]. Consider the finite continued fraction with k partial numerators p. and k partial denominators q. =1, 1, . .. , k, whose value is — , i.e., 1 X \ (1.1) \ = p : \ ^i + p 2 ^ + p. v A convenient way of writing (l.l) is \ _ P l P _2 P 3 P k. • • •— q- \ q i + % + q 3 + ■k A, and B are determined, from the recursions : A. = q.A. _ + p. A. l ^i l-l ^l i-2 (1.2) B. = q B. , + p.B _ „ , i i l-l ^i i-2 i = 2, 3, ... , k, with initial values : A Q . A ± = Pl B =l B^^. It is clear that A, and B, can be separately and simultaneously determined in two binary arithmetic units in k-1 addition times if the p. and q. are The digit set for p. and q. for the purposes of this paper is {l/2,l}> it will be stated later that the continued fraction \ chosen to be simple in the binary sense. the purposes of this \ ea rr action assumes all values in the limit, over the interval (1-5) %i < lim £ < j 2 . d ~ , B, — k->co k The range defined in (1.3) includes the range of normalized floating point binary fractional parts, [1/2,1]- This property indicates that a suitabl ■ 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 — . B k The main reason for selecting p., q.e{l, l/2},i=l, 2, ... is that the four multiplicative operations required for each iteration in (1.2) are 3 reduced to "shift" and "add" operations. These operations will be called "short" operations throughout this paper, mainly because the time required to perform these operations is shorter than the time required to perform "long" operations, e.g., Multiplication, Division. The purpose of this paper is to develop algorithms for fast evaluation of certain elementary functions by using "short" operations in several registers simultaneously. In order to be able to do so we make use of functional consistency which will be defined at the end of section 2. Determination of selection rules for p and q in each iteration is an important step for the development of the algorithm. Selection rules were extensively studied by Trivedi [5], where a complete set of such rules were developed for the quadratic equation. The set of selection rules that is used in this paper is described in section 3. In section k we generalize our results to a higher degree polynomial and in section 5 we show two more cases where our analysis is applicable. 2. BILINEAR TRANSFORMATIONS AND THE RICCATI EQUATION. In this section we develop a special case of the analysis of Wynn [8]. The general continued fraction will "be regarded as a sequence of bilinear transformations of the form: (2.1) f = -£— , k-1, 2, ^k k+1 where f v (x) is a function of x. Therefore p l P P P n 2.2) f, = 1 q l + q 2 + ' Wl A +f A . i~ -*\ n n+1 n-1 , _, (2.3) = B+ f B ' n =1 ' 2 > ••• ' n n+1 n-1 where the functions A and B satisfy the recursion n n (2.10 A = q A +p A n ^n n-1 n n-2 I = q B + p B n = 1, 2, ... , n n n-1 n n-2 with the initial values A = A - p 1*1 1 B = n 1 q ] For the purposes of this paper p., q. , i = 1, 2, . .., will be selected from the digit set (l/2, 1} so that the recursion (2.1)-) can be performed by using only "short" operations. The main purpose of this section is to show that there exist functions for which bilinear transformations of the form (2.1) can be used, and such that the functions f, , k =1, 2, ..., are consistent, i.e. equal. Consider the Riccati equation (2.5) y{ + a x y^ + b 1 y 1 + c ] _ = where y is a function of the variable x, and a,, b, and c, are functions of x or constants. The property of this equation as noted 'by P. Wynn [7] is that if the dependent variable y. is replaced by the bilinear transformation (2.1) then the functions f , k = 1, 2, ... also satisfy the Riccati equation (2.6) y^ + Vk + Vk + C k = ° ' st We develop below the recursion for the coefficients of the (k+l) equation th by means of the coefficients of the k equation. Let y^ + a k y2 + b k y k+ c fc = be the k-th Riccati equation. P k From (2.1) we have y fe = , p fe , q fe e {1, l/2] k k+l then since y' = - p k y k+l we have k , A ^2 " P k y k + 1 P k , p k ( VW K +y k+l> k k+1 p / If we multiply by -(q,+y, -, ) / to normalize the coefficient of y ' , we get / k C k 2 ,. 