ExerciseNonlinear equationsintermediate

Exercise: Newton on x=cos²x

Full application of Newton's method to x=cos²x from x0=0.3: table of iterates, residuals, increments and ACOC tending to the theoretical order 2.

Setup

Solve $x=\cos^2 x$ , that is, $f(x)=\cos^2 x-x=0$ , with initial estimate $x_0=0.3$ , stopping criterion $|x_{k+1}-x_k|<10^{-9}$ and at most 20 iterations. Newton needs the derivative:

f'(x)=-2\cos x\,\sin x-1=-\sin 2x-1

Iterations

iter	$x_k$	$\|f(x_k)\|$	$\|x_k-x_{k-1}\|$	ACOC
1	0.691570	0.098293	0.39157	n/a
2	0.641989	5.3803·10⁻⁴	0.049581	n/a
3	0.641714	2.1349·10⁻⁸	2.7463·10⁻⁴	2.5143
4	0.641714	3.3663·10⁻¹⁷	1.0898·10⁻⁸	1.9505
5	0.641714	8.3691·10⁻³⁵	1.7184·10⁻¹⁷	1.9999

Newton iterations from

x_0=0.3

(the last two rows require extended-precision arithmetic).