40 85 90 1988 Computing - PDF document

1 40, 85 - 90 (1988) Computing by Springe
40, 85 - 90 (1988) Computing by Springer-Verlag 1988 Note on Moore's Interval Test for Zeros of Nonlinear Systems Shen Zuhe, Nanjing, and A. Neumaier, Freiburg i. Br. November 11, 1986; -- Zusammenfassung on Moore's Interval Test for Zeros of Nonlinear Systems. study relations between Moore's interval test and Miranda's theorem. As an application we combine the (real) Newton iteration f: D c N"--+ R" be a continuously differentiable function and let F' be an inclusion monotone interval extension of the derivative f' on the interval vector X Shen Zuhe and A. Neumaier: On the other hand, a well-known theorem by Bolzano states that whenever a continuous real function changes sign in some interval it has a zero in this interval. An n-dimensional generalization of this theorem is due to C. Miranda. We state an equivalent but slightly more general result: 1 (X) ~ Int (X) (2) X contains a unique solution x* is a regular zero the ordinary iterative method 11= x(~_ rf(x~k~) to the unique solution x* from any starting point x (~ in X. Int (X) denotes the interior of X and a zero x* of(l) is called regular iff' (x*) is nonsingular. Theorem 2 + = {x ~ x, x~ = ~, + di}, (X);- = {x ~ X, x~ = ~i- d~} the n pairs of parallel, opposite faces of the interval vector X = 2- d, 2 + d, d2,..., dn) T. If is a permutation (r,, r2,..., r,) of n) such that f~ (x) fi (Y) 0 for all x ~ (X) + , y ~ (X)~ (i =

2 1,2,..., n) f has at least one solution
1,2,..., n) f has at least one solution x* in X. purpose of this note is to show that Miranda's theorem can be combined with the iteration (3) to determine an approximate solution to the system (1) with specified accuracy. Miranda's theorem can be violated for interval vectors X in arbitrarily small neighbourhoods of a regular zero x* ~ X. Take n = 2 and consider 1-x27 2e" Then X contains the regular zero x*=0 as an interior point, but (4) cannot be satisfied for a permutation (r,, of (1,2). example shows that to use Miranda's theorem as a local existence test we must precondition the equation in a suitable way. As for the Krawczyk operator we consider (x)= U(x), (5) Y is a nonsingular n x n-matrix. Then g(x) has the same zeros as j'(x). Moreover, if 2 is an approximation of a regular zero x of f, then the choice Y=f' (~)- ~ yields (x) = g(f(x*) +f' (x*)(x - x*) + o (e)) (f' (2) ~ f' (x*))(x - x*) + o (e) (1 + o ~*) + o (s) +o(O Note on Moore's Interval Test for Zeros of Nonlinear Systems 87 for all x e ~" with ! x - x* LI e, s small. This implies that (4) holds (with g in place of f) for any interval vector X = 2-d, 2 +dl with ~-x*l
I1 d LI sufficiently small. To verify (4) numerically, Moore and Kioustelidis 41 gave the following test which we shall use for bounding the error of an approximate solution to the system (1). Theorem 3: Let X, (X) + and (X):fl be defined as in Theorem

3 2. Let g(x)= l~f(x). Then system (1) has
2. Let g(x)= l~f(x). Then system (1) has at least one solution in X=2-d,2+d if" the .following conditions are satisfied for all i = 1,2,..., n: gz (2 4- dz e~) g~ (2 - d~ e~) ___ 0, (6) I YF' ((X)i +)~j dj I g~ (2 + d, e,)l, (7) jP~ Z IrF'((X)~)~jldj()~ (8) Jt-~ We would like to know that the conditions of Theorem 3 can be satisfied locally. But first we show that Theorem 3 is stronger than the existence statement of Theorem 1. Theorem 4: If K(X)cX then (6)-(8) hold. Proof: First we note that K (X)c X is equivalent with and therefore with g (2) + (I - YF' (X)) - d, dl ~ - d, d g (Sc)l 4-11 - YF' (X)l d d, (9) or, in components, gi (2)1 +IU- I d~ + ~ I I dj_d,. by the mean value theorem, gi (2 4- d i ei) ~ gi (2) ~ YF' (X)~ dl 4- di + g, (~) -T- I - YF' di, whence -b g~ (fc + d, e,) &#x 000;__ d, -I g, (2)1 -I - YF' (X) n d, &#x 000;_ ~ YF'(X),j I which implies (6)- (8). A slight variation of the proof allows an application to the iteration (3). Theorem $: Suppose that (dl, .... ,dn) T a positive vector (the error bound required in advance). If K (Xo) c Int (Xo), where Xo = x - 7d, x + 7d for some 7 � 1, then the conditions (6), (7), (8) are satisfied for all i = 1, 2,..., n with X (k) = x (k) - d, x (k) + d in place of Xo for all iterates x (k) of(3) with sufficiently large k and any starting point x (~ in Xo, and, for such k, the i-th component x! k) of the approximate so

