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5/0.84 
 
 no. U6 
 cop. 3 
 
 REMARKS ON THE ROUND- OFF ERRORS IN ITERATIVE PROCESSES 
 FOR FIXED- POINT COMPUTERS * 
 
 by 
 J„ Descloux 
 
 UNIVERSITY OF ILLINOIS 
 DIGITAL COMPUTER LABORATORY 
 URBANA, ILLINOIS 
 
 Report No. Il6 
 
 May 10, 1962 
 
 * This work was supported in part "by the National Science Foundation 
 under Grant Gl6^89o 
 
re 
 
 -j >— 3 
 
 REMARKS ON THE ROUND-OFF ERRORS IN ITERATIVE PROCESSES 
 FOR FIXED-POINT COMPUTERS 
 
 Abstract 
 
 This report develops some ideas suggested by A. H. Taub on the round- 
 off errors in iterative processes. It will be shown that in certain cases the 
 convergence of the process can be improved by using special method for rounding 
 
 Part I is concerned with simple first order processes. In part II, the 
 
 2 
 Aitken's 5 process will be considered. 
 
 PART I. ERRORS IN FIRST ORDER ITERATIVE PROCESSES 
 
 Introduction 
 
 Let G , ... G be m real functions of the real variables x , 
 ... x . For any set of m numbers p , . . . p , we shall use the vectorial 
 
 notations: 
 
 =? ( (1) (m)> 
 
 P = (P V--P i-4 
 
 |p| = y(p ') + ... (p ) 
 
 We consider the iterative process 
 
 *n+l = G (*r? > n = °> 1} '" (l) 
 
 and suppose there exists a vector r and a number b (0 < b < l) such that 
 
 |G(x) - r| <b |x - r | ; (2) 
 
 the condition (2) insures the convergence of the x ' s to r. 
 
 n 
 
 We want to realize the process (l) as a fixed-point computer under the 
 two conditions: a) for representing each of the x , we use only one "word"; 
 we consider the content of the word as an integer ; b) we may use higher pre- 
 cision for computing the values of the functions G^ , ... G^ . 
 
 -1- 
 
We distinguish two types of errors: 
 
 l) truncation errors ; even using double precision, we cannot expect to evaluate 
 
 (i) 
 
 the functions G exactly; 
 
 2) round-off errors ; according to the condition a) , the value found for G 
 
 must be rounded to an integer. 
 
 Truncation errors 
 
 Let r^(x), ... IT m )(x) approximate the functions G^ (x) , . ..G^ m '(x): 
 
 H (i) (x*) = G (i) (x*) + | (i) (x*); 
 
 I (x ) is called the truncation error; it is supposed to satisfy the inequality 
 
 U (i) (x)| <a (i) ; a (i) = constant. (3) 
 
 The iterative process 
 
 V _,- = H (V ) (*0 
 
 n+1 K n J v ' 
 
 is considered as an approximation of (l) and gives some information about r. 
 
 — > — > 
 
 Theorem 1 . For any V , the sequence V given by (k) is bounded and all its 
 
 -> 
 
 points of accumulation V satisfy the inequality 
 
 .— > 
 
 ->i 
 
 i—» 
 
 l a 1 -> 1 (l) ( m )\ 
 
 V - r < -t— r 1 a = (a' '. ... a' ') 
 
 Theorem 2 . The process (h) is the best possible in the following sense: for 
 
 given a and b, there exist m functions H (x) , . . .H (x ) for which 
 it is impossible to find an algorithm using only H, a, b , providing 
 closer points of accumulation to r than the algorithm (k) 
 
 — > 
 
 — >\ , -» — » 
 
 Proof : Let G(x) = bx + a, 
 H(x ) = bx 
 G'(x) = bx* - a 
 
 -» 
 
 H(x) is an approximation for both G(x) and G'(x) with limits r = — T and - 
 
 -, ■ a 
 r = l^b 
 
 -2- 
 
If any sequence W has a point of accumulation W such that 
 
 |W - ?| < '* ' 
 
 l-b ' 
 then by the triangular inequality, 
 
 |W - r'l > 
 
 M 
 
 1-b 
 
 and the process (k-) presides in this case a better information. 
 
 Round-off errors 
 
 For the computer, the process (l) can be written in the form 
 
 .(i) = rn(i), 
 
 (i) 
 
 y n+1 = [^-(y n ) + |-] R ; 
 
 (5) 
 
 (i) • 
 
 y is an integer, 
 n 
 
 [ ] is called a rounding procedure. [x]„ is any integer function of x satis- 
 K K 
 
 fying the inequality: 
 
 |[x] R - x| < 1. 
 We consider two particular types of rounding procedures: 
 
 1) normal rounding: [x] = [x + 0.5] ; 
 
 2) anomalous rounding: W/r for l x l < 1> ItxJ/J > l x l 
 
 for |x| > 1, |[x] J < |x| 
 
 -> -> 
 Theorem 3 . Let G and £ satisfy the equations (2) and (3) . If 
 
 £1 = CG (l) ( yn ) ♦ !*]„, 1-1,2...., 
 
 (6) 
 
 then for any y , there exists N such that 
 
 K - ri 
 
 vn 
 
 1 ^ bb + 271^7 for n > N ; 
 
 furthermore, for given a and b, there exist a function G and the errors | for 
 which the bound is attained. 
 
Now, we restrict ourselves to the particular case m = 1, i.e., the 
 process (l) becomes scalar. Equations (l), (2), (3), and (5) can be written 
 as: 
 
 x _,, = G(x ) (7) 
 
 n+1 n 
 
 |G(x) - r| < b |x-r| (8) 
 
 ^n + l - [G ^n> + ^n ] R (9) 
 
 It I <a (10) 
 
 1 n 1 — 
 
 Theorem k . Let G(x) and | satisfy the equations (8) and (10) . If 
 
 y _ = y + [G(y ) + g - y ]. , (ll) 
 
 J n+1 J n w n' n J n A ' 
 
 then for any y , there exists N such that 
 
 II ri 
 
 y , - r < ■=— r- + 1 for n > N. 
 