2C k q k, , ^ Vk , y k + i - 17 y k + i " ( V -p— ) y y-i" (a k p k + W -p— ) = ° k k k and the recursion that follows is Vi = - TT (2.7) b^ n = -t>. £-£ v ' k+1 k p. k 2 C k q k c. , _ = - a. p, - "b. q, - k = 1 . 2 , k+1 k k k^k p n ' ' k We note that all the operations involved in (2.7) require only "short" operations, since both p and q are simple binary constants. Lemma 1: [k] 2 A = b, - ka.-, C-, = constant k = 1. 2, k k k Proof: We use the recursion (2.7) and get iv i 2 2 n 4b ? c. q, 4C. q, . 2 , ,2 k k k k^k , Vi " ^k + iVi = \ + -T~~ + —2— - k W k p k iv i 2 2 4b n c. q n he. q. k k k k k ,2 , - — 7- - \ - K e k We define below the term functional consistency. Definition : For a substitution of the form P k f k (x) = T+f — nr > k = 1 > 2 > • ■ • > where f, (x) is a function of x, p, and q n are constants: if f (x) and k ' k k k f. , (x) is the same function, then we have functional consistency for f(x). 3. SOLUTION OF ax 2 + bx - c = We show now how the solution of a quadratic equation with two distinct roots of opposite sign, and in particular the square root problem, can be found by the technique of section 2. Let (3.1) a^ 2 + b.^ - c ± = be a given quadratic equation. The substitution we use is of the form P.- (3-2) x. = l q.+x.,.. ^l l+l where p. , q. e (1, l/2) i = 1, 2, th For the k step we have ,2 *V p k a 1 _ + \_ _J£ — - c,_ = k = 1, 2 } or = k / s2 k q ,+x, , , k ^k +X k + l } ! ' c k i + ( 2C A -Vk )x k + i + C A - \4 - VA The recursion that follows is: a k+l ~ °k (3-3) b, ,-, = 2c. q. - b. p. w y/ k+1 k^k k k C k + 1 = ^k + (a k P k + VkK k = lf 2 > and the resulting quadratic equation is k-u Viii + \ + i x k + i - c k + i =0 This method of approximating the solution of (3-1) can be used if we develop a technique for selecting p and q k =1, 2, from the coefficients of the k-th quadratic equation, i.e., a, , b and c, . K. Lemma 2; The following lemma will be stated without a proof. th Let the k approximation to the solution of (j.l) be X l - % + q 2 + • • • q k Then for the digit set p , q fc € {1, l/2) we have in the limit M = max x = v2 •s/2-1 m = mm x, = — - — . Using the result of the Lemma it can be seen that the functional consistency of the procedure can be achieved in each step if (3.5) m < x fc ^ M k = 1, 2, ... By imposing condition (3.5) we need only one set of selection rules for p and q , k = 1, 2, ... for the range [m,M]. 10 We develop now a set of selection rules for p n and q, , k k k = 1, 2., •'-, for the quadratic equation [5]. We write below a version of (3»l). Let C l (5 ' 6) X l = ^ + a A where it is assumed that c, > 0, b , ^ 0^ a, > 0,, and m g x, ^ M. We will find p, and q such that (3 - 7) x i ■ ^" where m ^ x g M and p . , q e {1, 1/2) . Clearly, we have four possibilities, and for each pair of p.. and q n we get a different x„. ■^1 H l & 2 We take now the inverse approach. We assume that condition (5.5) exists for x , and find the range for x., for each pair of p and q . We start with a pair p = 1 and q = l/2. From (3*7) we have For x = v2 (lower bound for x, ) 11 I 2(2/2-1) (3.8) x a? = " v " "~ y = 0.522 , 1 1/2 +J2 T and for x = (v2-l)/2 (upper bound for x,) (3-9) x, ^ i-s — = J2 = i.kih . 1/2 + 1 /2-1 The result is that for x., in the range defined by (3'&)-(~5-9) > we can choose p = 1 and q = l/2. Since x is the unknown we use (3.6) in order to find the allowable range for p = 1 and q = 1/2. We have c l (3.10) l.klk il — — > 0.522 . b + a x, Since (3«1°) is possible for any x in the range (3-8)-(3« 9), we conclude that for the range (3.11) 0. 