4 lution x (k) of (1) is accurate to withi
lution x (k) of (1) is accurate to within +_ d i for each i = 1,2, ..., n. Shen Zuhe and A. Neumaier: before, K (Xo) c Int X o implies 19 (2)I +I- 7d yd. (9 a) Therefore I - d d which shows that (9) holds if2 is sufficiently close to the zero x*. But by Theorem 1, lira x (k) = x*; hence we can continue as in the last proof with 2 = x (k), provided that k is sufficiently large, an illustration we consider the two-dimensional nonlinear system (10) from 4. The matrix of partial derivatives of the system (8) is 1 \ 1 natural interval extension (x) F'(X)= (211 -2X2' X1, X2 are intervals. Starting with the interval X (~ given by \0.6, \0.75/ and an approximate inverse Y of (m /0.434783 0.434783~ Y= ~0.289855 -0.376812`/' we have ,, 0.559793, 0.677173~ K (X ~~ 0 -~\ .744275, 0.830363/' and clearly K (X (~ c Int (X~~ Therefore, following Theorems 1, 3 and 5, we can use the point Newton iteration and the computational test (6)-(8) for the accuracy of the approximate solution. For example, let us suppose that we want to find an index d such that the approximate solution x r is accurate to within +_ 10 -6 for each i= 1, ..., n. Now, starting with the initial (0.65~ rn (X (~ = \0.75`/ and the nonsingular matrix g in (9), the first iteration gives (0.618478~ x (1) = m (X (~ - = \0.787318,/" If x (1) would satisfy- our accuracy requirements, then X(1) (0,618477, 0.618479~ = \0.787317, 0.787319'

5 Note on Moore's Interval Test for Zeros
Note on Moore's Interval Test for Zeros of Nonlinear Systems 89 contain a zero. With Xm + 0.618479 \ ( 0.618477 ( h -- t0.787317, 0.787319)' (X(1))1- = \0.787317, 0.787319/' =(0.618477, 0.618479) (0.618477, 0.618479~ (Xd))~ \ 0.787319 J' (X(~))~- = \ 0.787317 )' we are in a position to check conditions (6), (7) and (8). We find that (x(1)+dl 10 1, gl (x(1) -dl e0=0.3152070.10 -1, -1, e2)= -t, and d 2 = -0.1788139. l0 -11 , 0.1788139.10-11, 0.1788139-10-1~, YF' dl = - 0.5960464.10-12.0.5960464- 10- lZ, YF' dl = - 0.5960464.10 -12 0.5960464.10 - 12. While conditions (7) and (8) are satisfied at this step, condition (6) is not. Hence x (~) was not accurate enough to satisfy our test. The following table shows some data of the successive iteration. It is clear that conditions (6), (7) and (8) are satisfied in the third step. The approximate solution x(3) = (0.6180340~ \0.7861514/ is accurate to within ___ -6 as and /0.0000001\ x(3, (0.0000015" i/o.o0oo009/' k 1 2 3 gl (x(k) +dl el) ( x(k)-dl el) (x(k) + d2 e2) (x(~) - d2 e2) YF' ((X (k)) )312 dz I ) 12 dz YF' ((x(~))2)=1 dl r'r' ((x%;)322 l (x(k))l (x(% .31 .17 .17 10-~ 10 =1 10-1 10-1 10 11 10-12 .59 10 -12 .59 10 -22 .0023863 .0013927 ,6184783 ,7873188 .44 10 .3 .44 10 -3 .11 10 -2 .11 10 -a .18 10 11 .18 10 -it .56 10 -12 .56 10 22 .6180341 .7861523 .11.10 10 -6 .19- 10 -5 -6 -11 .17.10 11 .59.10 12 .59 10 -12 .6180340 ,7861514