 J n+1 ' 1-b 
 
 Let us compare the theorem h with the theorem 3 for m = 1. In both 
 cases, the bounds of errors have a common part which can be recognized from 
 theorems 1 and 2 as provided by the truncation errors . The part due to the 
 round-off errors is independent of b for the anomalous rounding; in particular, 
 if a = 0, the error is less than 1 and if the limit r is an integer, it is 
 reached after a finite number of steps. When the convergence is slow, i.e., 
 b ~ 1, the errors can be very large for the normal rounding, even if a = 0; 
 however, if b < 0.5, the normal rounding provides slightly better results than 
 the anomalous rounding. 
 
 Remark. The condition (2) insures a first-order convergence for the process (l) 
 If we assume higher convergence, i.e., if 
 
 |G (x) - r I < b |x - r | P , p > 1, 
 
 we get results which are quite similar, but generaly not simple to formulate. 
 Rather roughly, the theorem k becomes: if y is computed by (ll), then 
 
 ly - rl < B + 1, 
 
 where B is due to the truncation error. 
 
 -h- 
 
Proofs 
 
 — > — > 
 
 Lemma . Let V and V satisfy the equation (k-) under the assumption (2) and (3) 
 
 i->i _» i-*i 
 
 a) if |V - r | < J3- , then |v. - ?| < M 
 
 1 ' o ' — 1-b ' '1 ' — 1-b 
 
 M 
 
 o * ' ' 1-b ' "*"*" ' "1 * ' ' ' "o 
 
 b) if |V - r I > -\^£ , then Iv, - r I < |v - r* 1 
 
 Proof. Since V, = G (V ) + | : 
 1 00 
 
 
 
 
 .— » 
 
 
 
 
 l v i 
 
 — > 
 
 V ■ 
 
 
 — »l 
 - r | 
 
 < 
 
 |a | 
 1-b 
 
 — » 
 
 V 
 
 - r | 
 
 < 
 
 1— >i 
 
 |a | 
 
 ■"*-"* ->| IT* I „ , I'lT ->l |-»l 
 
 < |G(V ) - r I + |T I < b |V - r| + |a| (12) 
 
 a) |V - r| < ■*—. rj- ; we have by (12) 
 
 b , I a 
 ^1^ + 1 | = tb ' ^- e ' d - 
 
 .-» -* \a\ 
 
 b) V - r > ■!—£ ', ve have by (12): 
 ' 1 o ' 1-b 
 
 i - * — >i i - * — >i / \ i - * — »i 1— >i \~~ * — >i 1— >i 1 — >i 1 — >i 
 
 V n - r < V - r - (l-b) V -r + a < V - r - a + a = V -r ; 
 '1 ' — ' o ' /| o l ' ' ' o ' ' ' ' ' ' o ' 
 
 q.e .d. 
 
 I-* ">l la I 
 Proof of the Theorem 1. 1st Case: there is N such that V, T - r < 
 
 N ' - 1-b ' 
 
 by lemma a the same inequality holds for all n > N and the theorem is 
 
 proved . 
 
 i->l 
 
 2nd Case: for all n= 0, 1, 2. ...: |v - r*| > 4ht ', 
 
 ' ' ' ' n ' 1-b 
 
 1 — >i 
 by lemma b ; the positive sequence |V - r| is monotone decreasing and 
 
 converges therefore to a limit I. 
 1— *i 
 
 Q j V 
 
 Suppose that i = -?—£■ + d where d > 0; since b < 1, there exists y such 
 |y- - r| < JS +§ ; by 12: 
 
 n 
 
 ->i .. d i-»i , i-»i a 
 
 Vn - r < — — a + d + a = -^—r}- + c = i, which is a contradiction. 
 • 1 ' 1-b ' ' ' ' 1-b 
 
 Proof of the Theorem 3 - Since |[x] - x| C 0.5, we can write the equation (6) 
 
 in the form 
 
 -5- 
 
y (i) = G (i) (?) M W i = 1, 2, ... m 
 
 where It/ 1 ) I < a' 1 ' + 0.5 , 
 
 1 n ' 
 
 and therefore U | < lal +0.5 vm . 
 
 — > — > i— >i i— >i r 
 
 Replacing £ by tj and |a| by |a| +0.5 vm , we can apply the theorem 1: 
 
 for any £> there exists N such that 
 
 i-> ->i . 11*1 + 0-5 vm _ . .__ 
 
 y - r < J — L r rr- for n > N; 
 
 ,J n ' 1 - b 
 
 but since the y s are integers, there exists a particular e for which the 
 preceding inequality implies 
 
 i-> -* |i*| + .5 Vm ^ T .-. 
 y - r < J — L ^ — — ; for n > N, as desired. 
 
 1 n ' — 1 - b 
 
 We have still to show an example valid for every a and b where the bound of 
 error is attained. Let 
 
 G (i) (3 =bx (l) - a (i) - 0.5 
 
 —> — > — ¥ 
 
 and suppose that for the particular vector y = we have £ = a . 
 
 rm, ~* n * \^*~~ " ~*\ |a*l + v m • 0.5 „ . _ 
 
 Then y = and y - r = J — ■— - — — ; — for n > 0. 
 
 J n |J n ' 1 - b - 
 
 Proof of the Theorem h . We use the two simple properties of the anomalous 
 rounding procedures: 
 
 1) x - 1 < [x] < x + 1 
 
 2) if p < x < q and q - p > 1, then 
 
 p<p+[x-p].<q , provided that p is an integer, 
 and p < q + [x - q]. < q , provided that q is an integer. 
 
