UNIVERSITY OF 
 
 ILLINOIS LIBRARY 
 
 AT URBANA-CHAMPAIGN 
 
Digitized by the Internet Archive 
 in 2013 
 
 http://archive.org/details/useofriccatiequa674triv 
 
y^ y 7y UIUCDCS-R-7 I )-67 J + 
 
 The Use of Riccati Equation in 
 Digital Computer Arithmetic 
 
 by 
 Kishor S. Trivedi 
 
 August 1974 
 
 THE LIBRARY OF THE 
 
 OCT 4 1974 
 
 UNIVERSITY OF ILLINOIS 
 AT URB/ 
 
 URBANA, ILLINOIS 
 
 This volume Is bound without 
 
 ■no.676,vol.i.A-L. 
 
 which is/are unaval lable. 
 
lJUtiO 
 
 IUCDCS-R-7 1 f-67 i + 
 
 The Use of Riccati Equation in 
 Digital Computer Arithmetic 
 
 Kishor S. Trivedi 
 
 August 1974 
 
 THE LIBRARY OF THE 
 
 OCT 4 1974 
 
 UNIVERSITY OF ILLINOIS 
 AT U r 
 
 DEPARTMENT OF COMPUTER SCIENCE 
 UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN 
 
 URBANA, ILLINOIS 
 
UIUCDCS-R-T 1 )-^?^ 
 
 The Use of Riccati Equation in 
 Digital Computer Arithmetic 
 
 by 
 
 Kishor S. Trivedi 
 
 August 197^ 
 
 Department of Computer Science 
 University of Illinois at Urb ana- Champaign 
 Urbana, Illinois 
 
 This work was supported in part by the National Science Foundation 
 under Grant No. US NSF GJ 382 Ok. 
 
o/o. 
 
 1. INTRODUCTION 
 
 It was demonstrated by DeLugish [1] that number representations 
 other than the traditional positional notation are useful. This led to a 
 search for new number representations. 
 
 There seem to be several basic requirements for a proposed 
 representation of numbers to be useful for implementation in hardware [2], 
 These are the following: 
 
 Requirement 1: Conversion to the conventional positional 
 notation should be simple. 
 
 Requirement 2 : The set of numbers representable should be a 
 complete set [3]. In other words, the range of numbers representable should 
 be an interval of the real numbers. 
 
 Requirement 3 : It should be possible to define a broad class of 
 algorithms that are easily soluble for the representation of numbers used. 
 Hardware- compatibility among the algorithms is also desirable. 
 
 Requirement h: For algorithms which require trial and error 
 procedure, it should be possible to devise techniques such that the 
 coefficient-selection is practical (cf., quotient-digit selection in 
 division). 
 
 For continued fractions, a simple conversion procedure to positional 
 notation is well known [2,1+]. By an appropriate choice of the digit sets 
 for the partial numerators and partial denominators, the requirement of 
 completeness is satisfied [2,5]. The requirement of the identification of 
 a broad class of algorithms seems to be a difficult problem. A limited 
 class of quadratics can be solved using continued fractions [2], We will 
 
discuss a very powerful method by which continued fraction algorithms for 
 many elementary functions and for the integration of many functions can be 
 obtained. The requirement of an appropriate selection procedure is 
 another difficult problem. For the solution of quadratics, a selection 
 procedure has been obtained [2,5]. 
 
 The use of Riccati equation for continued fraction arithmetic will 
 be discussed in this report. We will first show the large number of functions 
 that fit into this analysis. Later we will discuss the derivation of 
 suitable selection procedures. The use of Riccati equation for this purpose 
 was first proposed in [6] . 
 
2. RICCATI EQUATION 
 
 2.0 Introduction 
 
 Riccati equation can be written as : 
 
 y' + a(x)y + b(x)y + c(x) = 0. (2.1) 
 
 Let L be the set of all Riccati equations of this form. Wynn has shown that 
 the set L is closed under the bilinear transformation y = p/(q+z) where p, q 
 are constants [7]. Starting with £ e L, by a repeated application of the 
 bilinear transformation, we can obtain a continued fraction expansion for 
 the solution to the initial Riccati equation £ . 
 
