I am trying to teach myself numerical analysis and I am trying to find out how you determine the order of convergence for newtons method. On the wiki there are multiple formulas for newton methods convergence chapter, some with a q value and some without and I am incredibly confused.
I have the nonlinear equation: $f(x) = -cos + x^3 + 2x^2 + 1 = 0$ I have the values from the iterations with initial guess -2 and calculated the error between iterations as $e_n = |x^* - x_n|$ but I'm not sure what to do after this to determine order of convergence. Would appreciate any insight.
Newton iteration values:
Iteration 1: $x1=−2.000153634$
Iteration 2: $x2=−2.000152282$
Iteration 3: $x3=−2.000152293$
Iteration 4: $x4=−2.000152293$
Error between iterations: $e_n = |x^* - x_n|$: e0= |-2.000152293-(-2)|=0.000152293
$e1=|-2.000152293-(-2.000153634)|=0.000001341$
$e2=|-2.000152293-(-2.000152282)|=0.000000011$
$e3=| -2.000152293-(-2.000152293)|=0$
$e4=| -2.000152293-(-2.000152293)|=0$