 Since the y ' s are integers, the theorem results from the three statements: 
 "'n ° ' 
 
 I. if |y o - r| <^ , then \y ± - r| < ^ + 1; 
 
 n ' if A < l y o " r l < A + 1 ' then l y l " r l < 1Tb + 1; 
 
 y o " r l - T-b + X> then ' y l " r ' < l y o " r ' 
 
 -6- 
 
Statement I: by lemma a: 
 
 r - 
 
 1-b - J o 
 
 < y„ + G(y J +| - y < r + 
 
 o o o — 
 
 1-b ' 
 
 by property 1: r - 
 
 a 
 
 —- - 1 < y + [G(y ) + | - y ]. < r + -7— + 1, 
 1-b J o w o' b o J o A 1-b 
 
 i.e. jy. - r| < r + — T" + 1 > q.e.d, 
 
 a 
 
 Statement II : We suppose r + -r-~T < Y < r + — jT + 1 ("the proof is analogous 
 
 when r - -r— — - 1 < y < r - — — ) ; by lemma b: 
 1-b J o 1-b 
 
 p = r - -7— r- - 1 < y + G(y )+| -y <y = q : 
 1-b J o J o o o ^o 
 
 Since 
 
 
 y > r, q - p > 1 and we apply the property 2: 
 
 r - t^t- - 1 < y + [G(y +| -y] A <y <r+ -r^r + 1 
 1-b J o w o b o J o A J o 1-b 
 
 1 .e 
 
 > ly-L 
 
 r < r + — - + 1. q.e.d. 
 ' 1-b 
 
 Statement III : We suppose y > r + — — - (the proof is analogous when 
 
 y o $ r " l^b 
 
 ) ; by lemma b: 
 
 p=2r-y <y + G(y ) + £ -y <y = q : 
 
 by property 2, since q - p > 1: 
 
 2r - y < y + [G(y ) + I - y ]. < y 
 J o J o yj o / b o J o A J o 
 
 1 .e 
 
 |y 1 - r| < |y Q - r 
 
 q.e .d, 
 
 -7- 
 
PART II. ROUND-OFF ERRORS IN THE AITKEN'S 6 2 PROCESS 
 
 Let G(x) a real continuous function of the real variable x such that 
 
 (1) 
 
 the sequence x defined "by 
 
 x = G(x ) 
 
 n+1 n' 
 
 converges to the limit x = r, 
 
 By the Aitkens 5 process, we define another sequence: 
 
 3n+l v 3n' 
 
 V = 0( W 
 
 V 
 
 V V -V 
 3n 3 n +2 3 n +I 
 
 3n+2 " V x +V -2V . 
 3n 3n+2 3n+l 
 
 (2) 
 
 Let us suppose we want to realize the process 2 on a fixed-point 
 computer with the following conditions: a) We use only one "word" for repre- 
 senting the V.'s; we may consider the content of the word as an integer ; 
 b) we may use higher precision for computing G(V.). 
 
 We cannot expect to compute G(V.) without error; furthermore, if usin^ 
 higher precision, the result must be rounded to an integer . 
 
 Definition o A rounding procedure denoted by [x] is any function of the real 
 variable x satisfying the inequality 
 
 [x] 
 
 R 
 
 x < 1. 
 
 We shall use the following particular rounding procedures: 
 
 1) [x] : rounding away from zero ; it is defined by the inequality 
 
 \[xf\ > |x| 
 
 2) [x] : rounding toward zero ; it is defined by the inequality 
 
 |[xf I < 1*1 
 Example . Let G(x) = 7/8 x and V = 8; by (2), we have 
 
 v i -7 
 
 v 2 = 6,125 
 
 V 5 = °- 
 
 -8- 
 
If we ■want to represent the V. 's only "by integers and if we use the 
 normal rounding procedure, we shall find: 
 
 v l 
 
 = 7 
 
 \ 
 
 = 6 
 
 s 
 
 = 00 
 
 It will he shown that this situation can he improved "by using the 
 following integer process: 
 
 W, , = W_ + [G(W_ ) + |_ - W_ f 
 3n+i 3 n 3 n 3 n 3 n 
 
 VL _ = w + [G(W_ -) + |_ , - W_ ] 
 3n+2 3n 3 n +l 3^+1 3n 
 
 W_. _ = W_ + 
 3n+3 3n 
 
 (w_ - w_ _) 
 
 3n 3n+l 
 
 2W^ . - W, - W_ OJ 
 3n+l 3n 3n+2- ) 
 
 (3) 
 
 I, and ^ . are the errors of computation of G(W_, ) and G(W_, .,): 
 3n 3n+l 3 n 3 n +l 
 
 since the numerator and the denominator are integers, it is possible with the 
 
 help of the remainder to compute W without any error; if the numerator and 
 
 the denominator are simultaneously equal to zero, then W = W = W and 
 
 we set VL , = W_ . 
 3n+3 3 n 
 
 Theorem 1 , We suppose there exists the numbers < h < 1, < c < 1, 5 > 0, l > 1, 
 such that : 
 
 1) |x n - r| <b \x - r| 
 
 1 ' - ' o 
 
 where x and x n satisfy the relation (l) and r - Z < x <r+ 
 o 1 — o — 
 
 2) |V, - rl < c |V -rl 
 1 3 o 
 
 where V and V, satisfy the relations (2) and r - I < V < r + I 
 o 3 J v ' — o — 
 
 3) |G(x) - G(y)| < 6 | x - y| 
 where r-£<x^y<r+i. 
 
 k) The errors £ and (U -, in (3) satisfy the inequality 
 3 n 3 n +l 
 
 l^ 1 - a " d "¥ (l + c)(l + 6) * 
 
 -9- 
 
Then , for any W "belonging to the interval [r - I + 1, r + i -1], there 
 exists a finite number N such that 
 
