The origin of the interpolation with rational functions can be found in the previous work done by Edmond Halley. Halley's formula is known as one-point third-order iterative method to solve ${\displaystyle \,f(x)=0}$  by means of approximating a rational function defined by

${\displaystyle h(z)={\frac {a}{z+b}}+c.}$ 

We can determine a, b, and c so that

${\displaystyle h^{(i)}(x)=f^{(i)}(x),\qquad i=0,1,2.}$ 

Then solving ${\displaystyle \,h(z)=0}$  yields the iteration

${\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})}{f'(x_{n})}}\left({\frac {1}{1-{\frac {f(x_{n})f''(x_{n})}{2(f'(x_{n}))^{2}}}}}\right).}$ 

This is referred to as Halley's formula. This geometrical interpretation ${\displaystyle h(z)}$  was derived by Gander (1978), where the equivalent iteration also was derived by applying Newton's method to

${\displaystyle g(x)={\frac {f(x)}{\sqrt {f'(x)}}}=0.}$ 

We call this algebraic interpretation ${\displaystyle g(x)}$  of Halley's formula.

Similarly, we can derive a variation of Halley's formula based on a one-point second-order iterative method to solve ${\displaystyle \,f(x)=\alpha (\neq 0)}$  using simple rational approximation by

${\displaystyle h(z)={\frac {a}{z+b}}.}$ 

Then we need to evaluate

${\displaystyle h^{(i)}(x)=f^{(i)}(x),\qquad i=0,1.}$ 

Thus we have

${\displaystyle x_{n+1}=x_{n}-{\frac {f(x_{n})-\alpha }{f'(x_{n})}}\left({\frac {f(x_{n})}{\alpha }}\right).}$ 

The algebraic interpretation of this iteration is obtained by solving

${\displaystyle g(x)=1-{\frac {\alpha }{f(x)}}=0.}$ 

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).

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