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 d+1. Each of these methods is characterized by the number d, which is known as the order of the method. The algorithm is iterative and has a rate of convergence of d+1.
Contents
These methods are named after the American mathematician Alston Scott Householder.
Method
Householder's method is a numerical algorithm for solving the nonlinear equation f(x) = 0. In this case, the function f has to be a function of one real variable. The method consists of a sequence of iterations
beginning with an initial guess x0.
If f is a (d+1) times continuously differentiable function and a is a zero of f but not of its derivative, then, in a neighborhood of a, the iterates xn satisfy:
This means that the iterates converge to the zero if the initial guess is sufficiently close, and that the convergence has order (d+1).
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 d. The Ostrowski index expresses the error reduction in the number of function evaluations instead of the iteration count.
Motivation
An approximate idea of Householder's method derives from the geometric series. Let the real-valued, continuously differentiable function f(x) have a simple zero at x=a, that is a root f(a)=0 of multiplicity one, which is equivalent to
Here
Alternative motivation
Suppose x=a is a simple root. Then near x=a, (1/f)(x) is a meromorphic function. Suppose we have the Taylor expansion:
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 this idea.
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,
The third order method is obtained from the identity of the third order derivative of 1/f
and has the formula
and so on...
Example
The first problem solved by Newton with the Newton-Raphson-Simpson method was the polynomial equation
The Taylor series of the reciprocal function starts with
The result of applying Householder's methods of various orders at x=0 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,
As one can see, there are a little bit more than d correct decimal places for each order d.
Let's calculate the
And using following relations,
1st order;Derivation
An exact derivation of Householder's methods starts from the Padé approximation of order (d+1) 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 d has d+1 coefficients that depend on the function f, the Padé approximation also has d+1 coefficients dependent on f 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,
One could determine the Padé approximant starting from the Taylor polynomial of f using Euclid's algorithm. However, starting from the Taylor polynomial of 1/f 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
Now, solving the last equation for the zero
This implies the iteration formula