I have a function $f(x) = e^x - (2-x)^3$
And I am using the Secant method to find a root between $0$ and $5$.
I know that the value of this root is $t = 0.7261444\ldots$
Here's my output from matlab for 14 iterations where I show
$k$, $x_k$,$|t-x_k|$,$|t-x_k|^{1.68}$,$|t-x_k|^2$
k | x(k) | |t-x(k)| | |t-x(k)|^1.68 | |t-x(k)|^2 1 | 0.00000000e+000 | 7.261e-001 | | 2 | 5.00000000e+000 | 4.274e+000 | | 3 | 1.91872122e-001 | 5.343e-001 | 3.489e-001 | 2.854e-001 4 | 3.17334769e-001 | 4.088e-001 | 2.225e-001 | 1.671e-001 5 | 6.42309464e-001 | 8.384e-002 | 1.554e-002 | 7.028e-003 6 | 7.12431582e-001 | 1.371e-002 | 7.420e-004 | 1.880e-004 7 | 7.25679662e-001 | 4.648e-004 | 2.518e-006 | 2.160e-007 8 | 7.26141901e-001 | 2.565e-006 | 4.049e-010 | 6.580e-012 9 | 7.26144465e-001 | 4.793e-010 | 2.205e-016 | 2.297e-019 10 | 7.26144466e-001 | 5.551e-016 | 2.347e-026 | 3.081e-031 11 | 7.26144466e-001 | 0.000e+000 | 0.000e+000 | 0.000e+000 12 | 7.26144466e-001 | 1.110e-016 | 1.571e-027 | 1.233e-032 13 | 7.26144466e-001 | 1.110e-016 | 1.571e-027 | 1.233e-032 14 | NaN | NaN | NaN | NaN
Now I am told in a question that the 'rate of convergence of a sequence $\{\epsilon_0,\epsilon_1,\epsilon_2,\ldots\}$ is $q$ if
$\lim_{k \rightarrow \infty} \frac{\epsilon_{k+1}}{\epsilon_k^q} = u$
for some constant $u$.
And I am told to make the program output $|t-x_k|$,$|t-x_k|^{1.68}$,$|t-x_k|^2$ and hence deduce that the rate of converge is greater than $1$, less than $2$ and roughly $(1 + \sqrt 5)/2 = 1.618$, the golden mean.
I can't see how I supposed to deduce this...I tried getting a few values for $\frac{|t-x_{k+1}|}{|t-x_k|^{1.68}}$ ... and this seems to go to $0$ and then the precision becomes too small for the computer to handle. I don't really know what I am doing here trying to deduce that the convergence is 1.6.18. Can anyone explain how I am supposed to be making this deduction?
EDIT: If I check $\phi=(1 + \sqrt 5)/2\approx1.618$, $|t-x_{k+1}|/|t-x_k|^\phi$ up to iteration 9 I get
3 | 5.094e-002 4 | 1.127e+000 5 | 3.564e-001 6 | 7.569e-001 7 | 4.802e-001 8 | 6.331e-001 9 | 5.330e-001
Can this be taken that the sequence is converging to a fixed number $u$?