Simple rational approximation (SRA) is a subset of interpolating methods using

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

s. Especially, SRA interpolates a given function with a specific rational function whose

pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets * Pole star, a visible star that is approximately aligned with th ...

s and zeros are simple, which means that there is no multiplicity in poles and zeros. Sometimes, it only implies simple poles. The main application of SRA lies in finding the zeros of secular functions. A

divide-and-conquer algorithm In computer science, divide and conquer is an algorithm design paradigm. A divide-and-conquer algorithm recursively breaks down a problem into two or more sub-problems of the same or related type, until these become simple enough to be solved dire ...

to find the

eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

and

eigenvectors In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...

for various kinds of

matrices Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

is well known in

numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods th ...

. In a strict sense, SRA implies a specific

interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has ...

using simple rational functions as a part of the divide-and-conquer algorithm. Since such secular functions consist of a series of rational functions with simple poles, SRA is the best candidate to interpolate the zeros of the secular function. Moreover, based on previous researches, a simple zero that lies between two adjacent poles can be considerably well interpolated by using a two-dominant-pole rational function as an approximating function.

One-point third-order iterative method: Halley's formula

The origin of the interpolation with rational functions can be found in the previous work done by

Edmond Halley Edmond (or Edmund) Halley (; – ) was an English astronomer, mathematician and physicist. He was the second Astronomer Royal in Britain, succeeding John Flamsteed in 1720. From an observatory he constructed on Saint Helena in 1676–77, Hal ...

. Halley's formula is known as one-point third-order iterative method to solve

\,f(x)=0

by means of approximating a rational function defined by :

h(z)=\frac+c.

We can determine a, b, and c so that :

h^(x)=f^(x), \qquad i=0,1,2.

Then solving

\,h(z)=0

yields the iteration :

x_=x_-\frac \left(\right).

This is referred to as Halley's formula. This ''geometrical interpretation''

h(z)

was derived by Gander (1978), where the equivalent iteration also was derived by applying Newton's method to :

g(x)=\frac=0.

We call this ''algebraic interpretation''

g(x)

of Halley's formula.

One-point second-order iterative method: Simple rational approximation

Similarly, we can derive a variation of Halley's formula based on a one-point ''second-order'' iterative method to solve

\,f(x)=\alpha(\neq 0)

using simple rational approximation by :

h(z)=\frac.

Then we need to evaluate :

h^(x)=f^(x), \qquad i=0,1.

Thus we have :

x_=x_-\frac \left(\frac\right).

The algebraic interpretation of this iteration is obtained by solving :

g(x)=1-\frac=0.

This one-point second-order method is known to show a locally quadratic convergence if the root of the equation is simple. SRA strictly implies this one-point second-order interpolation by a simple rational function. We can notice that even third order method is a variation of Newton's method. We see the Newton's steps are multiplied by some factors. These factors are called the ''convergence factors'' of the variations, which are useful for analyzing the rate of convergence. See Gander (1978).

References

*. *. *. *{{citation , last = Gander , first = Walter , publisher = Stanford University, School of Humanities and Sciences, Computer Science Dept. , title = On the linear least squares problem with a quadratic constraint , year = 1978. Interpolation