 W, - r < 1 + t^t- for n > N. 
 1 3 n 1-b 
 
 Theorem 2 . We make the assumptions : 
 
 l) The convergence of the process (l) is alternating, i.e. for 
 r-£<x<r+i: 
 
 < r - G(x) < x - r 
 
 < G(x) - r < r - x 
 
 G(x) - r 
 
 if 
 
 X 
 
 - r > 0, 
 
 if 
 
 X 
 
 - r < 0, 
 
 if 
 
 X 
 
 = r; 
 
 2) 'The errors £ and £_ 
 ' Jn 3n+l 
 
 in (3) satisfy the inequality 
 
 I . < a < — , where a is a fixed number. 
 J ■ - - 3 
 
 Then , for any W belonging to the interval [r - I + — , r + I - —} , there 
 exists a finite number N such that 
 
 |W, - rl < 1 + a for n > N, 
 1 3 n 
 
 Remark . The assumption (l) of the theorem 2 is sufficient for providing the 
 
 convergence of the V 's satisfying the equations (2) for any 
 
 r-£<V < r + i. 
 — o — 
 
 It is easy to prove the inequality 
 
 V, - r < 
 3 n 
 
 V - r 
 1 o 
 
 Proof of the Theorem 1 
 
 Notations 
 
 r is the root of the equation F(x) = and the limit of the process (l) 
 
 W , W , W , W satisfy the equations (3) with errors i and | , 
 
 , u , u are defined by 
 
 -10- 
 
u„ = 
 
 u^ = 
 
 G (W Q ) 
 
 G (W 1 ) 
 
 W o W 2 
 
 > 
 
 < 
 
 o 2 1 
 
 The integers p, q_, s, t are defined "by 
 
 q = p + 1; s = q + 1; t = s + 1; q < r < s. 
 
 Lemas 
 
 The following lemmas, except lemma 1 and 2, are valid only under the 
 assumptions of the theorem 1. 
 
 Lemma 1 . The relations 3 are invariant for the transformation 
 
 W. = -V. , g'(x) = -G(x) , i ± = -i ± , 
 
 i.e., if W', Wl, W' are computed from W' = -W by replacing G by G' and £. by £! 
 in 3, then W^ = -W , W' = -W , W' = -W . 
 
 Proof. W* = W + [G'(W') - W + £'1* 
 — 1 o L x o' o o 1 
 
 = W + [-(G(W ) - W + I )]* 
 o L v v o' o o /J 
 
 = -W - [G(W ) - W + if -- 
 o L x o' o o J 
 
 W • 
 
 the proofs, based on the properties 
 
 [-x] = -[x] and [-x] = -[x] 
 
 and the same for W' and W' . 
 
 x o X 2 " X l 
 
 Lemma 2. Let x , x n , x^ and real numbers and x^ = ■ — r— 
 
 — — — o' 1' 2 3 x + x - 2x 
 
 ax 
 
 o 
 
 > ; ^ > 0; 
 
 2 1 
 bx (x 1 - x Q )(x 1 - x 2 ) 
 
 Sx 2 - 
 
 1 
 
 (x + x_ 
 o 2 
 
 2x 1 )' 
 
 b) If x > x. , there is the following scheme of variations for x 7 , 
 o 1 ° 3 
 
 as function of x 
 
 2' 
 
 ■11- 
 
 UNIVERSIT 
 
 ILLINOIS LI8RARY 
 
f 2x x - Xq f x f 
 
 t * 
 
 x T + °° 
 
 o ' - 
 
 / x x ;|(x o + Xl ) ; 
 
 c) If x > x > x , then x > x > x ; 
 
 if x Q = x 2 
 
 if x ± = x 2 
 
 then x = - (x Q + x ± ) ; 
 
 then x^ = x_ . 
 3 
 
 d) Considering x as a function of x . x , x , one 
 
 has 
 
 x + xx - 2x x ~dx dx 
 
 2 x " x 3 ^ x o ~ ^ x 3 ~ 
 
 Lemma 3 . If W Q > r + 1, then W < W Q . 
 
 Proof. By assumption 1; 
 
 u n - r < b(W - r) 
 1 — v o 
 
 u < W - (I - b)(W - r) < W - (l - b) 
 
 l—o o ' — o 
 
 u n + £ n < W - (1 - b) + a < W - (1 - b)(l 
 1 1 — o — o 
 
 1 (1-^)(1-C) N < „ 
 
 W l = W o + t u l + h - W ol < W o ^ e ' d ' 
 
 Lemma k . Let V > r + 1, V } V , V satisfy the relations (2) and 
 
 let V.,, V rt , V^ be such that 
 123 
 
 V < V < V + d (1 + 8) 
 V 2 > V 2 - d (1 + 8) 
 
 V, = 
 
 V V - V, 
 
 02 1 
 
 5 V + V - 2V n 
 
 o21 
 
 then 
 
 V^ > 2r - V 
 3 o 
 
 ■12- 
 
Proof . By assumption 2: V, > 2r - V . By assumption (l) : V < V . 
 Let x = V + a 
 
 x 2 = v 2 - a 
 
 f s V o X 2 - x l 
 
 y v o+ x 2 - 2Xl 
 
 We have: x,(0) = V^ > 2r - V 
 3 3 o 
 
 x 3 ( j ) = - - 
 
 Since x^(oj) is continuous in the interval 
 
 V o + V 2 " 2Y l\ 
 0, — J , there exists 
 
 V + V -2V 
 
 < B < — i for which x,(b) = 2r - V . 
 
 3 3 o 
 
 It is easy to check that V n + S < V . 
 
 1 o 
 
 By lemma 2a , for every V < V < B and V > V - B , one has 
 
 V^ > x,(B) = 2r - V . 
 3 3 o 
 
 For proving the lemma k, we have to show that d(l+8)<p. 
 
 v q (v 2 - p) - (v 1 + b) 2 
 
 3 satisfy the equation: y + y . p _ g(y + p ) = 2r - V q . 
 