522b x + 0.522 2 a 1 < q ^ >J~2 1 o ± + 2a we choose p = 1 and q = 1/2. Similarly we write below the ranges for each of the remaining possibilities : For (3.12) (s/a-l)!) ,+ (\/2-l) 2 a $ c ^ 2(N/2-l)b 1 +if(N/2-l) 2 a , choose P l = lj q l = 1; (5.13) Z&± b x + (2^1)^ * c x S ^ + | v choose 1 1 P l = 2' q l = 2' 12 2 r r d (3.14) -^T\+ (-^~") a x ^ c x ^ (^2-1^+^2-1)^, choose P l = \> q l = 1 ' The result is that the entire range is divided into four sections, (3. Il)-(3«l4) > and for each section we can choose a pair of p and q and such that condition (3*5) f° r x be satisfied. Clearly, if we have to do two multiplications for each selection range in order to find p.. and q. , our procedure is inefficient. In the analysis that follows we make use of an important feature of the ranges defined in (3. 11)- (3. 14); this is the existence of overlapping between any two consecutive ranges. This means that in the overlap regions we have a freedom of selecting the pairs p, and q between two sets of such constants. We will use this freedom in order to simplify our selection algorithm by defining a line of selection inside the overlap region and such that the coefficients of b, and a will be simple in the binary sense. Before we define the selection lines, we note that rate of convergence of the method was found to be strongly dependent on these lines. The first set of selection lines can be the upper lines in each range, for Cl - 0.4l4b 1 % 0.1713a 1 then v ± = | , q_ ± = 1; (3.15) c 1 - 0.707b 1 g 0.5a 1 then P;L = | , ^ = p c - 0.828b g 0.686a then p = 1 , q = 1 ; otherwise p = 1, q = — . Experimentally this set of rules gave the best rate of convergence. 13 To simplify the constants that appear in (3. 15) we use simple binary constants with at most two non-zero binary digits, We have, c x - 0.37^ * 0.15625 ai then ? ± = ^ , q ± = 1 ; c - 0.625b 1 ^ 0.5a 1 then ?! = 2 ' q l = 2 ' (3-16) c - 0.75b ^ 0.625a then p = 1 , q = 1 otherwise p = 1, q = o • 3 These selection rules involve only short operations. The procedure can be carried out now in the same way for x ? and so on. The algorithm described in this section involves the following steps, for the k iterations: (1) Equation (3-^) is given, then use selection rules (3.16) to find p and q . (2) Use the results of step (l) and iterate on (2.10. (3) Use the recursion (3-3) an( 3- find (3-1)-) for k+1. (k) Check if A, /R reached the required precision. This check can be done only once if the number of iterations required to achieve certain precision is known. The analysis below for the rate of convergence gives the necessary information to find such numbers. If the required precision is not reached proceed to step (l) for k+1. Two examples are shown in appendix A. The rules of selections which were used are (3.16). Ik The following theorem assures convergence of the algorithm. Theorem 1: Let (3-17) x 1 = A B p n q + x , , n = 1. 2, . . . , jl ■ c n n+1 "be the exact solution of (j.l)_, which was found "by substitution (3.2), with the set of selection rules (3. 16) and recursion (3-3) f° r "the coefficients of (3*4). n "til Let — , be the n approximation to x . Then for every n e > 0, exist N such that for all n > N, A A 5 n = - " J" < e n th Proof: First observe that if x. satisfies the i equation (3»4) > then x is a solution of (3«l)- We will study now a relation between 5 and 5 . because of n n-2 a well known property of odd and even convergents of a continued fraction. Define T n = — a .5, k, ..... n-2 then we only have to prove that T < 1. We have [2] (Chapter l): A/B - A /B (-l) n+2 PiP^. .-P P ,-,/B B ' n 7 n \ / -^1-^2 n^n+1 7 n n " A/B - A 77b ~ / n \n / , \ /„ _ n- 2 ' n-2 (-1) P 1 P 2 ..-P n _ 1 (VlV P n + l ) / BB n-2 n n+1 n-2 VrVVi n 15 Since p n+1 = x n+1 = p n /x n - % , %+1 we conclude that: 1 and B =qB , + p B _ , n n n-1 n n-2 T = n p - a* n ti n 1 ^ 1 X p n n B n-1 P n + \ B I n-2 P n B n-2 All the quantities in T are positive , T < 1, and the result follows. We will use now the analysis of Theorem 1 to study the rate of convergence, by finding an upper bound to T . 