Householder's method

In mathematics, and more specifically in numerical analysis, Householder's methods are a class of root-finding algorithms that are used for functions of one real variable with continuous derivatives up to some order. Each of these methods is characterized by the number, which is known as the order of the method. The algorithm is iterative and has an order of convergence of.
These methods are named after the American mathematician Alston Scott Householder. The case of corresponds to Newton's method; the case of corresponds to Halley's method.

Method

Householder's method is a numerical algorithm for solving the equation. In this case, the function has to be a function of one real variable. The method consists of a sequence of iterations
beginning with an initial guess, where the superscript in parentheses indicates the number of times the function is differentiated.
If is a times continuously differentiable function and is a zero of but not of its derivative, then, in a neighborhood of, the iterates satisfy:
for some
This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence has order or better. Furthermore, when close enough to, it commonly is the case that for some. In particular,

if is odd and then convergence to will be from values greater than ;
if is odd and then convergence to will be from values less than ;
if is even and then convergence to will be from the side where it starts; and
if is even and then convergence to will alternate sides.

Despite their order of convergence, these methods are not widely used because the gain in precision is not commensurate with the rise in effort for large. The Ostrowski index expresses the error reduction in the number of function evaluations instead of the iteration count.

For polynomials, the evaluation of the first derivatives of at using Horner's method has an effort of polynomial evaluations. Since evaluations over iterations give an error exponent of, the exponent for one function evaluation is, numerically,,, for, and falling after that. By this criterion, the case is the optimal value of.
For general functions the derivative evaluation using the Taylor arithmetic of automatic differentiation requires the equivalent of function evaluations. One function evaluation thus reduces the error by an exponent of, which is for Newton's method, for Halley's method and falling towards 1 or linear convergence for the higher order methods.

Motivation

First approach

Suppose is analytic in a neighborhood of and. Then has a Taylor series at and its constant term is zero. Because this constant term is zero, the function will have a Taylor series at and, when, its constant term will not be zero. Because that constant term is not zero, it follows that the reciprocal has a Taylor series at, which we will write as and its constant term will not be zero. Using that Taylor series we can write
When we compute its -th derivative, we note that the terms for conveniently vanish:
using big O notation. We thus get that the correction term that we add to to get a value of that is closer to is:
Thus, is.

Second approach

Suppose is a simple root. Then near, is a meromorphic function. Suppose we have the Taylor expansion:
around a point that is closer to than it is to any other zero of. By König's theorem, we have:
This suggests that Householder's iteration might be a good convergent iteration. The actual proof of the convergence is also based on these ideas.

The methods of lower order

Householder's method of order 1 is just Newton's method, since:
For Householder's method of order 2 one gets Halley's method, since the identities
and
result in
In the last line, is the update of the Newton iteration at the point. This line was added to demonstrate where the difference to the simple Newton's method lies.
The third order method is obtained from the identity of the third order derivative of
and has the formula
and so on.

Example

The first problem solved by Newton with the Newton-Raphson-Simpson method was the polynomial equation. He observed that there should be a solution close to 2. Replacing transforms the equation into
The Taylor series of the reciprocal function starts with
The result of applying Householder's methods of various orders at is also obtained by dividing neighboring coefficients of the latter power series. For the first orders one gets the following values after just one iteration step: For an example, in the case of the 3rd order,

d	x₁
1	0.100000000000000000000000000000000
2	0.094339622641509433962264150943396
3	0.094558429973238180196253345227475
4	0.094551282051282051282051282051282
5	0.094551486538216154140615031261962
6	0.094551481438752142436492263099118
7	0.094551481543746895938379484125812
8	0.094551481542336756233561913325371
9	0.094551481542324837086869382419375
10	0.094551481542326678478801765822985

As one can see, there are a little bit more than correct decimal places for each order d. The first one hundred digits of the correct solution are.
Let's calculate the values for some lowest order,
And using following relations,

x	1st	2nd	3rd order	4th order
x₁	0.100000000000000000000000000000000	0.094339622641509433962264150943395	0.094558429973238180196253345227475	0.09455128205128
x₂	0.094568121104185218165627782724844	0.094551481540164214717107966227500	0.094551481542326591482567319958483
x₃	0.094551481698199302883823703544266	0.094551481542326591482386540579303	0.094551481542326591482386540579303
x₄	0.094551481542326591496064847153714	0.094551481542326591482386540579303	0.094551481542326591482386540579303
x₅	0.094551481542326591482386540579303
x₆	0.094551481542326591482386540579303

Derivation

An exact derivation of Householder's methods starts from the Padé approximation of order of the function, where the approximant with linear numerator is chosen. Once this has been achieved, the update for the next approximation results from computing the unique zero of the numerator.
The Padé approximation has the form
The rational function has a zero at.
Just as the Taylor polynomial of degree has coefficients that depend on the function, the Padé approximation also has coefficients dependent on and its derivatives. More precisely, in any Padé approximant, the degrees of the numerator and denominator polynomials have to add to the order of the approximant. Therefore, has to hold.
One could determine the Padé approximant starting from the Taylor polynomial of using Euclid's algorithm. However, starting from the Taylor polynomial of is shorter and leads directly to the given formula. Since
has to be equal to the inverse of the desired rational function, we get after multiplying with in the power the equation
Now, solving the last equation for the zero of the numerator results in
This implies the iteration formula

Relation to Newton's method

Householder's method applied to the real-valued function is the same as applying Newton's method
to find the zeros of the function:
where we are computing a -th derivative and raising to a power of.
In particular, gives Newton's method unmodified and gives Halley's method.