Abbreviated title : Verication for nonsmooth equations 2

Transcription

1 A Verication Method for Solutions of Nonsmooth Equations Xiaojun Chen School of Mathematics University of New South Wales Sydney, NSW 2052, Australia February 1995 Revised March 1996 Abstract A Verication Method for Solution of Nonsmooth Equations. This paper proposes a verication method for the existence of solutions of nonsmooth equations. We generalize the Krawczyk operator to nonsmooth equations by using the mean-value theorem for nonsmooth functions. We establish a semi-local convergence theorem for the generalized Newton method for nonsmooth equations. The proposed method is a combination of the generalized Krawczyk operator the semi-local convergence theorem. Zusammenfassung Eine Methode zur Verizierung von Losungen nicht-dierenzierbarer Gleichungen Dieser Artikel schlagt eine Methode zur Verizierung der Existenz von Losungen nicht-dierenzierbarer Gleichungen vor. Wir verallgemeinern den Krawczyk-Operator auf nicht-dierenzierbare Gleichungen indem wir den Mittelwertsatz fur nicht-dierenzierbare Funktionen benutzen. Wir entwickeln einen semilokalen Konvergenzsatz fur das verallgemeinerte Newton-Verfahren, angewendet auf nicht-dierenzierbare Gleichungen. Die vorgeschlagene Methode ist eine Kombination aus verallgemeinertem Krawczyk-Operator und dem semi-lokalen Konvergenzsatz. Key words: verication of solutions, nonsmooth equations AMS(MOS) subject classication : 65H10, 65G10, 90C33 This work is supported by the Australian Research Council. 1

2 Abbreviated title : Verication for nonsmooth equations 2

3 1 Introduction Nonsmooth equations provide a unied framework for the study of a number of important problems in numerical analysis mathematical programming. Although several techniques have been designed for solving nonsmooth equations [3], [4], [6], [7], [11], [12], [13], [14], [20], [21], [23], [24], [25], [26] little has been done to verify the existence of solutions of nonsmooth equations. One likely reason for this omission is the diculty of nondierentiability. Verication methods for solutions of smooth equations have been studied for several decades. These methods utilize the dierential mapping in algorithms numerical analysis [1], [2], [9], [10], [17], [22], [27], [28]. Although these methods are ecient for smooth equations, they are not applicable for nonsmooth equations. Example 1.1 Let f(x) = 0:780 0:563 0:913 0:659 x1 x 2? 0:217 0:254 ; g(x) = x 1 x F (x) = min(f(x); g(x)); where \min" denotes the componentwise minimum operator on a pair of vectors. The system of nonsmooth equations F (x) = 0 is equivalent to the linear complementarity problem [21]: f(x) 0; g(x) 0 f(x) T g(x) = 0: The point x =(1, -1) is the unique solution of F (x) = 0. The function F is nondierentiable at x. Using the numbers in f(x), we set ^x=(0.0, 0.254/0.659) x =(0.0, 0.217/0.563). It is easy to verify F (^x) = (?1:517 10?6 ; 0:0); F (x) = (0:0; 1:776 10?6 ); k^x? x k 1 = 1 + 0:254=0:659; kx? x k 1 = 1 + 0:217=0:563: Furthermore F is not dierentiable at x. The function f in this example was given by Forsyth (1970) [28]. A common numerical practice is to stop an algorithm for solving F (x) = 0 whenever the norm kf (x k )k is less than a given tolerance. However, as Example 1.1 shows, kf (x k )k 0 does not guarantee that x k is a good approximate solution. The goal of this paper is to give a practical verication method for solutions of nonlinear equations: F (x) = 0; (1.1) where the mapping F : < n < n is assumed to be locally Lipschitz continuous. 3