 Let £ Q be given by: y^ = a Q y Q + b Q y Q + c Q , and let y Q = P^/^+y^. 
 Let this transformation be called T : L -* L. T (1 ) = ! is given by 
 
 y{ + ^ y\ + b 1 y x + c x = 0. 
 
 The recursion relations for the coefficients a , b , c in terms 
 of a Q , b Q , c are, 
 
 a i = c c/ p i' 
 
 \ = b + 2 °0 V P l' / (2.2) 
 
 c i = a o p i + b o q i + c o 4/ p i 
 
 Note here that, we have changed the form of £ to avoid negative 
 
 2 
 signs in recursions (2.2). In general, let ^ = (y^ m - a 2m y 2m + b 2m y 2m + c 2m ) 
 
 2 
 and let £ n , _ = (y' , _ + a„ n y_ . + b_ . y , + c„ , = 0). Assume that, 
 2m+l w 2m+l 2m+l J 2m+1 2m+l ,/ 2m+l 2m+l y ' 
 
I = T T , ... T., (iL) has been obtained. Then the coefficients of 
 n n n-1 l v 
 
 I ., = T , (i ) are given by the following recursions: 
 n+1 n+1 n 
 
 a , , = c /p , , 
 n+1 n n+1 
 
 b ,=13 +2c qn/p.n 
 n+1 n n ti+1' ^n+1 
 
 c , = a p -,+b q _ + c q , /p i • 
 n+1 n ^n+1 n n+1 n n+1 7 n+1 J 
 
 As a result of these transformations, we have expanded y to n+1 terms as 
 
 (2.3) 
 
 follows 
 
 p l p 2 p n+l 
 
 Let P -i/Ql^ -i denote the finite continued fraction obtained by setting 
 
 y = in equation (2.4). If we assume that |y | < M where M is a fixed 
 
 constant then clearly, the fraction P /Q, converges to y . By setting, 
 
 P_ =0, Q = 1, P_, = p n and Q_ = q_, , the recursions for P ., and Q n 
 11 \L 1 n+1 Ti+1 
 
 are [4] : 
 
 P n =q n P+p n P, "> 
 
 n+1 ^n+1 n ^n+1 n-1 I 
 
 ( (2.5) 
 
 Vl = Vl \ + P n+1 Vl J 
 
 Thus if we have a method to correctly choose p , q for every n then, 
 we have an algorithm to solve the Riccati equation. 
 
 2.1 Power of the Method 
 
 We will now discuss the number of functions that can be obtained by 
 the method of Riccati equation. 
 
2.1.1 Constant Coefficients 
 
 Let us consider a subset L of L such that 
 
 2 , 
 
 L = {y' + ay + by + c = | a, b, c e R) i.e., the set of all Riccati 
 
 equations with constant coefficients. Consider £ e L given by, 
 
 
 
 
 
 2 2 
 
 y 6 = a 7 + b y + °0* De P endin S on the si § n of A = b n - k a n c n , the 
 
 
 
 0' 
 
 solution y n (x) of £ can be written as, 
 
 v -L 
 
 ■c\ 
 
 if A < and a ^ 0; 
 
 y W 
 
 V 2a o ° 
 
 jf A = 0, a n f 0; 
 
 , (x) = ^ (tan „(§ x + A Q ) - ^) 
 
 if A > 0, a Q ^ 0; 
 
 b Q x 
 y (x) = A Q e + c Q x if a Q = 0, 
 
 Depending on the values of the coefficients a_, b , c and the initial 
 condition t„ = y n (0), many different functions may be evaluated as shown 
 in the following table. 
 
 a o 
 
 b o 
 
 c o 
 
 A 
 
 *0 
 
 y W 
 
 1 
 
 
 
 1 
 
 -k 
 
 
 
 tan x 
 
 -1 
 
 
 
 -1 
 
 -k 
 
 oo 
 
 cot X 
 
 -1 
 
 
 
 
 
 
 
 00 
 
 1/x 
 
 -1 
 
 
 
 1 
 
 k 
 
 00 
 
 cot h x 
 
 -1 
 
 
 
 1 
 
 h 
 
 
 
 tan h x 
 
 
 
 +1 
 
 
 
 >o 
 
 1 
 
 +x 
 e— 
 
 Table 2.1 
 
2.1.2 Variable Coefficients 
 
 Consider a subset L, of L so that, 
 
 o 
 
 1^ = [y< = a (x) y + b(x) y + c(x)|a(x) = k(x) a, 
 
 b(x) = k(x) b, c(x) = k(x) c, and a, b, c 
 
 are constants). 
 
