The Systematic Formation of High-Order Iterative Methods

Fixed point iteration and the Taylor-Lagrange formula are used to derive, some new, efficient, high-order, up to octic, methods to iteratively locate the root, simple or multiple, of a nonlinear equation. These methods are then systematically modified to account for root multiplicities greater than one. Also derived, are super-quadratic methods that converge contrarily, and super-linear and super-cubic methods that converge alteratingly, enabling us, not only to approach the root, but also to actually bound and bracket it. The Systematic Formation of High-Order Iterative Methods

iterative functions; The Taylor-Lagrange formula; High-order iterative methods; Undetermined coefficients; Contrary and alternating convergence; Root bracketing

Fixed Point Iteration
Consider the paradigmatic fixed point iteration to locate fixed point a, F(a)=a of contracting function F(x). We write x 1 −a=F(x 0 ) a and have by power series expansion that implying that if 0<|F'(x)|<1 near x=a, namely, if F(x) is indeed contracting, then the fixed point iteration converges linearly, and if F'(a)=0, then the fixed point iteration converges quadratically, and so on.
Suppose now that we are seeking root a, f(a)=0, f'(a) ≠ 0, of function f(x). To secure a quadratic iterative method we rewrite f(x)=0 as the equivalent fixed point problem for weight function w(x),w(a) ≠ 0, which we seek to fix to our advantage. For a quadratic method we need w(x) to be such that for x near a. Since f (a)=0, we choose to ignore w'(x) f(x) in the above equation, to have w(x)=−1/f'(x), and with it, Newton's method which is actually quadratic where f'=f'(a) ≠ 0, f"=f"(a) < ∞, and where x 0 is the iterative input and x 1 the iterative output.
From the two zero conditions we obtain, after ignoring f(x) w"(x) in the second of equations (7), the system of equations which we solve for w(x) as to have Halley's method which is, indeed, cubic provided that f(a)=0, but f'(a) ≠ 0 Requesting that F(a)=a, F'(a)=0, F"(a)=0, F'"(a)=0, we similarly obtain the method provided that f'=f'(a) ≠ 0.
Higher order single-point methods are readily obtained by

A Recursive Determination of the Higher Order Iterative Function
There are various ways to recursively generate a new higher order iterative function F(x) of eq. (1) from a known lower order one. Traub [2] has suggested such a rational recursive formula. If, for example, with which the iterative method x 1 =F 3 (x 0 ) to locate fixed point a, F(a)=a becomes cubic Instead of rational formula (15) we suggest the product formula with which we still have third order convergence For example, for Newton's method F 2 (x)=x-f(x)/f'(x). Using formula (18) we obtain by it the method which is, indeed, cubic Iterative method (20) is also obtained from Halley's method of eq. (10) using the approximation is such that and the iterative method x 1 =F 4 (x 0 ) to locate fixed point a is quartic It is well known that the modified Newton's method converges quadratically to a root of any multiplicity m ≥ 1. From equation (24) we derive the third order method Indeed, assuming that we obtain for the method in eq. (28) where A=g(a), B=g'(a), C=g"(a), and m is the multiplicity index of repeating root a [3].
From eq. (29) we have by which we may, knowing an x close to a, estimate m.

A One-Sided Third-Order Two-Step, or Chord, Method
Having computed x 1 =x 0 -f 0 /f ' 0 we return to correct it as the midpoint method which is now cubic, or third order . 24 See also Traub [2] page 164 eq. (8-12).
Convergence of this method is also one sided.

Construction of High-Order Iterations by Undetermined Coefficients
Halley's method, or for that matter any other higher order method, can be constructed by writing δx, x 1 =x 0 + δx, as a power series of u 0 =f 0 /f ' 0 , or even of merely f 0 =f(x 0 ). For example, we write the quadratic and then sequentially fix the undetermined coefficients P and Q for highest attainable order of convergence.
Thus, at first we have from eq. (39) that and we set P=−1/f' 0 . With this P we have next that and we set with which the polynomial method of eq. (20) is recovered.
Doing the same to the rational method we determine by power series expansion that cubic convergence is assured for P=−1/f00, Q=0, R=1, S=−f" 0 /(2f '2 0 ), with which the classical Halley's method of eq. (10) is recovered.
The polynomial in r method is also quartic The multistep method is quintic . 24

Contrarily converging super-quadratic methods
We write for undetermined coefficient P, and have We request that for parameter k, by which the iterative method in eq. (54) turns into The interest in the method is that it ultimately converges oppositely to Newton's method, as is seen by comparing eq. (60) with eq. (6).
The average of Newton's method and the method of eq. (59) is cubic,

Alternaingly Converging Super-Linear and Super-Cubic Methods
We start by modifying Newton's method to have indicating that, invariably, the method converges, at least asymptotically, alternatingly. For k > 0, if x 0 − a > 0, then x 1 − a < 0, and vice versa. For a higher-order alternating method we rewrite the originally quartic method of eq. (46) as for the undetermined coefficient Q, and have that This super cubic method converges alternatingly if parameter k > 0.

Correction for Multiple Roots by Undetermined Coefficients
We rewrite Newton's method as for undetermined coefficient P, and have that for a root of multiplicity m ≥ 1 where A=g(a), B=g'(a) for g(x) in eq. (29). Quadratic convergence is restored, as is well known, with P=m.
Next, we rewrite the method in eq. (37) as and seek to fix correction coefficients P and Q so that convergence remains cubic even in the event that root a is of multiplicity m > 1. By power series expansion we determine that where A=g(a), B=g'(a),C=g"(a) for g(x) in eq. (29) The method is also cubic where A=g(a), B=g'(a), C=g"(a) for g(x) in eq. (29) [11][12][13].

Correction of Halley's Method for Multiple Roots
We rewrite Halley's method of eq. (10) for the undetermined coefficient P and Q as and determine by power series expansion that for convergence remains cubic for a root of any multiplicity m ≥ 1 where A=g(a), B=g'(a),C=g"(a) for g(x) in eq. (29) [11,12].

Use of the Taylor-Lagrange formula
We write the second order version of the Taylor-Lagrange formula and take f(x 1 =x 0 + δx)=0, ξ=x 0 to obtain the iterative method We approximate the solution of the increment equation or, for that matter, any such higher order algebraic equation, by the power series The methods , 6 3 provided that f'(a) ≠0.
The method converges cubically as well to a root of any multiplicity m ≥ 1 where A=g(a), B=g'(a),C=g"(a) for g(x) in eq. (29).

Starting with
we obtain the iterative method

Unknown Root Multiplicity
The two single-step methods converge contrarily to root a of any multiplicity m where A=g(a), B=g'(a) for g(x) in eq. (29). Their average is a cubic method where A=g(a), B=g'(a),C=g"(a) for g(x) in eq. (29).