4 If F is dierentiable, a popular method for solving (1.1) is the Newton method x k+1 = x k? F 0 (x k )?1 F (x k ): (1.2) Several verication methods for solutions of smooth equations are based on the Newton method (1.2). Very recently, Alefeld-Gienger-Potra [1] presented a new stopping criterion for the Newton method (1.2) by combining the properties of the Krawczyk operator a corollary of the Newton-Kantorovich theorem. When this criterion is satised they used the last three Newton's iterates to compute an interval vector that is very likely to contain a solution of (1.1). In the nonsmooth case, F 0 (x k ) may not exist. The generalized Newton method uses generalized Jacobians of F to play the role of F 0 in the Newton method (1.2). From Rademacher's theorem, the local Lipschitz continuity of F implies that F is almost everywhere dierentiable. Let D F be the set where F is dierentiable. B F (x) = f lim xi x x i 2D F F 0 (x i )g: (1.3) The generalized Jacobian of F at x 2 < n in the sense of Clarke [8] is equal to the convex hull B F (x) = co@ B F (x); (1.4) which is a nonempty convex compact set. Note that the local Lipschitz continuity of F implies that F is F -dierentiable at x if only if it is G-dierentiable at x [8]. Therefore there is no dierence between F -dierentiable G-dierentiable in this paper. We say that F is BD-regular at x if all V B F (x) are nonsingular that F is BD-regular in a set D if F is BD-regular at any point x 2 D [21], [23], [25]. Here BD sts the subdierential dened in (1.3). This paper generalizes the technique in [1] to a BD-regular nonsmooth operator F in a domain gives a stopping criterion for the generalized Newton method : x k+1 = x k? V?1 k F (x k ); (1.5) where V k B F (x k ). Calculating an appropriate element V k B F (x k ) is based on the denition of the function F. For instance, see Example 5.1 Example 5.2 in section 5. The verication method in [1] is based on the Krawczyk operator the Newton-Kantorovich theorem, which requires a Lipschitz continuous dierentiability assumption on the function F. In this paper, we generalize the Krawczyk operator to nonsmooth equations. We establish a semi-local convergence theorem for the generalized Newton method (1.5). Our verication method is based on the generalized Krawczyk operator the semi-local convergence theorem. The remainder of the paper is organized as follows. In section 2 we discuss the generalized Krawczyk operator for nonsmooth equations. In section 3 we give a semi-local convergence theorem for the method (1.5). In section 4 we propose a verication method for solutions of nonsmooth equations. In section 5 we report numerical experiments. 4

5 S(x 0 ; r) denotes an open ball with the center x 0 2 R n the radius r > 0. S(x 0 ; r) denotes the closure of S(x 0 ; r). Interval vectors are denoted by [x]; [y]; ::: : Interval matrices are denoted by [L]; [M]; :::. The diameter of an interval [x] = [x; x] is d([x]) = x?x: The midpoint of an interval [x] = [x; x] is m([x]) = 1 2 (x+x): 2 A generalized Krawczyk operator Let F be dierentiable in an open domain D < n let [x] be an interval vector in D. The Krawczyk operator is dened by K([x]; x; A) = x? A?1 F (x) + (I? A?1 F 0 ([x]))([x]? x); (2.1) where x is a vector in [x], A is an nn nonsingular matrix F 0 ([x]) is an interval arithmetic evaluation of the derivative. If K([x]; x; A) [x], then F has a zero x in K([x]; x; A). To generalize the Krawczyk operator to nonsmooth equations, we use the following mean-value theorem. Theorem 2.1 [8] Let F be Lipschitz continuous on an open convex set D in < n, let x y be points in D. Then F (y)? F (x) 2 (xy)(y? x); where xy is the line segment between x y, Let Then Theorem 2.1 implies (xy) = cofv (u); u 2 xyg: ([x]) = cofv (x); x 2 [x]g: F (y)? F (x) 2 ([x])(y? x); for x; y 2 [x]: (2.2) Let [L [x] ] be an interval matrix such that ([x]) [L [x] ]. Then for any x; y 2 [x] < n there holds F (x)? F (y) 2 [L [x] ](x? y): (2.3) Let A be an n n nonsingular matrix let x be a vector in [x]. We call the mapping B([x]; x; A) = x? A?1 F (x) + (I? A?1 [L [x] ])([x]? x) (2.4) a generalized Krawczyk operator. One will naturally ask how to calculate the interval matrix [L [x] ]. If F is continuously dierentiable, the [L [x] ] can be dened by the interval arithmetic evaluation of the derivative F 0 ([x]). In the nonsmooth case, calculating [L [x] ] is based on the 5

