ExerciseNonlinear equationsadvanced

Exercise: numerical comparison of iterative methods

Newton, Halley, Ostrowski, Traub, midpoint, Jarratt and double Newton on test functions with tolerance 10⁻¹⁰⁰: iterations, residuals and ACOC confirming the theoretical orders.

Experimental conditions

Seven methods are compared under identical conditions: stopping criterion $|x_{k+1}-x_k|<10^{-100}$ or 60 iterations, with extended-precision arithmetic (400 digits), essential for such extreme tolerances. The test functions and their roots are $f_1(x)=\sin x-e^{-x}$ with $\alpha\approx 0.58853274$ and $f_3(x)=(x-1)^3-1$ with $\alpha=2$ .

Results for f₁ (x₀=0.1)

Method	iter	$\|f(x_{k+1})\|$	$\|x_{k+1}-x_k\|$	ACOC
Newton	8	6.5531·10⁻²⁰⁵	1.0865·10⁻¹⁰²	2.0000
Halley	6	0	5.3661·10⁻¹⁸⁷	3.0000
Ostrowski	5	1.2716·10⁻⁴⁰⁸	6.7766·10⁻¹⁹⁹	4.0000
Traub	6	1.2716·10⁻⁴⁰⁸	9.3924·10⁻¹⁶⁶	3.0000
Punto medio	6	1.2716·10⁻⁴⁰⁸	2.9422·10⁻¹⁹²	3.0000
Jarratt	5	1.2716·10⁻⁴⁰⁸	5.1327·10⁻¹⁹⁸	4.0000
Newton doble	5	1.2716·10⁻⁴⁰⁸	4.7250·10⁻²⁰⁵	4.0000

Results for

f_1(x)=\sin x-e^{-x}

with

x_0=0.1

Results for f₃ (x₀=1.5)

Method	iter	$\|f(x_{k+1})\|$	$\|x_{k+1}-x_k\|$	ACOC
Newton	11	2.8174·10⁻³⁵⁹	3.0646·10⁻¹⁸⁰	2.0000
Halley	7	0	1.7850·10⁻²¹⁴	3.0000
Ostrowski	6	0	7.3471·10⁻²³⁹	4.0000
Traub	58	1.2798·10⁻³⁹³	5.9750·10⁻¹³²	3.0000
Punto medio	7	2.1994·10⁻³⁹⁹	9.2824·10⁻¹³⁴	3.0000
Jarratt	6	0	7.3471·10⁻²³⁹	4.0000
Newton doble	6	0	3.0646·10⁻¹⁸⁰	4.0000

Results for

f_3(x)=(x-1)^3-1

with

x_0=1.5