Finding error of secant method empirically

[jjr]

Homework Statement

I need to find the roots of a given equation using the secant method and matlab. I have already found all the roots, but I am also asked to find the rate of convergence for the method empirically, meaning that they want me to find the order of convergence through the set of errors generated in by the program.

Homework Equations

The Attempt at a Solution

I know very well that the order of convergence of the secant method is α ≈ 1.618, but I am not sure how I should go about showing this empirically. In my case the roots are found within the desired error tolerance (1e-10) in about 6-8 iterations. I've tried taking the logarithms of the errors and plotting them versus the iteration number n, to draw out the exponential behaviour, but I can't get anyone clear answer... Any suggestions for putting me on the right track? Let me know if I need to clarify anything.

J

 

[BvU]

Check out these notes which Anatolii Grinshpan put on the net.

 

[jjr]

Thank you! I'm still not sure how I would go about finding the exponent given a set of errors, but I think it will be sufficent to show that $\frac{|x_{n+1}-α|}{|x_n-α|^p} ≈ C$ , making $p ≈ 1.618$ a starting assumption.

 

[BvU]

I think so. What I wanted to point out is that the analysis is based on ignoring higher order terms, and even then you need $f''$ and $f'$ in $\alpha$. Depending on your test functions (and on whether you have an exact solution for $\alpha$ or only the last iteration) you end up in the noise very quickly. So with only a few iterations and any deviation popping up twice (in point n and also in point n+1), you can't expect a perfectly smooth log-log plot.

so there's no need to be so precise about the 1.61803398874989...

Conceptually I always remembered this as: Newton is quadratic, secant approaches Newton if the secant approaches the derivative. But it's a numerical derivative, so before it's really good enough you are in the noise anyway. So somewhere between 1 and 2.

 

[paranormal]

: As a scientist, it is important to understand the limitations and assumptions of different methods used in scientific research. The secant method is a numerical method used to find the roots of a given equation, and its order of convergence is an important factor in determining its accuracy and efficiency. In order to find the error of the secant method empirically, you will need to use a set of errors generated by your program and plot them against the iteration number. This will allow you to observe the behavior of the errors and determine the order of convergence by looking at the slope of the curve. Additionally, you can also compare the rate of convergence with the theoretical value of α ≈ 1.618 to validate your results. It is important to note that the accuracy of your results may be affected by the choice of initial points and the specific equation you are solving. Overall, by analyzing the empirical error data, you can gain a better understanding of the performance of the secant method and its limitations.