6 generalized (x). For instance, we consider the linear complementarity problem Mx + p 0; Nx + q 0; (Mx + p) T (Nx + q) = 0; (2.5) where M; N 2 < nn, p; q 2 < n. The linear complementarity problem (2.5) is equivalent to the nonsmooth equations F (x) = min(mx + p; Nx + q) = 0: (2.6) The nonsmoothness happens when there is an i, 1 i n such that (Mx + p) i = (Nx + q) i M i 6= N i. Here M i N i are the ith rows of M N respectively. By the denition of the generalized Jacobian (1.3)-(1.4), where i (x) = >: M i if (Mx + q) i < (Nx + p) i N i if (Nx + q) i > (Nx + p) i [l i ; l i ] if (Mx + q) i = (Nx + q) i ; l i = (minfm i;1 ; N i;1 g; ; minfm i;n ; N i;n g) l i = (maxfm i;1 ; N i;1 g; ; maxfm i;n ; N i;n g): Hence the interval matrix [L [x] ] for the function F in (2.6) can be dened by [L [x] ] = [l; l]; l i;j = minfm i;j ; N i;j g; l i;j = maxfm i;j ; N i;j g: Example 1.1 is a linear complementarity problem, the interval matrix for the concrete example has the form [0:780; 1] [0; 0:563] [L [x] ] = : [0; 0:913] [0:659; 1] Notice that the interval matrices satisfying (2.3) are not unique. The optimal interval matrices for the generalized Krawczyk operator have the smallest diameter d([l [x] ]). Now we give an interval test for solutions of nonsmooth equations. Theorem 2.2 Assume that F is Lipschitz continuous in an open domain D. 1. If B([x]; x; A) [x] D, then there is a solution of (1.1) in B([x]; x; A): 2. If B([x]; x; A) \ [x] = ;, then there is no solution of (1.1) in [x]. Proof: The application of Theorem 2.1 yields that for any y 2 [x] y? A?1 F (y) = x? A?1 F (x) + y? x? A?1 (F (y)? F (x)) 2 x? A?1 F (x) + y? x? A?1 (xy)(y? x) x? A?1 F (x) + (I? A?1 [L [x] ])(y? x) x? A?1 F (x) + (I? A?1 [L [x] ])([x]? x) = B([x]; x; A): (2.7) Since B([x]; x; A) [x], Brouwer's xed point theorem [19] implies that there is a solution x of (1.1) in [x]. From (2.7), x = x? A?1 F (x ) 2 B([x]; x; A). Thus Statement 1 holds. Statement 2 is a direct corollary of Statement 1. 6

7 3 The semi-local convergence theorem In this section we give a semi-local convergence theorem for the generalized Newton method (1.5). This theorem gives sucient conditions for the convergence of the method (1.5) starting at x 0. Moreover, it provides an error estimate. Theorem 3.1 Let F be BD-regular in a ball S(x 0 ; r). Assume that there exists a constant 2 (0; 1) such that for any x; y 2 S(x 0 ; r); V x B F (x); V y B F (y), kv?1 y (F (y)? F (x)? V x (y? x))k kx? yk: (3.1) Let V 0 B F (x 0 ) 0 = kv?1 0 F (x 0 )k. If 0 (1? )r; then the sequence fx k g generated by the method (1.5) with starting point x 0 is well-dened, remains in S(x 0 ; 0 =(1? )) converges to a solution in S(x 0 ; 0 =(1? )). Furthermore, we have the error estimate kx k? x k 1? kx k? x k?1k; k = 1; 2; ::: : (3.2) Proof: Let r = 0 =(1? ). First we show that the sequence fx k g lies in S(x 0 ; r). From kx 1? x 0 k = kv?1 0 F (x 0 )k = 0 < 0 =(1? ) = r; we have x 1 2 S(x 0 ; r). Suppose that x 0 ; x 1 ; :::; x k 2 S(x 0 ; r). Then kx k+1? x k k = kv?1 k F (x k )k = kv?1 k (F (x k )? F (x k?1)? V k?1(x k? x k?1))k kx k? x k?1k k kx 1? x 0 k = k 0 : Hence kx k+1? x 0 k kx kx i+1? x i k 0 kx i=0 i=0 i 0 1? = r < r; which implies x k+1 2 S(x 0 ; r). Therefore, the sequence fx k g lies in S(x 0 ; r). For any two integers k 0 p > 0, we have kx k+p+1? x k k k+p X i=k kx i+1? x i k 0 k+p X i=k i k r: This shows that fx k g is a Cauchy sequence. Hence the generalized Newton method (1.5) converges to a point x 2 S(x 0 ; r). Since F is Lipschitz continuous in S(x 0 ; r), kv k k is uniformly bounded. Hence thus F (x ) = 0. kf (x )k = lim k1 kf (x k)k lim k1 kv kkkx k+1? x k k = 0; 7

