OPERATIVE DIVISION 597 



select for iteration and inversion? Evidently (by trial or otherwise) X<2 and 

 f'X<f'2 = 3. By Section V1I1 (6), p. 590, both the iteration and the invert will 

 ultimately converge if 2 qg>f'X i where q = 1 or 2 and g=y. It is simpler to use no 

 modifier if possible and to take^= each critical point in turn. If g -l, 2qg may 

 not be >f'X ; but the condition will hold if q = 2. We therefore iterate 0-//2 

 on the base£-= 1 ; or calculate X from the corresponding invert of x*—x- 1, which 

 is, by Table II, 



^=1 + 1/2+1/8-1/128+1/1024 . . . =r6i8i6. . . . 



The iteration gives the same result, and by the usual solution X= 1-618034. . . . 

 The root being ascertained, we have /'.¥= 2-236 . . . ; so that if we had taken 

 q = 1 and inverted x-x~ l = 1, we sho ild have obtained an ultimately divergent 

 series. 



The reciprocal equation (x = l/z) is ** + z- 1 + O. This has only one positive 



critical point (=1) which is >Z, while Z>^. Now/'Z must be </'(i) = 3. We 



must take q = 2 ; so that 2qg>f'Z. Hence the iteration of o -//2 and the invert 

 of z" + z=i will be valid, giving Z = 061816 . . ., which will be the reciprocal 

 of X. 



Observe that if the other sign of g = ± *]i be taken, the negative root of the 

 original equation is obtained by a valid series. 



(2) x*-x'-x-i=o(q = 2>,g=i,2qg*>f'X,X=i-%5? ) .. .). 



(3) x 3 -\ox i -x-\^o{q=l,g= 10, 2qg 2 >f\ll), X= 10-1087 • • .)• 



(4) x^-x*- iocur- I =o(q = 2,g= V ioo, 2 qg 2 >/\n\ X= 10-51719 . . .). 



(5) x 3 -x 2 -x-iooo = o(q = 3,g = ^1000, 2qg*>f{i\\ X = 10-37912 . . .). 



(6) x 3 -2x-s = o(q = 3,g=$S, 2^^>/'A', ^ = 2-0945 . . .). 



(7) x?~x*-Zl -90 = (? = 2,g= V81, 2qg i >/'X, X= 10). 



(8) x 3 - 2X° - 2x + 5 = 8 (q = 3, g = ^ 8 , 2qg i >f'X, X = 3). 



The reader should also examine the corresponding reciprocal equations {x= i/z). 

 Equation (6) has been also dealt with on Part I, p. 232, and Part II, p. 394. In 

 order to avoid the irrational subject ^5 we may add 3 to both sides of the equation 

 and invert x 3 -2.r+3 = 8, as suggested in Part II, p. 401 ; and here also 

 2 Qg* — 2A> f'X, (The rule for finding the modifier there given applies to midaxial 

 iteration.) Examples (7) and (8) are from p. 581. In the latter the critical points 

 2, y/2, ^3 will not give convergent iterations and series, and we are therefore 

 obliged in this case to use a modifier. The one suggested is 5, but 26 may be tried. 



(9) x b -x*-x z -x 2 -x-i =0. 



Compare Examples (1) and (2) above. All the critical points are unity and 

 prospective ; and X<2, and/ X </'{?) = 31. Thus none of the critical points will 

 of themselves suffice for^, since we must have 2^ 4 >3i, where q= 1, 2, 3, 4, or 5. 

 We must therefore use a modifier calculated from 2q{i + mf >/'X— say m = 1 with 

 q = 2. But, though all these series will be ultimately convergent, yet owing to the 

 large number of terms in/, the first few terms of the inverts in Table II will not 

 nearly reach the root, unless we take the invert in invariants (Note VI, p. 596). 

 The reciprocal equation is not better; but arithmetical iteration, say of O— //30 

 upon g=2 } easily gives X= 1-965948 .... If we still require a series for the 

 remainder of the root we can shift the origin by 1-96, say, and obtain a rapidly 

 approximating simple ascending invert (Part I, Section III). 



39 