 Recursions for a , , b , and c . can be derived from the recursions (2.3) 
 n+1 n+1 n+1 v ' 
 
 and are as follows : 
 
 a.^ = c,/p 
 
 -\ 
 
 i+1 r -^i+1 
 
 b. _ = b. + 2c. q. ,/p. , 
 1+1 l i i+r *i+l 
 
 - 2 
 'i+1 " "i ^i+l ' "i ^i+1 ' C i ^i+l' ^i+l' 
 
 -, = a. p. , + b. q. n + c. q. ,/p. ,. 
 
 (2.6) 
 
 -2 
 
 Depending on the sign of A„= b - ^-a c , the solution to £ is 
 
 given by: 
 
 ^ \d 
 
 y (x) = 
 
 *b 
 
 2a, 
 
 ^0 f 
 (tan(— ^ Jk(x) dx + A 
 
 ) --£=) 
 
 s/- 
 
 *0 
 
 if Aq < 0, a Q / 0: 
 
 y» = - 
 
 + A, 
 
 a n k(x) dx 2a n 
 
 if A Q = 0, a Q / 0; 
 
 Yr 
 
 A 
 2a 
 
 (x) = - — ° (tan h(-^ /k(x) dx + A Q ) - -~r. 
 
 
 
 J% 
 
 if Aq > 0, a Q / 0; 
 
y (x) = A Q e 
 
 |k(x) 
 
 dx c 
 
 
 
 - -^ if a Q = 0, b Q / 0; 
 b 
 
 and 
 
 7, 
 
 } (x) = c Q Jk(x) dx + A Q if 
 
 a o = b o = °' 
 
 Clearly, a large class of functions can be evaluated with this method, 
 
 2.1.2.1 The Case With Aq = 
 
 In this section, we will concentrate on a subset L of L such that, 
 L, = {I € L JAq = 0}. Any £ e L can be rewritten as: y* = k(x)(a*y+b*) 
 
 where, a* = v a, b* = a*(— ). With this modification, we have reduced the 
 
 2a 
 
 number of coefficients from three to two. The recursions on a*, b* can now 
 
 n n 
 
 be written as follows : 
 
 *\ 
 
 a * _ t)-x-A/p 
 n+1 ■ n ^n+1' 
 
 K + i - < a S p n+ i + K w)^ 
 
 n+1 
 
 J 
 
 (2.7) 
 
 The solution £ e L is given by, 
 
 y (x) = 
 
 (ag) 2 (A Q - j*k(x) dx) 
 y Q (x) = (b*) 2 Jk(x) dx + A Q 
 
 a 5 
 
 if a* / 0, 
 
 if a* = 0. 
 
 Note that, we can integrate the given function k(x) by this method by 
 
 setting a* = and b* = 1. 
 
8 
 
 2.2 Implementation Considerations 
 
 Let us assume that simple selection procedures are available for 
 all the functions to be evaluated as detailed in section 2.1. We now give 
 steps of an algorithm T which will evaluate these functions. 
 Algorithm T: 
 Step 1 : [Initialize] 
 
 Set P Q - 0, Q Q - 1, P_ 1 - 1, Q_ 1 - O; 
 
 Set initial values of coefficients according 
 
 to the function to be evaluated.; 
 
 Set i - 0; 
 Step 2 : [Select] 
 
 (P- Tt % i ) "*" Select (x, coefficients, function); 
 Step 3 '■ [Recursions] 
 
 P i + i^ Vi p i + p i + i p i-i ; 
 Vi - Vi \ + p i+ i \-i> 
 
 Recur se using equations (2.3), (2.6) or 
 
 (2.7) whichever is applicable. 
 Step k : [Test] 
 