8 The error estimate (3.2) follows from kx k+p+1? x k k k+p X i=k px i=0 kx i+1? x i k i+1 kx k? x k?1k 1? kx k? x k?1k: Corollary 3.2 Suppose that the conditions of Theorem 3.1 hold. Assume, further, that for any x; y 2 S(x 0 ; r) with F (x) = F (y), there are a V x B F (x) a V y B F (y) such that min(kv?1 y V x k; kv?1 x V y k) < 1 : (3.3) Then S(x 0 ; r) only contains one solution x of (1.1). Proof: Suppose that there are two solutions x ; y 2 S(x 0 ; r). Suppose that V x B F (x ) V y B F (y ) satisfy (3.3). Then B = V?1 y V x is nonsingular with B?1 = V?1 x V y. Furthermore, the following inequalities hold: kb(y? x )k = kv?1 y (F (y )? F (x )? V x (y? x ))k ky? x k (3.4) kb?1 (x? y )k = kv?1 x (F (x )? F (y )? V y (x? y ))k kx? y k: (3.5) By (3.4)-(3.5), we have Hence kx? y k = kbb?1 (x? y )k kbkkb?1 (x? y )k kbkkx? y k kx? y k = kb?1 B(x? y )k kb?1 kkb(x? y )k kb?1 kkx? y k: This implies kx? y k min(kbk; kb?1 k)kx? y k: kx? y k = 0; i.e., x = y : Hence x is the unique solution of (1.1) in S(x 0 ; r). We say that F is semismooth at x if lim V 2@F (x+th) t#0 8 fv hg

9 exists for any h 2 R n. If F is semismooth at x then kf (x + h)? F (x)? V x hk = o(khk); where V x B F (x). If F is BD-regular at x, then there are a neighborhood N x of x a constant c such that F is BD-regular in N x for any y 2 N x ; sup V 2@ B F (y) kv?1 k c. Furthermore if F is semismooth BD-regular at x, then the generalized Newton method (1.5) superlinearly converges to a solution x of (1.1) [23], [25]. Semismoothness was originally introduced by Miin [16]. There are a number of functions which are semismooth but not dierentiable, e.g. the piecewise smooth functions [21], [23], [25]. If F is semismooth in a neighborhood N x of x F is BD-regular at x, then for any 2 (0; 1), there exists an ball S(x ; r) N x such that the conditions of Theorem 3.1 hold. 4 A verication method Alefeld-Gienger-Potra [1] established the following theorem on the Krawczyk operator. Theorem 4.1 [1] Assume that the mapping F : D < n < n is dierentiable the derivative has an interval arithmetic evaluation F 0 ([x]) for all [x] D such that kd(f 0 ([x]))k 1 kd([x])k 1 ; [x] D; (4.1) for some 0. If A 2 F 0 ([x]), then the inequality holds with = ka?1 k 1 : kd(k([x]; x; A))k 1 kd([x])k 2 1 The verication method proposed in [1] is based on Theorem 4.1 the Newton-Kantorovich theorem. It is well-known that the Newton-Kantorovich theorem assumes that F is dierentiable in D kf 0 (x)? F 0 (y)k kx? yk; for x; y 2 D (4.2) for some > 0. Note that conditions (4.1) (4.2) cannot be simply generalized to when F is not dierentiable in D. Proposition 4.2 Assume that F is Lipschitz continuous in D. If for all [x] D, then F is continuously dierentiable in D. kd([l [x] ])k 1 kd([x])k 1 ; (4.3) 9