1 x p n n Max T = Max 1 , q n B n-1 + P n B n-2 i n 1 - mm — x p n n 1 + min n n-1 ] P B J n n-2 Mm — x = -=, mm x = i£-i = O.IO35 p n 2 n k n Min /q b A n n-1 P ' B _ n n-2 „ 1 . n-1 = p min * 2 B n-2 but — is a continued fraction whose value is B n-2 Vl + Vl V2 ^2_ 3 n-2 " Vl V2 + V 5 + ' * ' q l Therefore, Min < _ + = B . - 2 2 2 n-2 Finally we have, \T2-1 k "5 - n/2 Max T ^ = = ^ = 0.2929 n J~2 )++ sT2 1 + T 16 Therefore for n sufficiently large, the error 5 is reduced by a factor which is less than or equal to 0.2929 for each pair of additional iterations. 17 k. SOLUTION OF A HIGHER DEGREE POLYNOMIAL We show now how one solution of the cubic equation can be found by the method of section 3* Let (l+.l) a,x:r + b x + c x - d = be a given cubic equation. We use the substitution (3. 2) and we th get for the k step: 3 2 _i , ^k \ 7~. 3 + b k 7~ E + c k q-^-T ■ "k = p k . . p k p k (^ +1 ) 3 k Vvi) 2 kqk+Xk+1 or - (a i p k + ¥kVwk q k )=0 The recursion relations between the coefficients of the k cubic equation and the (k+l) cubic equation are therefore: Vi =d k Vi = 5 W C A {k ' 2) Vi =5d kV 2c k p kWk ^'vMv^kVVk k =i, 2, ... . The resulting cubic equation is Viii + V1V1 + V1V1 - Vi = ° 18 For the selection rules we use an analysis similar to that of section 3. First observe that the bounds given in (3-£S)-(3-9) for the case p = 1 and q = l/2 are valid. Therefore we can write an expression, similar to (3. 10) for the cubic equation: ± (If. 3) ^2 ^ g " °- 522 a l k l + Vl + C l The result is that for 0.522c 1 + 0.522 2 b x + O.522V, ^ d ± % Jkc^ + 2b ][ + 2\T2a x we choose p = 1 and q = l/2. For the remaining cases we have (s/2-l)c 1 +(N/2-l) 2 b 1 +(N/2^l) 5 a 1 g d- ^ 2 (vfe-l) c H(«/2-l) ^+8(^2-1)^ choose p =1, q = 1; 2^2-1 , /2nT2-1\ 2 , /2"V~2-l\ 5 ^ , ^ \fe 1. ^ \fe -7— c i + [^T~J b i [~T~] a i " d i " T c i + 2 b i + T a i choose p = l/2 ; q n = l/2; 2 1 %i) b + [%i] a^^ (n/2-1) Ci + (^2-1)^ + (^2-l)\, choose p = l/2, q, = 1. We are ready to write now a set of selection rules similar to (3.15), i.e., the upper line in each range will be our selection line: L9 d - O.Wc. - 0.1713^ ^ 0.071a 1 then p 1 = l/2, q ± = 1; (k.h) d 1 - 0.707^ - 0.5*^ £ 0.3535a 1 then P;L = 1/2, q 1 = l/2; d 1 - 0.828c - 0.686b =g 0.5683a then p = 1, q = 1; otherwise p = 1, q_ = l/2 . We note that as in (3.16) the constants which appear in (k.k) can he simplified. For the proof of convergence and the rate of convergence we can use the analysis of section 3> and therefore we develop a method to approximate one positive solution of a cubic equation. The procedure can he generalized now to higher degree polynomials. The necessary steps are (1) to write the recursion for the coefficients of the polynomial (2) to develop the selection rules by using an argument similar to (3. 10) and (^.3) (3) to simplify the coefficients in the selection rules. The result is an algorithm which always converges to one positive solution. In appendix B we give two examples for approximation of a positive root of a cubic equation. 20 5- CONTINUED FRACTION EXPANSIONS OF CERTAIN FUNCTIONS BY THE USE OF THE GENERAL SOLUTION OF THE RICCATI EQUATION Let (5.1) y' + ay 2 + by + c = be a Riccati equation, with a, b ; and c constants. In order to find the solution of (5.1) we integrate by parts : ^— = -dx ^' 2 > ay 2 +by+c and the solutions are (5-3) / dy 1 2ay+b- vb -Ij-ac ay +by+c v b -l+ac 2ay+b+ vb^-lj-ac Arctgh b ^ 2ay , when b 2 -l^ac > 0; vb -4ac vb -k ac P 2 when b -^ac = 0; b+2ay ' arctg — , when b -i|-ac < 0. vJ+ac-b vi+ 2 ac-b The above solutions can be used now for the continued fraction expansion of the inverse functions which appear explicitly in the solution. We start with the case (5.^) y = tg x 21 The Riccati equation for (5.