 2 
 By the well-known translation invariance of the 8 process, we have: 
 
 (V o -r)(V 2 -r-p) - (V 1 -r+p) 2 
 (V Q -r) + (V 2 -r-p) - 2^-r+p) + (V o " r) = °" W 
 
 We set: (V, - r) = V(V - r) -b < b' < b 
 
 1 N o — — 
 
 (V, - r) = c' (V - r) -c < c' < c 
 3 o — — 
 
 Solving {k), we find: 6 = (V Q -r) {-(2 + h')J ^ + ^ ' 2 ^^T ^ C ' ' 6V c ' | 
 
 -13- 
 
2 ( l -"b ' ) ^ 
 the derivative of the quantity under ^ ~~ ' with respect to c' is — — > 0; 
 
 (l-c«) 
 
 .\2 
 
 if we replace c' "by -c and (V -r) by 1, we get the inequality: 
 P> {-(2+V) + J ? + 2b' 2 ^^ 3C + fib'c | ; 
 
 We set A = (2+b : ) 2 and B = J ' so that > Vp - *Ja . 
 
 It is easy to show that l6 > B > A; by the mean value theorem: 
 
 / B - nTa > a. (b-a) > (1 -^V c) (1 r^ )2( ^ c) 
 
 aTi 
 
 B 
 
 4(l+c) 
 
 Ml+c) 
 
 = d(l+6) , 
 
 i.e 
 
 d(l+6) < p as desired. 
 
 Lemma 5 • If W > u.. > w n > r , there exists z such that 
 - oil 
 
 G(z) = W n and W > z > W n . 
 
 1 o 1 
 
 Proof. G(x) is a continuous function with G(W ) = u n and G(r) = r: 
 o 1 
 
 fo 
 
 r u, > W., > r, there exists W > z > r for which G(z) = W n . By 
 11 o 1 
 
 assumption 1, z > W. 
 
 1 
 
 Lemma 6 . For any z satisfying the conditions 
 
 W >r + l>z>r and G(z) = W n , 
 o — 1 
 
 we have W p > W . 
 
 Proof. Let u 
 
 zu 2 - WJ 
 
 3 z + u 2 - 2W 
 
 (5) 
 
 By assumption 2: u_, > 2r - z > r - 1: 
 
 "by lemma 2d, if we replace in (5) u by r - 1 < u and z "by r 
 letting W unchanged , we have to replace u by y with u > y 
 
 + 1 
 
 (r + 1) y - W 1 
 r + 1 + y - 2W n 
 
 r - 1, 
 
 -14- 
 
W 1 - 7 = 
 
 1 - (W 1 - r) 2 -l 
 
 <2 > 
 
 consequently: u > J > W 
 
 1 " 2 ' 
 
 W = W + [u_ - W + | n Y > W + [W_ 
 
 2 o'2 o "1 — o 1 
 
 w„ - if , 
 
 by assumption 1: W < Z < W ; 
 
 therefore W n - W < and [W n - W 
 1 o 1 o 
 
 w 2 > W i » q.e.d. 
 
 I ] ' - W l " W o ' 
 
 Lemma 7, IfW >r+l+ 
 
 o - 
 
 — , then W„ < W : if the sign = ' holds, 
 
 1 - b ' 2 — o ' 
 
 W - W n > 2, 
 
 O -L — 
 
 Proof . By assumption 1: 
 
 |u x - r| < b(W Q - r), 
 
 |w_ - r| < b(W - r) + a + 1, 
 
 1 1 o ' 
 
 i i 2 
 
 u_ - r < b (W - r) + ab + b, 
 '2 o 
 
 u_ + i_ < r + "b (W - r) + ab + b + a, 
 2 1 o 
 
 u 2 + | 1 - W q < (b 2 - i)(W q - r) + ab + b + a < (b 2 - l)(l + 3-^) 
 
 2 
 + ab+b + a^b - 1 + b < 1, 
 
 consequently: [u + | - W J < 0, 
 
 W 2 = W o + [u 2 + l x - W o T < 0, q.e.d. 
 
 For proving the second part of lemma 7j> suppose that W - W , but W - W < 2, 
 
 By lemma 3> "the only possibility is W = W - 1 for which W - r > - — — r- . 
 By assumption 1: |u - r| < b(W - r), 
 
 u 2 < r + b(W x - r), 
 
 u 2 + i x - W 1 < (b - l)(W L - r) + a < (b - l) Y^Tb + a = °> 
 
 U 2 + h < W l 
 
 W = [u + I ] < W < W , which is a contradiction. 
 2 2 IK — 1 o 
 
 -15- 
 
Lemma 8. If W > r + 1 + and W, < 2r - W , then 
 
 o - 1 - b 1 — o' 
 
 W_ > W_ > 2r - -W - 1 
 
 2 1 o 
 
 Proof. By assumption 1 
 
 u > r - b(W - r), 
 
 W = [u + | ] > r - b(W -r)-a-l = 2r-W +(l- b)(W - r) 
 
 W 
 
 ' > 2r - W + (l - b)(l + 
 1 o 
 
 - a - 1; 
 a \ 
 
 a - 1 = 2r - W - b > 2r - W - 1. 
 
 o o 
 
 1 - V 
 
 For the second part of the lemma, we use again the assumption 1: 
 
 u 2 > r - b(r - W 1 ), 
 
 u 2 > W 1 + (1 - b)(r - W x ) > (1 - b)(l + 3-f^) y 
 U 2 + ^L - U 2 
 
 a > W + 1 - b > W , 
 
 u + e -W>W - W 
 2 ? 1 o 1 o' 
 
 by lemma 3 • W - W < 0; 
 
 [u 2 + | x -W o ]"> Wl -W o , 
 W 2 = W q + [u 2 + l x - W q ]* > W x , q.e.d. 
 