10 Proof: For any point x 2 D, [x] = [x; x] = x is a degenerate interval. Since d(x) = 0, (4.3) implies d([l [x] ]) = 0. By the denition of [L [x] ], the (x) reduces to a singleton. From Proposition in [8], F is continuously dierentiable in D F 0 (x) (x): Similarly, the following assumption k@f (y)k kx? yk; for x; y 2 D implies that F is continuously dierentiable in D. The verication method proposed in this paper is based on the generalized Krawczyk operator (2.4) Theorem 3.1. Assume that the generalized Newton method (1.5) converges to x F is semismooth in a neighborhood N x of x. Then the method (1.5) is superlinear convergent for any 2 (0; 1) there is an integer k 0 > 0 such that for any k k 0, kv?1 k+1(f (x k+1 )? F (x k )? V k (x k+1? x k ))k kx k+1? x k k kx k+1? x k 1? kx k+1? x k k: Therefore, we consider the interval [x] = fx j kx? x k+1 k 1 k g = x k+1 + [? k e; k e]; where k = kx k+1? x k k 1 e = (1; 1; :::; 1) T. The diameter of [x] is d([x]) = 2 k e the midpoint of [x] is m([x]) = x k+1. Using [x], x k+1 an n n nonsingular matrix A, we construct a generalized Krawczyk operator B([x]; x k+1 ; A) = x k+1? A?1 F (x k+1 ) + (I? A?1 [L [x] ])([x]? x): Then the diameter the midpoint of B([x]; x k+1 ; A) have the form: d(b([x]; x k+1 ; A)) = d((i? A?1 [L [x] ])([x]? x k+1 )) m(b([x]; x k+1 ; A)) = x k+1? A?1 F (x k+1 ): Noticing ([x] + [y])[z] [x][z] + [y][z] [x]? x k+1 = [? k e; k e], we calculate (I? A?1 [L [x] ])([x]? x k+1 ) = ( nx j=1 (I? A?1 [L [x] ]) 1j ; : : : ; nx j=1 (I? A?1 [L [x] ]) nj ) T [? k ; k ]: Theorem 4.3 Assume that F is Lipschitz continuous. Then if only if B([x]; x k+1 ; A) [x] (4.4) ka?1 F (x k+1 )k 1 (1? k(i? A?1 [L [x] ])ek 1 ) k : (4.5) 10

11 Proof: First we show that (4.4) is equivalent to ka?1 F (x k+1 )k kd((i? A?1 [L [x] ])([x]? x k+1 ))k 1 k : (4.6) Let p(x) = x? A?1 F (x): Then p(x k+1 ) = m(b([x]; x k+1 ; A)): If (4.4) holds, then ka?1 F (x k+1 )k 1 = km([x])? p(x k+1 )k (kd([x])k 1? kd(b([x]; x k+1 ; A))k 1 ); thus (4.6) holds. If (4.6) holds, then for any x 2 B([x]; x k+1 ; A); kx? x k+1 k 1 kx? p(x k+1 )k 1 + kp(x k+1 )? x k+1 k kd(b([x]; x k+1; A))k 1 + ka?1 F (x k+1 )k 1 k ; thus (4.4) holds. Now we show that (4.5) (4.6) are equivalent. As [x]? x k+1 is a symmetric interval, Theorem 10 in Chapter 2 of [2] implies kd((i? A?1 [L [x] ])([x]? x k+1 ))k 1 = kd(( = k(j = max i nx j=1 nx nx j=1 j (I? A?1 [L [x] ]) 1j ; : : : ; (I? A?1 [L [x] ]) 1j j; : : : ; j j=1 (I? A?1 [L [x] ])j2 k : nx nx j=1 j=1 Hence the three inequalities (4.4)-(4.6) are equivalent. (I? A?1 [L [x] ]) nj ) T [? k ; k ])k 1 (I? A?1 [L [x] ]) nj jk 1 2 k Now we are ready to propose the verication method. Algorithm 4.4 Let be a given tolerance. Let x k be a cidate for a solution to (1.1), dene z = x k? V?1 k F (x k ) = kz? x k k 1. Choose an n n nonsingular matrix A. Dene [x] = z + [?e; e]; e = (1; 1; :::; 1) T, calculate [L [x] ]. If ka?1 F (z)k (4.7) k(i? A?1 [L [x] ])ek 1 1? ; (4.8) then (1.1) has a solution x 2 B([x]; z; A): The cidate x k can be obtained by an appropriate method for solving nonsmooth equations [3], [6], [7], [11], [12], [20], [21], [22], [24], [25], [26]. The inequality (4.7) is a stopping criterion for an algorithm for solving the nonsmooth equations (1.1). We cannot guarantee that if (4.7) is satised, then the interval [x] will contain a solution x of (1.1). However, it is reasonable that the interval [x] is very likely to contain a solution x. Furthermore, the inequality (4.7) can be checked easily. 11

