# Convergence and precision of root-seeking programs

1. How might I find a root-seeking program's "order of convergence"?

2. For an iteration program, how might I examine the effect of the rounding-error? (If MATLAB displays the iterates to 4 d.p., does that mean it is working with that level of accuracy or does it work with more digits internally?)

Thanks.

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Welcome to math.sx! You should try to write a more descriptive title. As for matlabs display of numbers, you can change it with format long or with fprintf if that's what you mean – Mikael Öhman Sep 5 '11 at 8:04

## 1 Answer

1. You will need to find out which algorithm the program implements. The bisection method has a linear convergence, and Newton iterations has quadratic convergence.

2. MATLAB works with normal 8 byte floating point numbers (double), which has precision down to about 16 decimals (or rather, exactly 52 decimals position in binary). In matlab, the function eps(x) will give you the smallest possible increment for the value x. The value displayed corresponds to an increment of the least significant decimal. Note that you cannot exactly describe even just 0.1 in binary, as you can try for yourself in MATLAB: fprintf('%.20e\n',0.1) will result in 1.00000000000000005551e-01 which is the closest binary representation.

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Thanks, Mikael. So if there are N iterates for the Newton Raphson method, does the round-off error accumulate through each iterate? – ans Sep 5 '11 at 9:40
"Newton ... has quadratic convergence." -that is, if the root being sought is simple. If the root is a multiple one, the convergence is linear at best. – J. M. Sep 5 '11 at 9:41
Thanks, J.M.. Does the rounding error in the program make convergence slower? And if so, how might I estimate its effect? – ans Sep 5 '11 at 9:53
@ans: The precision errors does not accumulate during iterations in any method I could think off. It just puts a limit on how close you can get, or cause problems if the tangent is (nearly) singular. – Mikael Öhman Sep 5 '11 at 11:11
I would even say that these root-finders are "self-correcting": if you are within a "basin of convergence", the convergence rate follows the rate expected of it up until you're near the precision of your system. – J. M. Sep 5 '11 at 11:38