 After 'sufficient' number of iterations 
 
 GO TO Step 5; otherwise set i <- i+1, 
 
 and GO TO Step 2; 
 Step 5 : [ e valuat e ] 
 
 y (x) = f(x) = R i+1 /Q i+1 ; 
 
 END T; 
 
 In any such iterative algorithm, the number of iterations required 
 and the execution time required per iteration are two important considerations, 
 
In each iteration, steps 2, 3 and h are executed. Clearly, step 2 
 
 and 3 require more attention. We can assume that if the procedure Select 
 
 is known, it can be implemented in a combinational network and therefore, 
 
 will require very little time. In step 3, we see that all the assignments 
 
 are independent of each other and therefore, can be executed in parallel. 
 
 Thus, given sufficient hardware, step 3 can be speeded up considerably. 
 
 Each individual recursion requires additions (subtraction), multiplications 
 
 and sometimes division also. Since multiplication and division are relatively 
 
 slower operations, we would like to avoid them if possible. If we restrict 
 
 the coefficients p. and q. to be integral powers of two these multiplications 
 
 and divisions will be reduced to shifts, which is relatively a faster 
 
 operation. If we use the recursions (2.7), then we further require that 
 
 p. = 1 for V i. For the sake of simplicity, we will assume that p. = 1 
 
 in every case and q. € S where, S = f 2 J I -j is an integer}. Since the 
 
 l q ' q l ' J 
 
 number of shifts available is finite we further require that, 
 
 S = {2^11 < j < J ] where J and J are fixed integers, 
 q l '-q - d - q J q -q & 
 
 The number of iterations to be carried out can be decided on the 
 basis of allowable error in the result. 
 
 2.3 Initial Condition 
 
 Associated with the solution of any differential equation, there are 
 one or more arbitrary constants which are evaluated using the boundary 
 conditions imposed. Depending on the Function f(x) to be evaluated, we 
 choose a particular £ e L (and the corresponding coefficient values) and 
 the associated initial condition y n (0) = t so that y n (x) = f(x). Clearly, 
 the initial condition on I. (i > l) is dependent on t . In particular, 
 
10 
 
 y n-l = Vr/^V*^ WhiCh im P lies > 
 
 t n=p/(q.+t) which implies, 
 n-1 n in n 
 
 t = p /t . - a (2.8) 
 
 n r n' n-1 ti 
 
 As we will see in Chapter 3, t is needed as an argument in a selection 
 
 procedure for p and q ... Therefore, we need to evaluate t in every 
 
 iteration. This, however, implies that a division be carried out. We can 
 
 avoid the division by the following technique. 
 
 Let t = d /e then, from equation (2.8), 
 n n 7 n 
 
 d P e 
 
 n , n n-1 
 
 e , 
 
 n n-1 
 
 From which, 
 
 d = p e ,-qd .. 
 n r n n-1 in n-1 
 
 (2.9) 
 
 e = d ., 
 n n-1 
 
 and d = t and e = 1. 
 
 If the selection procedure can choose with the help of d and e 
 
 n n 
 
 (does not explicitly require t ) then we have solved our problem. Now in 
 step 3 of algorithm T, we have to carry out recursions (2.9) as well. 
 
 2.k Convergence 
 
 If the digit sets S and S are finite and positive then it can easily 
 
 p q 
 
 be shown that m < y < M V n, i.e., y is bounded [5]. The convergence 
 P /0 to y n (x) can then be guaranteed if a correct selection procedure is 
 available [5]. Therefore, we will discuss the question of selection 
 procedure in the next chapter. 
 
11 
 
 3. ATTEMPTS AT FINDING SELECTION PROCEDURES 
 
 3.0 Introduction 
 
 It is quite clear that the selection procedure will be different for 
 different functions. We will say that a selection procedure is consistent 
 if at every step in the evaluation of a given function, the same selection 
 procedure can be used. 
 
 The definition of consistency implies that the function represented 
 
 by y. (x) is independent of i. From the solutions of Riccati equation 
 
 (given in Chapter 2), it is easy to see that this will be true if A. = A 
 
 for V i. This can be shown as follows r8l : Let I. , e L and T. (£. . ) = £. 
 
