By A. M. Ostrowski
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Additional info for Solution of Equations in Euclidean and Banach Spaces
17) is apolynomiul of degree not exceeding n-1. -lu x ) + Rn-l (y,x) that La-l x) = y ( x ) . We see in particular that the polynomial y ( x ) of degree < n satisfying (1 B. 17) is uniquely determined. 12. We are now going to prove that there always exists a polynomial y ( x ) satisfving ( 1 B. 17). We are going to prove even more. , m,- l), we prove that there always exists a polynomial y (x) of degree (1B. ,mK-l). 19) 13. Indeed, if we write y ( x ) as y(x) = uox"-l + u1x"-2 + + Un-1, Eqs. 19) represent a set of n linear equations in the unknowns u, and we have only to prove that the determinant of this set is not zero.
3) where by (x, xl, x2) we mean the opeminterVal having two of these points as end points and the third point not outside. 5) as follows: We see thatf(x3) will be small if x1 and x2 are close enough to some zero of f(x), since then y , and y2 are small. 2) is called the rule of false position,or regulafalsi. USE OF INVERSE INTERPOLATION 3. We shall now obtain a direct estimate of the error of the approximation by x3 to a zero off(x) from the theory of inverse interpolation. Let x = cp (y), the inverse function of y =f (x), be defined in the y-interval corresponding to J,.
In thefrst case we haue 3. Proof. Part I. Suppose I$’(l0)l < 1. Then, if we choose a p with I$’(co)l< p < 1 we have for any x within a convenient V(c,) Let an x1 E V(Co)be the starting point. , lxv+l-col < PvIxl-col + 0 ( v + w). 5) Hence lois a point of attraction. From xv + lofollows Proof. Part ZI. Suppose that I$’(co)l > 1. Then, if we choose a p I$’(co)l > p > 1, we have for any x from a convenient V(c,) 4. , x2 is farther away from lo than xl. D. 5. Remarks. 5a) may not hold, and it is entirely possible that our next point may be To.