12 5 Numerical experiments First we illustrate Algorithm 4.4 by Example 1.1. Next we test our method by a test problem from [15]. As we will see in the numerical examples, the stopping criterion (4.7) works well in practice. The numerical results were obtained using Matlab 4.2c on a Sun 2000 workstation. The rounded error in Matlab is 10?16. To give safe error bounds, condition (4.8) was considered as k(i?a?1 [L [x] ])ek 1 < 1? - rounded error. Example 5.1 To illustrate Algorithm 4.4, we consider Example 1.1. Let x 0 = x = (0:0; 0:217=0:563) = 10?7. We use the generalized Newton method (1.5) to solve this problem. We choose A = I. The numerical result is as follows: ka?1 F (x 1 )k 1 kx 1? x 0 k 1 = 2: ?1 ; ka?1 F (x 2 )k 1 kx 2? x 1 k 1 = 4: ?17 : Let z = x 2 = (1: ;?1: ) T go to step 1 of Algorithm 4.4. = kx 2? x 1 k 1 = 1:3854; [x] = z + [?e; e], [L [x] ] = ; (I? A?1 [L [x] ])e = = 1? [0:780; 1:563] 1? [0:659; 1:913] [0:780; 1] [0; 0:563] [0; 0:913] [0:659; 1] 1? [0:780; 1]?[0; 0:563]?[0; 0:913] 1? [0:659; 1] = ; [?0:563; 0:220] [?0:913; 0:341] k(i? A?1 [L [x] ])ek 1 = 0:915 1? : 1?1 [?0:563; 0:563] [?0:915; 0:915] 1 1 From ka?1 F (x 2 )k 1 =kx 2? x 1 k 1 + k(i? A?1 [L [x] ])ek 1 1? 10?2, we know that there is a solution x of (1.1) in B([x]; z; A) = + [x]: We can make d(b([x]; z; A)) smaller by choosing an appropriate matrix A. For example, we choose A =diag(1=0:8; 1=0:7). Then (I? A?1 [L [x] ])e = B([x]; z; A) = 1? 0:8[0:780; 1:563] 1? 0:7[0:659; 1:913] 1?1 + = [?0:3760; 0:3760] [?0:5387; 0:5387] [?0:2504; 0:3760] [?0:3391; 0:5387] : 12

13 Example 5.2 Extended Powell Badly Scaled Function [15]. Recently Luksan [15] published a collection of 17 test problems for iterative solution of smooth equations. From among these we choose the extended Powell badly scaled function which is dened by with (x) = ( 1 (x); :::; n (x)) T i (x) = 10000x i x i+1? 1 i (x) = exp(?x i?1) + exp(?x i )? 1:0001 if i is odd if i is even, where n is an even number. Let x = (0; 1; 0; 1; :::; ) T 2 < n, f(x) = (x)? (x ) g(x) = x: We dene F (x) = min(f(x); x): The point x is a solution of the system F (x) = 0. As f i (x ) = g i (x ), fi(x 0 ) 6= gi(x 0 ) for all odd numbers i in f1,2,...,ng, F is not dierentiable at x. We chose V k = ((V k ) 1 ; :::; (V k ) n ) T with ( g 0 (V k ) i = i(x k ) if f i (x k ) g i (x k ) fi(x 0 k ) otherwise, i = 1; :::; n: (5.1) We then have V k B F (x k ) from the denition B F (x k ). The interval matrix is dened by [L [x] ] = [l; l], where l i;j = minff 0 i;j(x); g 0 i;j(x); x 2 [x]g l i;j = maxff 0 i;j(x); g 0 i;j(x); x 2 [x]g: We used the generalized Newton method (1.5) to solve this problem dene V k by (5.1). We chose A =diag (10 5 ; 10 2 ; 10 5 ; 10 2 ; ::::; ) = 10?7. The initial point x 0 2 (0; 1) was romly generated. The example is characterized by the parameters listed in Table 1. Table 2 gives some typical results for various values of n. From k + k < 1??5, we know that there is a solution in [x] = x k+1 + [? k e; k e]; where e = (1; 1; :::; 1) T : n number of unknowns equations k + 1 number of iterations until (4.7) is fullled with z = x k+1 k kx k+1? x k k 1 k ka?1 F (x k+1 )k 1 = k k k(i? A?1 [L [x] ])ek 1 Table 1: Parameters in Table 2 Acknowledgements The author thanks T. Yamamoto two referees for their valuable comments. 13

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