11 Three Interpolation Points

Three Interpolation Points

INTERPOLATION BY LINEAR FRACTIONS 1. Let f ( x ) be defined in J x . Assume that we are given three distinct interpolation points

xv E J x ,

f ( x v ) =$,

(v = 0,

(11.1)

First we build up a function which interpolates f ( x ) in these three points. If we choose a polynomial as our interpolating function, it will usually be quadratic; as such it has at least two zeros, and we will not know which of them to take (cf., however, the discussion in Chapter 17). We avoid this difficulty by considering instead a function which has only one simple zero, namely, (11.2)

Notice that in (1 1.2) we have three essential constants. We know from projective geometry that if w is related to x by (1 1.2), then the points x yxo, x l , x2 have the same cross ratio as the points w,f0,f1,f2; i.e., (11.3)

We assume thatfo# f l # f i and introduce A by

then we have from (1 1.3) w-f1 --

w-fo

- A-x-x1 x-x,

vo Zfl

Zf2,

xo f x1 # x2).

(1 1.5)

In this way we obtain our interpolating function w when the three interpolation points are distinct. 79

80

11

I

THREE INTERPOLATION POINTS

TWO COINCIDENT INTERPOLATION POINTS

2. We consider from now on the case where two of our x, are coincident; i.e., x2 = xo, x1 # xo. We assume further that we know f o , f l and fo’ = f ’ ( x o ) and that f o # fi.We can get the corresponding interpolating function by going to the limit in A. Rewrite (1 1.4) as follows:

(11.6) Then as x 2 -,xo, we have as the limits of A and (1 1.5)

It is immediately clear that w(x) given by (11.8) satisfies the conditions w(xo) =f o , w ( x l ) =f l . Differentiating both sides of (1 1.8) with respect to x, we verify at once also that w’(xo)=fo‘.

3. As in previous discussions, we shall use the inverse function to obtain estimates of error. Putting w =f ( x ) , let x = O(w) be the inverse function of f (x). We have then obviously

On the other hand, if we solve (11.8) with respect to x , we obtain a function cp (w) given by (11.9) Solving (1 1.9), we have (11.10) It is easily verified that generally for constant a, B, y, 6 (1 1.1 1) From (11.10) and (11.11) it follows that

* 2 A*v; -fo)(x1 -xo)

c ~ ‘ ” ( w )= 6(1-A )

N4

9

(11.12)

81

ERROR ESTIMATES

where

N = (l-A*)w+foA*-fl.

(1 1.13)

ERROR ESTIMATES 4. Let ~ ( w 7 ) @(w). If we apply (lB.23), our interpolating function is cp(w); consequently the factorf(”)(<)-f”)(<) in ( I B.23) must now be replaced by [@ (q)- cp (q)]‘”). Then we have

W)- cp(f) = ~ ~ f - f o ~ 2 ( f - f i ~ ~ ~ ‘ 3 ~ ~ ~ ~ - c ptlp E( 3( f~f 0~, ftl )l .~ l y

(11.14)

For f = 0, denoting by x2 the new approximation cp(0) to our root t;, we have from (11.14)

Taking w = 0 in (1 l.lO), we have x2

=

x1 fo A* -xo f i f o A* - f 1

’

(1 1.16)

Assume xo, x1 -+ t;. Then fo,fl -,0 and hence q -+ 0. Under this assumption, iff’([) # 0, we have from (1 1.7) (1 1.17) We consider first -= A*-1 x1

-xo

f(X1)-

~f(xo)+f’(~o)(~l-Xo)l f’(xo)(x1- x0j2

(11.18)

The bracketed expression in (11.18) gives the first two terms of Taylor’s expansion off ( x ) around xo. Hence, the numerator of the right-hand side of (11.18) is equal to the remainder term +(xl-xo)y(tl), t1E (xo,xl),and we have (11.19)

If XO, x1+ [,then (11.20)

11

82

1

THREE INTERPOLATION POINTS

Consider now (11.21) Under the assumptions xo + C, x 1 --t l,fo+ 0 and using (1 1.20), we get (11.22) and hence from (1 1.13), since w = 0, N x1 -xo

5.

+

-f'(C).

(11.23)

From(11.12) we have, by (11.17), (11.20), and (11.23),

(11.24) (1 1.25) From (2.5) and the table in Chapter 2, Section 8, follows @3)(')

= 3f"zr-s

(11.26)

-f(3)fr-4.

On the other hand, as 7 -+0, @(q) tends to c. Hence @(3)(7)

+

3fll(C)Y(5)-

-f(3)(C)f'(c)-4.

(1 1.27)

Further we have f ( X 0 ) =f ( X 0 )

f(XJ

- f (0= f ' ( 5 0 ) (xo - 0,

=f'(51)(x1--C),

to E 51

E

(xo, 0, (1 1.28) (X1,C).

(11.29)

Substituting these results in (1 1.15), we obtain

Equation (1 1.30) shows that we have here an approximation of the third order. USE IN ITERATION PROCEDURE

0. Our procedure obviously can be considered as a combination of the regulafalsi and the Newton-Raphson method. If we have already applied both

83

USE IN ITERATION PROCEDURE

methods, the Newton-Raphson method in the point xo and the regulafalsi in the points xo and x l , then the application of (1 1.16) requires no further horner and the results obtained using those methods once can be considerably improved. On the other hand, if we want to use this method of approximation as an iteration method, we would have to use consecutively the following triplets of interpolation points : (XO,Xl,Xl),

(XI,X2,XZ),

(XZ,X3,X3),

**-,

where in the first triplet x1 is used twice and xo only once. 7. Ifwe put

In-

1 IXr-TI

(1 1.31)

= Y,,

we obtain then from (11.30), observing that the values of xo and x1 must be interchanged, the relation

Yp+z = Y,, + 2Y,,+l + k,?

(11.32)

where k,,are bounded, if the expression to the right in (1 1.30) is ZO. But then we will prove in the following chapter that if x,, 3 T, y,, 3 00, we have

+

+ 1 J j = 2.414 ..., (11.33) Y, and since we spend two horners at each step, the efficiency index of this iteration is . = 1.55.. . . If we compare this with the efficiency indices of the regula falsi, 1.618 ..., and of the Newton-Raphson method, 1.414 ..., we see that this new iteration method is better than the Newton-Raphson iteration method but not as good as the regulafalsiused as an iteration method.

*y

Jm..

Example. If we apply the above method to the equation x z -2x- 1 = 0 discussed in Section 17 of Chapter 3, starting with the triple ( x o , x l , x l ) , xo = 3, x 1 = 2, we obtain the set of values’ given in the accompanying table. V

2 3 4 5

lo-’(x”0.026045 1977 0.024551 4320 0.024551 4815 0.024551 4815

2.09) 4011 3811 4232 4232

IXY-CI

2994 0802 6592 6591

3 6 3 4

1.4937. 4.95 9 .10-19

-

Y”+llYV

2.759 2.585 2.47

-

?Computed by Mr. Allen Reiter in the Mathematics Research Center of U.S.A., Madison, Wisconsin.

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