Effective High-Order Iterative Methods via the Asymptotic Form of the Taylor-Lagrange Remainder

The asymptotic form of the Taylor-Lagrange remainder is used to derive some new, efficient, high-order methods to iteratively locate the root, simple or multiple, of a nonlinear function. Also derived are superquadratic methods that converge contrarily and superlinear and supercubic methods that converge alternatingly, enabling us not only to approach, but also to bracket the root.

The difficult problem of finding the root of the nonlinear function () is replaced in Newton's method by the easy task of repeatedly finding the approximating root of the tangent line () =   0 ( −  0 ) +  0 .This is the essence of all other higher-order methods, to supplant the finding of the root of the original function by the repeated finding of the root of an approximating polynomial.
Having computed  1 by (16), it occurs to us to return and replace the initial  =  0 by the asymptotic to have the two-step, mid-point method which is cubic or third order See also Traub [3, page 164, (8)-( 12)].
The linearization reproduces out of ( 19) classical Halley's method which is cubic as well, but requires the second derivative   ().
We write the equation of the osculating parabola to () at  0 and seek its intersection with the -axis.The smaller root,  1 , of () is given by which is Halley's method of (24).

Construction of High-Order Iterations by Generalized Undetermined Coefficients
Halley's method or, for that matter, any other higher-order method such as that in (24) can be derived ab initio by writing ,  1 =  0 +  as a power series of  0 =  0 /  0 or merely  0 = ( 0 ), as in and then progressively fixing the undetermined coefficients  and , which eventually need not remain constant, so as to achieve the highest possible order of convergence.Thus, at first, we have from (28) that We substitute variable   ( 0 ) for the constant   () in (29) and try  = −1/  0 .With this  we have next that and we set with which the polynomial variant of Halley's method in (24) is regained.Doing the same to the rational method we verify that cubic convergence is achieved with  = −1/  0 ,  = 0,  = 1, and  = −  0 /(2 2 0 ) as in classical Halley's method in (22).
For other interesting applications of the method of undetermined coefficients, see [4,5].
For more on such recursive formulas, see

A Finite-Difference Approximation
Wishing to avoid the possibly computationally costly additional derivative in (19), we propose to approximate it by the central difference scheme Taking ℎ = −/2 leaves us with the approximation where  1 =  0 −  0 ,  1 = ( 1 ), by which (19) becomes the cubic chord or two-step method See also Traub [3, page 180, (8)-( 55)].We return to this method in the next section.
The second derivative approximation leads to the same result.

A Cubic, One-Sided, Two-Step, Secant Method
Having computed by Newton's method we propose to proceed and predict the next  2 by pseudo-Newton's method skirting the computation of a new   ( 1 ).In (53) We write (53) variously as with all three methods being cubic Notice the extra  2 in the last method of (55), added to recover the factor 1/4 in the last of error equations (56).
Convergence is here one sided: if For example, for () =  +  2 , we obtain by the first method of (55) the two oppositely or contrarily converging sequences by which root  = 0 is bounded as Method ( 55) is also obtained from the secant line passing through the two points, ( 0 ,  0 ), ( 1 =  0 −  0 ,  1 = ( 1 )), and then taking the root of () = 0 as the next Including   0 in the polynomial interpolation and passing a parabola through the available data ( 0 ,  0 ,   0 ) and ( 1 ,  1 ) should allow us to obtain a better approximation for   ( 2 ), and with it a higher-order method, as we will see next.

Quartic Two-Step Methods
Seeking a possibly higher-order method, we write the slope estimate of (53) as   1 = (1 − ) 0 for undetermined coefficient  and then advantageously determine it, to have which is the celebrated quartic method of Ostrowski [8].See also Traub [3, page 184, (8)-( 78)] and King [9].Quartic method (62) is also obtained by replacing  1 by 2 1 in the slope estimate   1 in (53).The method is quartic as well but with the simpler error equation Power series expansion changes rational method (62) into the polynomial in  method which is still quartic The quartic method of (66) is also obtained from the parabola passing through the data ( 0 ,  0 ,   0 ) and ( 1 ,  1 ) with the predicted  2 such that ( 2 ) = 0 or by taking 1 in (61) by the perturbed slope turns the method into the supercubic alternatingly converging method
However, if the root repeats, that is, if   () = 0, then the order of convergence of the method plummets from eighth order to first order.

Estimates for the Root Multiplicity Index
In this section we derive both first and second order estimates for the root multiplicity index .Also derived is an estimate for the relative size of the second term in the Taylor expansion of function () at root point .
Assuming that the power series expansion of function (), whose root  we are seeking, is of the form we obtain from it the first order estimate for the root multiplicity index as well as the second order estimate for For example, for  =  2 +  3 ,  = 2, we compute, at  = 0.1, the first order and second order approximations respectively.For  = 0.1 and  = 0.01, we compute from (82) We have also that From the first of the previous equations we have, for  = 1, which, with the Padé rational approximation becomes the first order estimate for the multiplicity index For example, for () =  3 +  4 ,  = {1.0,0.5, 0.1}, (89) yields the approximations  = {3.72,3.51, 3.14} for the exact  = 3.