 1 J l-l l l-l i 
 
 then from recursions (2.3), we have, 
 
 2 
 A. = b. -'4a. c. 
 
 11 11 
 
 = (b. , + c. , q./p. ) 
 i-I i-I r i 
 
 - 4 — — (a. p. + b. q. + c. _ q./p. ) 
 p. i-I i i-I n. i-I t/ i 
 l 
 
 ^2 ,,22/2,. / 
 
 = b. + 4 c. 1 q./p. + 4 b. c. 1 q./p. 
 i-I i-I r l i-I i-I t/ i 
 
 2 2 2 
 
 -4 a. , c. .. -4b. , c. , q_./p. -4c. , q_./p. 
 
 i-I i-I i-I i-I ^i' *i i-I ^±'^2. 
 
 1.2 , 
 = D. _ - 4 a. , c. , 
 
 i-I i-I i-I 
 
 "\-r 
 
 Thus we have shown that consistency requirement is satisfied. 
 
12 
 
 We now fix the digit sets for the partial numerators and the partial 
 
 t 
 
 denominators. We will assume p. = 1 for all i. We will assume 
 
 1 
 
 S = {1, l/2, l/k) . Note that this is a redundant digit set. Such a 
 
 redundancy is required for the selection procedure to be possible [5]. 
 
 The set of all infinite continued fractions representable with this 
 
 1 
 
 digit set is the interval F = [0.39* 1-56]. Given v. e F and y. = - 
 
 the constraint for choosing q. , is that q. , e S and y. n e F. Therefore, 
 
 tL+1 l+l q J l+l ' 
 
 if y. E F = [0.39, 0.72] then we can choose q^ = 1, if y. € Fp = [O.M35, 
 
 1.123] then we can choose q. .. = l/2, and if y. e F = [0.553, 1.56] then 
 
 we can choose q. -, = l/^+. Quite clearly, the intervals F , F and F„ are 
 
 not pairwise mutually disjoint, which shows the redundancy of the digit set 
 
 S . In the intersection F, = F, fl F„, q. n can be chosen as either 1 or 
 q 12 1 2 7 T.+1 
 
 l/2. Similarly, in the interval F p ~ = F p fl F„, q. , can be chosen as either 
 1/2 or l/h. Thus, if we know y n then we have a consistent method of 
 expansion of y into an infinite continued fraction. Since y is not known 
 in advance, this simple method will not work. 
 
 3.1 The Criteria for Selection 
 
 For further discussion, we will concentrate on the subset L of L. 
 
 2 
 Consider i e L,_ given by: y' = j k(x) (a y +b ) where j = 1 if n is even 
 10 n nnn 
 
 and j = -1 if n is odd. The solution to this equation is given by: 
 
 A = j g(x) + — j — 1 r v (3.1) 
 
 n d 6 ^ ' a (a y +b ) w ' 
 
 nnn n 
 
 Where, A is an arbitrary constant of integration and g(x) = / k(x) dx. A 
 
 d 
 
 can be evaluated by using the initial condition y (0) = t = — . Therefore, 
 
 n n e 
 
 n 
 
 e n 1 
 
 A- = 3 g(°) + ~r~ i — 3 — tz \ • Substituting this value of A in equation (3.1), 
 
 nd(ad+be) ° n 
 
 n n n n n 
 
13 
 
 we have, 
 
 d + j(g(x)-g(0)) b (a d +b e ) 
 ( ) n , , , n n n , 
 
 y n U; ~~ e - j(g(x)-g(0)) a (a d +b e ) 
 n n n n n n 
 
 A S (k(x), a , b , d , e , j). (3.3) 
 
 = n v v " n' n' n' n' J y VJ J; 
 
 Quite clearly, if the function S can be evaluated accurately for 
 every value of n then the selection procedure is trivial. But the computation 
 of the function S is at least as complex as that of y (x). Therefore, 
 this method is not useful. However, the redundant number representation 
 may allow us to use an approximation S of S in the selection procedure. 
 If the computation of S is simple enough then such a selection procedure 
 will be practical. 
 
