ExerciseNonlinear systemsadvanced

Exercise: numerical comparison on systems

Newton, Trapezoids, Golden Ratio, NA, Jarratt and RN on two test systems with tolerance 10⁻¹²: iterations, norms and ACOC, with RN (order 6) as the most effective method.

Experimental conditions

Six methods are compared (RN with $a=-\frac12$ , $b=\frac32$ , order 6) using extended-precision arithmetic (200 digits) and stopping criterion $\|x^{(k+1)}-x^{(k)}\|+\|F(x^{(k+1)})\|<10^{-12}$ , at most 40 iterations. Test systems: $F_1(x,y)=\bigl(e^xe^y+x\cos y,\;x+y-1\bigr)$ with $\alpha\approx[5.15723,-4.15723]$ , and $F_2(x,y,z)=\bigl(\cos y-\sin x,\;z^x-\frac1y,\;e^x-z^2\bigr)$ with $\alpha\approx[0.909569,\,0.661227,\,1.575834]$ .

Results for F₁ (x⁽⁰⁾=[2,−1])

Method	iter	$\\|F(x^{(k+1)})\\|$	$\\|x^{(k+1)}-x^{(k)}\\|$	ACOC
Newton	5	4.3406·10⁻¹⁷	6.2690·10⁻⁹	1.9989
Trapecios	9	1.9040·10⁻⁶⁴	6.7281·10⁻²²	2.9993
Golden Ratio	7	8.2086·10⁻⁴³	2.4788·10⁻²⁷	2.1867
NA	5	9.5523·10⁻⁸⁷	5.7738·10⁻³⁶	3.4151
Jarratt	6	2.8540·10⁻¹⁹⁵	6.1911·10⁻⁴⁹	3.9985
RN	4	2.6765·10⁻¹⁴¹	1.1130·10⁻²³	6.4561

Results for

F_1

with

x^{(0)}=[2,-1]

Results for F₂ (x⁽⁰⁾=[1,1,2])

Method	iter	$\\|F(x^{(k+1)})\\|$	$\\|x^{(k+1)}-x^{(k)}\\|$	ACOC
Newton	6	5.7716·10⁻¹⁷	7.5973·10⁻⁹	1.9760
Trapecios	6	6.1613·10⁻⁵⁷	1.4405·10⁻¹⁹	2.9999
Golden Ratio	6	1.0510·10⁻⁴¹	1.2430·10⁻²⁵	2.7407
NA	6	5.4870·10⁻⁹²	1.4939·10⁻³⁸	3.2701
Jarratt	4	5.0114·10⁻⁷⁷	6.9430·10⁻²⁰	3.9638
RN	4	7.7869·10⁻²⁰⁸	1.1636·10⁻⁴¹	6.0053

Results for

F_2

with

x^{(0)}=[1,1,2]