Correction for Multiple Roots by Undetermined Coefficients
We rewrite Newton's method as for the undetermined coefficient  and have that near a root of multiplicity  ≥ 1 Quadratic convergence is restored, as is well known, with  = .In the previous equation,  and  are the coefficients in the power series expansion of () in (80).
(98) Next, we rewrite the method in (50) as and seek to adjust correction coefficients  and  so that convergence remains cubic even to root  of multiplicity  > 1.By power series expansion we determine that upholds cubic convergence where , , and  are as in (80).Method (99)-( 100) is found in [17].See also [18].The method No such correction to account for multiple roots exists for the quartic two-step method of (62).

Correction of Halley's Method for Multiple Roots
We parametrize Halley's method of (22) with the undetermined coefficients  and  as and determine by power series expansion that for convergence remains cubic even to a root of any multiplicity  ≥ 1 Method ( 104)-( 105) is found in Hansen and Patrick [19].Method (24) becomes here for a multiple root with error equation (106).

From a Cubic to a Quartic Method by Taylor's Formula
We write the second order,  = 2, version of the Taylor-Lagrange formula and take ( 1 =  0 + ) = 0,  =  0 to obtain the iterative method We propose to approximate the solution of the quadratic increment equation or, for that matter, any such higher-order algebraic equation, by the power series and have upon substitution and collection from which we deduce, by annulling lower order terms, that and so on.The methods or are both cubic provided that   () ̸ = 0.As here  = 2, we take next recalculate   (), and verify that the second method in ( 114) is elevated thereby to quartic ) ( 0 − ) 4 +  (( 0 − ) 5 ) . (118)

Contrarily Converging Superquadratic Methods
We write for undetermined coefficient  and have We request that for parameter , or, in view of (51), that by which the iterative method in (119) becomes This superquadratic method converges from above if  > 0, and from below if  < 0.
The interest in the method is that it ultimately converges oppositely to Newton's method as seen by comparing ( 125) with (17).For example, for () = + 2 and starting with  0 = 1/2, we compute from Newton's method and from method (124), respectively, The average of Newton's method and the method of (124) is cubic The modified Halley method is also superquadratic and one sided According to error equation (130), if  < 0, then convergence is at least asymptotically from above; if  > 0, convergence is from below.

Alternating Superlinear and Supercubic Methods
We start by modifying Newton's method as to have the superlinear method that ultimately converges alternatingly if 0 <  < 1.

Repeated Fourth-Order Method
The repeated Newton method is also quartic Similarly, the repeated modified Newton method remains quartic even near a root of any multiplicity  ≥ 1 The repeated-step method not requiring prior knowledge of the multiplicity index  of the root is also quartic The two single-step methods converge contrarily Their average is a cubic method For instance, for () =  2 +  3 +  4 ,  = 2, we compute by the two methods in (153) the sequences (156)

Stacked Higher-Order Methods and Simple Root
Higher-order, single-step methods can be written as a builtup power series of  = /   1 =  0 −  2  0 2nd order where and so on.All evaluations in (158) are at  =  0 .Alternatively, the method may be written as a product, such as the built-up sextic method where with all evaluations in (160) done at  =  0 .

Higher-Order Methods and Multiple Root
The method and converges alternatingly to a single root if  > 0.
Similarly, the method converges supercubically, and alternatingly if  > 0 to a root of any multiplicity  ≥ 1.
Osada [20] suggests the concise cubic method, compared with method (107), where , , and  are as in (80).See also Chun et al. [21].The method is quartic as well even in the event that root  is of multiplicity  ≥ 1.In (169),  = () is as in (46).See also [7].Householder's [22] concise representation of the singlepoint iterative method of order  + 2 unfolds, for  = 2, into the quartic method of (167).An inexact value for the root multiplicity index plunges method (168) to mere first order.Indeed, replacing  in (168) by (1 + ) results in To have a cubic method that does not require prior knowledge of the root multiplicity, we replace  by  to have for which Still higher-order methods are readily thus generated.

Numerical Differentiation
All derivatives may, of course, be approximated by sufficiently accurate finite differences.Replacing   and   in Halley's method by the central finite-difference approximations we have, after some simplification, Following Steffensen's suggestion (see [23,24]), we take ℎ = ( 0 ) and have ) . (176)

Ratio and Mediant Corrections
In this simplest of all iterative methods, we start with / and add 1 to the numerator  if the ratio is an underestimate, or add 1 to the denominator  if the ratio is an overestimate.For example, seeking accurate rational approximations to the root of () = Only a limited number of integers  satisfy the inequality  >   .Consequently, there is only a limited number of integers  that satisfy the absolute value inequality of (191).End of proof.
Unlike the classical method of bisections that may continue indefinitely, the method of mediants that generates rational approximations as in (190) terminates in a finite number of steps if the number it seeks to trap is rational.Conversely, if the method of mediants does not terminate in a finite number of steps, then the number being trapped is surely irrational.
3− 2, we start with / = 4/4, carry out 625 such corrections, and secure the bounds Theorem 2. Let  = / be a positive rational number in lowest terms.For any number  > 0 there is only a finite number of rational numbers, / ̸ = /, such that