 Lemma 9' 
 
 If W > r > W n , then W > W n , 
 o — — 1' 2-1 
 
 Proof, 
 
 By assumption 1, u > W ; 
 
 u 2 + \ > \ 
 
 a > W^ 
 
 1 
 k ' 
 
 W^ = W + [n + | n - W ] > W + [W T 
 2 o 2 1 o — o 1 
 
 \'\f = \, q.e.d, 
 
 Scheme of the proof of Theorem 1 
 
 The theorem results from the two statements : 
 
 l) If |W - r| > 1 + n a . , then |w^ - r| < |w - r| for r 
 o — 1 - b ' '3 o 
 
 + i - 1. 
 
 I + 1 < W < r 
 — o — 
 
 2) If [W - r| < 1 + , then |w - r| < 1 + — 
 
 -16- 
 
We can restrict ourselves to the case W > r. Indeed, suppose that the state- 
 ments i) and n) are right for W > r and consider a particular value W < r. 
 We set W ' = -W , G'(x) = - G(-x), |.' = - |. and compute W ' W ' , W ' by the 
 equations 3 with the ' values. Since |.', G'(x), -r satisfy the same hypothesis 
 as I., G(x), r, ve have: 
 
 I) If |W ' + r| > 1 + 
 o 
 
 |W_ - r| < |W - r 
 1 3 o 
 
 — i then W'+r<W'+r and by lemma 1 
 
 1 - b 3 o 
 
 II) If |W ' + r| < 1 + — 
 JW 3 - rj <l +r ^ 
 
 — , then W ' + r < 1 + — 
 b '3 1 
 
 and by lemma 1 
 
 Statement I: 
 
 If W > r + 1 + 
 o — 
 
 I — - , then |W 3 
 
 r I < I W - rl 
 
 o 
 
 Proof 
 
 We distinguish three cases : 
 
 1) W > W 
 
 1 — o 
 
 2) W > W, > W„ 
 
 o 1 2 
 
 3) W > W n and W„ > W n 
 
 o 1 2—1 
 
 l) By lemma 3> this case never occurs. 
 
 2a) W > W > W and W n > u n = G(W ) . 
 o 1 2 1-1 o 
 
 o 2 
 
 V x 2 
 
 V + V - 2V n 
 o 2 1 
 
 We define V = W , V = u n = G(W ), V = G(u n ). V n 
 o o 1 1 o 2 13 
 
 Since W_, = W + [u n + | - W and W n < W : 
 
 1 o 1 " o o 1 o 
 
 W < u + a < u + d , 
 
 By assumption 3: |u -V | = |g(W ) - G(u )| < o |w - u | < 5 d; 
 
 Since W = W + [u_ + £.. - W and W^ < W we have: 
 
 2 o 2 1 o 2 o 
 
 W 2 > u 2 " a > u 2 " d > V 2 " d (! + 5 ) • 
 
 We set V. = W. and apply the lemma h, the V.'s keeping their signification 
 
 W n > 2r - W . 
 3 o 
 
 By lemma 2b : u < W ; 
 
 W_ = [u l < W o < W so that 
 3 3 R _ 2 o 
 
 |W 3 - r| < |W q - r|, q.e.d. 
 
 -17- 
 
2b 
 
 ) W > W., > W_ and W n < u = G(W ) 
 o 1 2 1 1 o 
 
 By lemma 9 : W > r, 
 
 By lemma 5, there exists Z such that 
 
 W > Z > W_ and W n = G(Z). 
 o 1 1 
 
 By lemma 6 : Z > r + 1 „ 
 
 Since W^=W + [u„ + i_ - W < W , we have W^ > u. - a > u_ - d. 
 2 o 2 1 o o' 2-2 -2 
 
 We set Z = V , W = V , u = 1 , W = V , W = V and apply the lemma K 
 
 ZW - W 2 
 
 V = * 77 ^77" > 2r - Z > 2r - W ; 
 
 3 Z + W^ - 2W n o 
 
 2 1 
 
 Since W > Z, by lemma 2a: 
 
 W W - W 2 
 
 W = — - — — > V > 2r - W . 
 
 * W + VL - 2W, ^ ° 
 o 2 1 
 
 By lemma 2b : u < W ; 
 
 W_ = [u_]_, < W < W so that 
 3 3 R ~ 2 o 
 
 W^ - r < W - r , q „e„d. 
 '3 o 
 
 3) By lemma 7> W _ < W , we distinguish: 
 
 3a) W > W > W, > 2r - W : 
 o 2-1 o ' 
 
 3b) W q = W 2 ; 
 
 3c) W > W^ and W n < 2r - W . 
 o 2 1 — o 
 
 3a) By lemma 2c: W > u > W ; since W , W are integers: 
 
 W. > W_ > W_, i.e., |w_ - r| < |w - rl q.e.d. 
 
 2 — 3 — 1 3 o 
 
 3b) By lemma 2c: u = |(W + W ); 
 
 by lemma 7 ': u. < ^(2W - 2) = W - 1; 
 3 — 2 o o 
 
 W_ = [u_]_ < W - 1 < W . 
 
 3 3 R - o o 
 
 By lemma 8: W > 2r - W - 1; 
 
 u 3 > r - § ; 
 
 W = W + [u_ - W ] >W + [r - =■ - W ] >r-^>2r-W; 
 3 o 3 o o 2 o — 2 o' 
 
 therefore: W > VL > 2r - W , i.e., |w„ - rl < |w - rl q.e.d. 
 o 3 o 3 o 
 
 -18- 
 
3c) By lemma 8: W > W_ > W_ > 2r - W - 1; 
 -> ' J o 2 1 o 
 
 by lemma 2c: W > W > u_ > W. > 2r - W - 1: 
 J o231 o 
 
 since W„ = W + [u_ - W ] : 
 3o 3o 
 
 W > W > 2r - W , i.e., W_ 
 o' 3 
 
 2 - 3 
 Statement II 
 
 r < W - r 
 o 
 
 q.e.d. 
 
