Higher-Order Interpolation Methods

6.2.3 Higher-Order Interpolation Methods

The secant method does not achieve quadratic convergence. Indeed, it can be shown that under very weak restrictions no iterative method using only one function evaluation per step can achieve this. In Steffensen’s method

f (x n + f (x n )) − f (x n ) x n +1 =x n −

f (x n )

, g(x n ) =

g(x n )

f (x n )

632 Chapter 6. Solving Scalar Nonlinear Equations two function evaluations are used. This method can be viewed as a variant of the secant

method, where the step size used to approximate the first derivative goes to zero as f (x). To show quadratic convergence we put β n = f (x n ) and expand g(x n ) in a Taylor series about x n ; we get

+ O(β 2 n ) 2 , where h n = −f (x n )/f ′ (x n ) is the Newton correction. Thus,

2 n ) . Using the error equation for Newton’s method we get

where ǫ n =x n − α. This shows that Steffenson’s method is of second order. Steffensen’s method is of particular interest for solving systems of nonlinear equations. For a generalization to nonlinear operator equations on a Banach space, see [213]. In the secant method linear interpolation through (x n −1 ,f n −1 ) and (x n ,f n ) is used to determine the next approximation to the root. A natural generalization is to use an interpolating method of higher order. Let x n −r ,...,x n −1 ,x n

be r+1 distinct approximations and determine the (unique) polynomial p(x) of degree r interpolating (x n −j , f (x n −j )) ,

j = 0 : r. By Newton’s interpolation formula (Sec. 4.2.1) the interpolating polynomial is

p(x) =f n + [x n ,x n −1 ]f · (x − x n ) + [x n ,x n −1 ,...,x n −j ]f M j (x),

The next approximation x n +1 is taken as the real root to the equation p(x) = 0 closest to x n and x n −r is deleted. Suppose the interpolation points lie in an interval J , which contains the root α and in which f ′ (x) point on each side of α, then p(x) = 0 has a real root in J . Further, the following formula for the error holds (Traub [354, pp. 67–75]):

(r + 1)!p (η n ) i

where ζ n ∈ int (α, x n −1 ,x n ) and η n ∈ int (α, x n +1 ) . (Recall that by int (a, b, . . . , w) we denote the smallest interval that contains the points a, b, . . . , w.) In the special case r = 2

we get the quadratic equation p(x) =f n + (x − x n ) [x n ,x n −1 ]f + (x − x n )(x −x n −1 ) [x n ,x n −1 ,x n −2 ]f.

6.2. Methods Based on Interpolation 633 We assume that [x n ,x n −1 ,x n −2

secant method. Setting h n = (x − x n ) and writing (x − x n −1 ) =h n + (x n −x n −1 ) , this equation becomes

(6.2.13) where

h 2 n [x n ,x n −1 ,x n −2 ]f + ωh n +f n = 0,

(6.2.14) The root closest to x n corresponds to the root h n of smallest absolute value to (6.2.13).

ω = [x n ,x n −1 ]f + (x n −x n −1 ) [x n ,x n −1 ,x n −2 ]f.

To express this root in a numerically stable way the standard formula for the roots of a quadratic equation should be multiplied by its conjugate quantity (see Example 2.3.3). Using this formula we get

2f n

x n +1 =x n +h n , h n =−

, (6.2.15) ω ± ω 2 − 4f n [x n ,x n −1 ,x n −2 ]f

where the sign in the denominator should be chosen so as to minimize |h n |. This is the Muller–Traub method .

A drawback is that the equation (6.2.13) may not have a real root even if a real zero is being sought. On the other hand, this means that the Muller–Traub method has the useful property that complex roots may be found from real starting approximations.

By (6.2.11) it follows that

It can be shown that the convergence order for the Muller–Traub method is at least q = 1.839 . . . , which equals the largest root of the equation µ 3 −µ 2 − µ − 1 = 0 (cf. Theo-

rem 6.2.1). Hence this method does not quite achieve quadratic convergence. For r > 2 there are no useful explicit formulas for determining the zeros of the interpolating polynomial p(x). Then we can proceed as follows. We write the equation p(x) = 0 in the form x = x n + F (x), where

−f r

n − j [x n ,x n −1 ,...,x n −j ]f M j (x)

(cf. Sec. 4.2.2). Then a fixed-point iteration can be used to solve for x. To get the first guess x 0 we ignore the sum (this means using the secant method) and then iterate, x i = x n

+ F (x i −1 ) , i = 1, 2, . . . , until x and x −1 are close enough. Suppose that x n −j −x n = O(h), j = 1 : r, where h is some small parameter in the

context (usually some step size). Then M

= O(h j ) ,M ′ j (x) = O(h −1 ) . The divided differences are O(1), and we assume that [x n ,x n −1 ]f is bounded away from zero. Then the terms of the sum decrease like h j . The convergence ratio F ′ (x) is here approximately

j (x)

M ′ 2 (x) [x n ,x n −1 ,x n −2 ]f = O(h).

[x n ,x n −1 ]f

Thus, if h is small enough, the iterations converge rapidly.

634 Chapter 6. Solving Scalar Nonlinear Equations

A different way to extend the secant method is to use inverse interpolation. Assume that y n ,y n −1 ,...,y n −r are distinct and let q(y) be the unique polynomial in y interpo- lating the values x n ,x n −1 ,...,x n −r . Reversing the rule of x and y and using Newton’s interpolation formula, this interpolating polynomial is

q(y) =x n + [y n ,y n −1 ]g · (y − y n ) + [y n ,y n −1 ,...,y n −j ]f [ j (y),

j =2

where g(y n −j ) =x n −j , j = 0 : r,

[ j (y) = (y − y n )(y −y n −1 ) · · · (y − y n −j ).

The next approximation is then taken to be x n +1 = q(0), i.e.,

x n +1 =x n −y n [y n ,y n −1 ]g + [y n ,y n −1 ,...,y n −j ]g [ j ( 0).

j =2

For r = 1 there is no difference between direct and inverse interpolation and we recover the secant method. For r > 1 inverse interpolation as a rule gives different re-

sults. Inverse interpolation has the advantage of not requiring the solution of a polynomial equation. (For other ways of avoiding this see Problems 6.2.3 and 6.2.4.) The case r = 2

corresponds to inverse quadratic interpolation:

(6.2.17) Note that it has to be required that y n ,y n −1 , and y n −2 are distinct. This method has the same

x n +1 =x n −y n [y n ,y n −1 ]g + y n y n −1 [y n ,y n −1 ,y n −2 ]g.

order of convergence as the Muller–Traub method. Even if this is the case it is not always safe to compute x n +1 from (6.2.17). Care has to be taken in order to avoid overflow and possibly division by zero. If we assume that n | ≤ |y n −1 | ≤ |y n −2 |, then it is safe to compute

s n =y n /y n −1 , s n −1 =y n −1 /y n −2 , r n =y n /y n −2 =s n s n −1 .

We can rewrite (6.2.17) in the form x n +1 =x n +p n /q n , where

p n =s n [(1 − r n )(x n −x n −1 ) −s n −1 (s n −1 −r n )(x n −x n −2 ) ],

q n = (1 − s n )( 1−s n −1 )( 1−r n ). The final division p n /q n is only carried out if the correction is sufficiently small.