 Let us denote z^ = sup. (F^), z^ = Inf. (F^), z = Sup. (Fpo) and 
 
 _z p ~ = Inf. (F pc .). Further let z and z be suitable constants such that 
 
 z e F-.p, z = F p „. We use the following selection procedure, Select 1. 
 
 Select 1: 
 
 If z.. < S < z then q. _ = l/2 : 
 1 — n — 2 T.+1 " ' 
 
 else 
 
 if S > z then q. - = l/k: 
 n 2 T.+1 ' ' 
 
 else 
 
 if S < z n then q. _ = 1; 
 n 1 H i+1 
 
 end Select 1; 
 
 Clearly, Select 1 will be correct if the following condition is 
 satisfied for all n. | S n - Sj < Min.fz^ - z ± , z^ - Zp, z ± - z^, Zp - Zp 3 ). 
 
 (3.*0 
 
Ik 
 
 Therefore, we should try to find a sufficiently simple approximation S for 
 S which satisfies the condition (3.*0. 
 
 A linear approximation to S is perhaps the simplest. Further, if 
 the coefficients in the linear approximation are powers of two then 
 computation of S will involve addition and shift operations only. 
 
 It may be noted that with S = {1, 1/2, l/l+}, F^ = [O.kQj, 0.72] 
 and F_ = [0.553, 1.123]. If we let z = 0.5 and z^ = 0.75, then even if 
 S is correct only to one digit, q can he chosen correctly. 
 
 Before proceeding further, we choose a particular function to be 
 
 -1 r 2 
 
 evaluated. We let y n (x) = Sin (x), k(x) = l/vl-x , a = 0, b = 1, d = 0, 
 
 e Q = 1. 
 
 3.1.1 More Details 
 
 For y n (x) = Sin" (x), 
 
 -1 
 
 d + i Sin (x) b (a d +b e ) 
 a n n n n n n fo >. 
 
 n ~ ^1 / U-p; 
 
 e - j Sin (x) a (a d +b e ) 
 n n n n n n 
 
 Therefore, S £ F no iff, 
 
 n 12 ' 
 
 (0.72 e -d ) 
 
 n n' 
 
 and 
 
 Let 
 
 jx < Sin( £ — ) 
 
 0.72 a d + (0.72 e +d )(a b ) + b e 
 n n n n n n' n n 
 
 (O.i+85 e n -d n ) 
 
 jx > Sin( p * — ) (3.6) 
 
 0.^5 a d + (0.^85 e +d ) a b + b e 
 nn nnnn nn 
 
 s e - d 
 M n (s) = Sin(— -£ S H __) 
 
 sa d + (s e +d ) a b +be 
 nn nnnn nn 
 
15 
 
 The selection overlap regions can be given by: 
 
 jx 6 [M n (0.i485), M n (0,72)] then q^ = l/2 or 1, 
 and 
 
 jx e [M n (0.553), M n (l.l23)] then q^ = l/^ or V 2 - 
 
 Since the computation involved in M (s) is intolerable, we must find 
 a simple approximation M (s) to M (s). If M (s) satisfies the condition (3.^0 
 then the selection procedure will be correct. There are several ways to 
 obtain the approximation M (s). 
 
 The first attempt consisted of trying to obtain a continuous linear 
 
 least square fit for jx = M (s) which is linear in a , b , d , e. But this 
 i o n \ j n n n n 
 
 procedure involved the computation of a number of h- tuple integrals of 
 functions which are inseparable in the variables of integration. Therefore, 
 this approach was abaondoned. 
 
 Second approach consisted of obtaining a tangential hyperplane 
 
 to the surface jx = M (s) (for a fixed value of s) at some point in the 
 
 5 
 five dimensional space R . Unfotunately, after some computation, it was 
 
 seen that for displacements in the 5-space, the tagential hyperplanes rotate 
 
 by large angles. 
 
 The third approach consisted in obtaining a discrete least square 
 
 fit of jx = M (s). This approach also could not produce an approximation 
 
 which will satisfy the error condition (3.*+). 
 
 The growth of coefficients a , b , d , e seems to be the reason of 
 
 n n n n 
 
 failure. 
 