 If r < W < 1 + - 
 
 — o 1 
 
 -^ , then |W 3 - r| < 1 + ^—-^ 
 
 Proof. 
 
 We distinguish two cases and describe for each of them all the 
 possibilities without any computation: 
 
 1) W Q - s; 
 
 2 ) W = t and s - r < a . 
 
 o 
 
 l) W = s 
 o 
 
 W 2 = s 
 
 W 
 
 3 
 
 la) W = t /W_ = q : W = s 
 
 w 2 = p 
 
 3 
 V =s 
 
 2a) W = s 
 
 W = s (u does not depend on the value of W ), 
 
 3 
 
 3 
 
 w 2 = t 
 
 3a) W = q: / W = s 
 
 W 2 = q 
 
 W 3 =s 
 W 3 = s 
 W 3 =q 
 
 ka) W = p 
 
 w 2 = t 
 
 W 2 = s 
 W ? = q 
 
 W 3 = s 
 W 3 =q 
 W„ = q 
 
 2) W = t and s - r < a 
 
 2a) W 
 
 w 2 = t 
 
 W^ = s 
 
 W„= t 
 
 W„ - s 
 
 2b) W = q 
 
 W« = t 
 
 VL = s 
 
 W, 
 
 W. 
 
 -19- 
 
w„ = t 
 
 2c) W n = p h. = 
 
 w o = q 
 
 w. 
 
 W 3 = s 
 
 w 3 = , 
 
 Proof of the Theorem 2 
 
 We shall use the same notations as in the proof of the theorem 1. 
 
 Lemmas 
 
 The following lemmas are valid only with the assumptions of the 
 theorem 2 . 
 
 Lemma 10 . If W > t, then W < s; the sign "=" holds only if s - r < — . 
 
 Proof . By assumption 1: u < r; 
 
 u n +! < r + - < t; 
 1 o - 3 
 
 W 
 
 . = W + [tl. + I - W 3 < t; 
 
 1 o loo 
 
 W can equal s only if r +^> s, i.e., s - r < — , q.e.d, 
 
 Lemma 11. If W > r + 1, then V/ > W n . 
 o — 2—1 
 
 Proof . a) Suppose W < q; then: 
 
 u 2 > v > 
 
 u 2 + h > P ' 
 
 W 2 = W q + [u 2 + l x - W q ] > P > q > W v 
 
 q.e.d. 
 
 b) Suppose W > s; since W > t, by lemma 10, W = s and s 
 by assumption 1: u > r -~^><l + ^T 
 by assumption 2: u + I > q; 
 
 1 
 r< 3 ; 
 
 W 2 = W q + [u 2 + ^ - W q ] > q 
 
 i.e. 
 
 W > s = W 
 
 q.e.d. 
 
 -20- 
 
Lemma 12 , 
 
 Proof, 
 
 If W > r, then W < W + 1, 
 o — ' 2-o 
 
 By assumption 1: |u - r| < (W - r); 
 
 |W 1 - r| - |[u x + | q ] r - r| < \u ± - r| + k/ 3 < (W Q - r) + h/3; 
 
 |u " r| < |W 1 - r| < (W q - r) + k/3', 
 
 U 2 < W o + ^ /3; 
 
 u 2 + fi x < w o + 5/3; 
 
 V7 = W + [u_ + I. -W f < W + [5/3] = W + 1, q.e, 
 o 2 1 o — o ' o 
 
 Lemma 13- 
 
 If w = w +1 and W < W - 3, then W n < W . 
 2 o l-o 3 o 
 
 Proof. 
 
 If we keep W constant, u is an increasing function of W by 
 
 lemma 2a: since W_ = W + [u_ - W , W_ will have the same 
 
 3o3o3 
 
 property and it suffices to prove the lemma for the case 
 
 W = w - 3» one finds: 
 1 o J ' 
 
 u = W - 9/7 < W - 1 
 3 o o 
 
 W Q = [uJ D < W - 1 < W , q.e.d. 
 3 3 R - o o' 
 
 Lemma ik . 
 
 If W„ = W > r + 1 - a, then W n < r and W_ < W . 
 2 o - 1 3o 
 
 Proof . Suppose W > r; 
 
 by assumption 1: u < r, 
 
 u_ + (L<r + a<W - 1, 
 
 2 1 — — o 
 
 W = [u + i ] < W - 1, which contradicts our hypothesis. 
 2 2 1 K — o 
 
 We prove the second part of the lemma; since W < r, we have 
 
 W - W, >-l, i.e.", W - W, > 2: 
 
 o 1 o 1 - 
 
 by lemma 2c: u = J(W + W n ) = W + ^(W n - W ) < W - 1: 
 3 2 o 1 o 21 — 
 
 w 3 = [u 3 ] R < W q - 1 < W q , q.e.d. 
 
 Lemma 15. If W. > W > r + 1 + a, then W n < r - 1 and W n < W , 
 2 o — 1 3o 
 
 -21- 
 
Proof, 
 
 Suppose W > r - 1; 
 
 by assumption 1: u < r + 1, 
 
 u^ + L<r + l + a<W. 
 2 1 - o 
 
 VL = [u_ + |, ]„ < W , which contradicts our hypothesis. 
 2 2 1 R - o' J ^ 
 
 We prove the second part of the lemma; since ¥ < r - 1, we have 
 
 W - W, > 2. i.e., W - ¥ > 3; 
 
 o 1 ' ' o 1 — ~" 
 
 by lemma 12 : ¥^ = W + 1: 
 
 J 2 o ' 
 
 by lemma 13: ¥ < ¥ , q.e.d. 
 