^) is y' = y 2 + 1 , y(o) = . We note that -A = Jkac-h 2 = 2 . Now we use a bilinear transformation of the form (2.1). The result is a differential equation of the type (2.6) with the recursion (2.7). By Lemma 1 it follows that the solution for each equation k is of the type (5.3) with- A > 0, and therefore we get for the k th step ~*\~\ = a S 2 where d is a constant of integration. The solution is therefore b k 1 \ = -2a^ " a^ tg ^V^ k =1 > 2 > ■" • Except for the first part of the solution which is a linear transformation, we see the consistency of the method, because if a set of selection rules are developed for tg x it can be used for each step and therefore evaluation of this function will be possible. Another important function which can be included is e . We have y' = y, A = 1 th The k " step solution is . a Vt + V 1 or V 1 1 y k = -Ta" -^- k = X ' 2 > a^e *-l) 22 Again we note that if a set of selection rules can X be developed for e then it is possible to carry the process for each step and therefore to find the continued fraction expansion for the exponential function. For the case where A = we have several possibilities; (a) b = a = 0, y' + c = with the solution y = -ex + d 2 (b) b = c = 0, y' + ay =0 with the solution 1 k ax+b 2 (c) a ^ 0, b / 0, c 4 °j and h-.lj.ac = 23 REFERENCES [1] Hardy, G. H. and Wright, E. M. , An Introduction to the Theory of Numbers, Oxford 1954. [2] Khovanskii, A. N. , The Application of Continued Fractions and Their Generalizations to Problems in Approximation Theory, P. Noordhoff N. V. , Groningen, 1963. [3] Robertson, J. E. , Quarterly Progress Report, Department of Computer Science, University of Illinois, July-September 197°; pp. 121-126. [4 ] Robertson, J. E. , Private Communication, January 1972. [5] Trivedi, K. S. , M.S. Thesis, Department of Computer Science, University of Illinois, Urbana, Illinois. To appear. [6] Wall, H. S. , Analytic Theory of Continued Fractions, D. Van Nostrand Company, Inc. , New York, 194-8. [7 ] Wynn, P. , Five Lectures on the Numerical Application of Continued Fractions, Mathematics Research Center, The University of Wisconsin, Madison, Wisconsin. 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HS(r^ffx't\jMirry^Nff cc rr a rr cr o O o ^ *4 —. rv IVI tM rv m rp rf xt o o o o o n c n c o C n c a t- n c o n C n o c o r^ ir. rv — — • r« r» rr ■c cc cr in rP xO rtl -— cr ff xt rr x^ an _4 LP ^v — r- cc i — i ■J- xt INI — IP rv ~> r— rp — rr xt rv — ip rv — rx. ooooooorsnoooooooooooooooorsoooooo oio or»oooooooo rsio o o o o o IBLIOGRAPHIC DATA 1. Report No. 2. 3. Recipient's Accession No. MEET UIUCDCS-R-72-510 Tit It- .mJ Subt itle 5- Report Date- APPLICATION OF CONTINUED FRACTIONS FOR FAST EVALUATION March 1972 OF CERTAIN FUNCTIONS ON A DIGITAL COMPUTER 6. Author(s) 8. Performing Organization Rcpt. Aranon Bracha No. Performing Organization Name and Address 10. Project/Task/Work Unit No. Department of Computer Science University of Illinois at Urb ana-Champaign 11. Contract/Grant No. Urbana, Illinois 61801 US NSF GJ 813 , Sponsoring Organization Name and Address 13. Type of Report & Period Covered National Science Foundation Research Washington, D.C. 14. . Supplementary Notes . Abstracts A method for fast evaluation of certain elementary functions on a digital computer, by using continued fractions is developed. The time required for this evaluation is drastically reduced by using short operations like add and shift instead of multiplication and division . Several cases are discussed and convergence to the solution of quadratic and cubic equations are shown. . Key Words and Document Analysis. 17a. Descriptors continued fractions, Riccati equation >• Identifiers/Open-Ended Terms c. COSATI Field/Group Availability Statement 19. Security Class (This 21. No. of Pages Report) UNCLASSIFIED 32 Release unlimited 20. Security Class (This 22. Price Page UNCLASSIFIED RM NTIS-35 ( 10-70) USCOMM-DC 40329-P71