16 
 
 k. CONCLUSION AND FUTURE RESEARCH 
 
 The method of Riccati equation is a very powerful method. Countless 
 number of functions can be evaluated in a uniform manner using this method. 
 Obtaining a suitable selection procedure seems to be a very difficult 
 problem. 
 
 Further investigation for obtaining selection procedures is needed. 
 Growth pattern of the coefficients of the Riccati equation need to be studied. 
 Methods of scaling so as to control the growth of coefficients need to be 
 investigated. 
 
17 
 
 LIST OF REFERENCES 
 
 [1] DeLugish, B. G. , "A Class of Algorithms for Automatic Evaluation of 
 
 Certain Elementary Functions in a Binary Computer," Ph.D. Dissertation, 
 University of Illinois, Urbana, June 1970; also Department of Computer 
 Science Report 399. 
 
 [2] Robertson, J. E. and K. S. Trivedi, "The Status of Investigations into 
 Computer Hardware Design Based on the Use of Continued Fractions, " 
 IEEE Transactions of Computers , Vol. C-22, No. 6, June 1973, pp. 555-560. 
 
 [3] Goldberg, R. R., "Methods of Real Analysis, " Blaisdell Publishing 
 Company, New York, 1965. 
 
 [h] Wall, H., "Analytic Theory of Continued Fractions," Van Nostrand, 
 Princeton, New Jersey, 1950. 
 
 [5] Trivedi, K. S., "An Algorithm for the Solution of a Quadratic Equation 
 using Continued Fractions," M.S. Thesis, University of Illinois, Urbana, 
 June 1972; also Department of Computer Science Report 525- 
 
 [6] Bracha, A., "Application of Certain Functions on a Digital Computer," 
 University of Illinois, Urbana, Technical Report, UIUCDCS-R- 72-510. 
 
 [7] Wynn, P., "On Some Recent Developments in the Theory and Application 
 of Continued Fractions, " Journal SIAM on Numerical Analysis , Vol. 1, 
 pp. 177-197, 196^. 
 
 [8] Robertson, J. E., private communication. 
 
BIBLIOGRAPHIC DATA 
 SHEET 
 
 1. Report No. 
 
 . Kcport No. . , , 
 
 UIUCDCS-R- 7^-67}+ 
 
 4. Title and Subtitle 
 
 The Use of Riccati Equation in 
 Digital Computer Arithmetic 
 
 3. Recipient's Accession No. 
 
 5. Report Date 
 
 August 197^ 
 
 6. 
 
 7. Author(s) 
 
 Kishor S. Trivedi 
 
 8. Performing Organization Rept. 
 No. 
 
 9. Performing Organization Name and Address 
 
 Department of Computer Science 
 University of Illinois 
 Urbana, Illinois 
 
 10. Project/Task/Work Unit No. 
 
 11. Contract /Grant No. 
 
 GJ 382 Qh 
 
 12. Sponsoring Organization Name and Address 
 
 National Science Foundation 
 Washington, DC 
 
 13. Type of Report &• Period 
 Coveted 
 
 14. 
 
 15. Supplementary Notes 
 
 16. Abstracts 
 
 This is a report of ongoing research in continued fraction arithmetic, 
 It is shown that a large number of functions can be evaluated using the 
 Riccati differential equation. However, the problem of properly selecting 
 the partial numerators and the partial denominators remains to be solved. 
 Several attempts to obtain such selection procedures are described. 
 Possible directions for further research are indicated. 
 
 17. Key Words and Document Analysis. 17o. Descriptors 
 
 Combinational Network 
 Computer Arithmetic 
 
 Continued Fractions 
 Function Evaluation 
 Representation of Numbers 
 
 Riccati Equation 
 Selection Procedure 
 
 17b. Identifiers 'Open-Ended Terms 
 
 17c. COSAT! Field/Group 
 
 18. Availability Statement 
 
 19. Security Class (This 
 Report) 
 
 UNCLASSIFIED 
 
 21. No. of P-tf 1 • 
 
 20. Security Class (This 
 Page 
 
 UNCLASSIFIED 
 
 22. Price 
 
 f-'ORM NTIS-35 (10-70) 
 
 USCOWM-DC 40329-1 
 
ffi . »tw 
 


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