 Lemma l6. If W > r + k/3, then ¥_ > 2r - W 
 o - ' -" 3 c 
 
 Proof, 
 
 a) Suppose ¥ > r; 
 
 by lemma 11 : ¥ > ¥ > r; 
 
 by lemma 2b : u > ¥ , 
 
 ¥_ = [u_]_, > W n > r > 2r - ¥ , q.e.d. 
 3 3R-1- o' 
 
 b) Suppose ¥ < r; we have the inequalities: 
 
 |u x - r| < (¥ q - r), 
 
 u > 2r - ¥ , 
 1 o 
 
 n + | > 2r - ¥ - \ , 
 1 o o 3 
 
 W l = [u i + l o\ > 2r ' W o " ^ 3 - \ ' 
 
 u 2 > r > 
 
 u 2 + I-l > r - i 
 
 ¥_ = ¥ + [u + L - W f > W + [r - -| - W ] > r - ^ = ¥ . 
 
 2o 2 1 o — o 3° — 32 
 
 By lemma 11, ¥ > ¥ ; by lemma 2a, for fixed ¥ , u_ decreases 
 when ¥_ and ¥ decrease; consequently: 
 
 u, 
 
 ¥ W„ 
 o 2 
 
 ¥ n 
 
 > 
 
 ¥ + ¥ 
 o 2 
 
 2W, 
 
 ¥ VL - ¥ n 2 
 o 2 1 
 
 ¥ + ¥ - 2¥ n 
 o 2 1 
 
 = B 
 
 -22- 
 
k 
 
 The fact that B > 2r - W for W > r + - results from the 
 
 o o - 3 
 
 three statements : 
 
 1) B is a continuous function of W for r < W < °° : 
 
 o — o ' 
 
 -W 
 
 2 ) for W -> oo W and therefore B > 2r - W when W 
 
 o 3 3 o c 
 
 is large enough. 
 
 1 + ^33 ^ , k 
 
 — r t ~ 
 o 
 
 r + 7 < r + — , is the only value in [r, °°] 
 
 for which B = 2r - W . 
 
 o 
 
 h 
 We have therefore established that u_ > 2r - W for w > r + — . Now 
 
 3 o o - 3 
 
 W„ = W + [u_ - W Y > 2r - W , q.e.d. 
 3 o 3 o o' 
 
 Lemma 17. If W > r + 1 + a, then W„ > 2r - W . 
 o — 2 o 
 
 r k 
 
 Proof . By lemma 16, we have only to establish the lemma for W < r + — , i.e., 
 
 1 
 W = t, s - r < -. 
 
 o 3 
 
 a) Suppose W > r; by the same argument used in lemma l6, we con- 
 clude that W n > 2r - W . 
 
 3 o 
 
 b) Suppose W < r; 
 
 o 
 
 u>2r-t>t--, 
 
 u 1 +l 1 >t-|=p, 
 
 W 
 
 [U 1 + Ur ^ P ■ 
 
 u 2 > r > 
 
 u 2 -i - r " 3 q 3 ; 
 
 W 2 = W o + [U 2 + 6 1 ~ W o ] > q + \ ' i • e • , W 2 - S; 
 
 by lemma 11 : W > W ; by lemma 2a : 
 
 2 2 
 
 W W - W n ts - p 
 o 2 1 ~ 1 
 
 U 3 " W + W - 2W n - t + s - 2p q + 5 ; 
 o 2 1 ^ 
 
 therefore: W_>s>r>2r-W, q.e.d. 
 3 - - o' 
 
 -23- 
 
Scheme of the proof of the theorem 2 
 
 The theorem results from the two statements : 
 
 1) If |w - r| > 1 + a, then |w_ - rl < |w - rl for r -,#+-< W < r + I - -. 
 
 1 o - 3 o 3-0- 3 
 
 2) If|w - r| < 1 + a, then |w - r| < 1 + a. 
 
 Using the same argument as in the proof of the theorem 1, we can 
 
 restrict ourselves to the case W > r. 
 
 o — 
 
 Statement I: If W > r + a, then W_ - r < W - r, i.e.. 
 o - 3 o 
 
 1 ) W < W 
 
 3 o 
 
 2) W., > 2r - W . 
 
 3 o 
 
 1) We prove that W < W ; we distinguish three cases: 
 
 a) W < W ; by lemma 11, W > W : by lemma 2c, u < W ; 
 
 since W_ = [u_] D , W_ < W < W . 
 3 3 *< 3 — ^ o 
 
 b) W^ = W ; by lemma 1^, W„ < W . 
 
 2 o 3 o 
 
 c) W > W ; by lemma 15, W < W . 
 
 2) By lemma 17, W > 2r - W . 
 
 2 o 
 
 Statement II. If r < W < r + 1 + a, then Uj - r < 1 + a. 
 - o 3 
 
 We distinguish two cases; for each of them, we describe all the 
 
 possibilities without any computations: 
 
 1) W = s 
 
 o 
 
 2 ) W = t and s - r < a . 
 
 o 
 
 l) W = s 
 o 
 
 lb) W = s : W = s (u does not depend on W ) 
 
 -2k- 
 
W„ = s 
 
 W^ = s 
 
 lc) W = q 
 
 W = q : W = q 
 
 Id) W = p 
 
 W, 
 
 w ^ = q 
 
 vi = 
 
 w„ = 
 
 2 ) W = t and s - r < a 
 
 2a) W = s 
 
 W^ = t 
 
 W„ = s 
 
 W = t 
 3 
 
 W = s 
 
 2b) W 
 
 W„ = t 
 
 W^ = s 
 
 W 3 =s 
 W = s 
 
 'W,= t + 1 . : W,= s 
 
 2c) W n = p J W^ = t 
 
 W, 
 
 = s 
 
 W. 
 
 ■25-