Can I eliminate roundoff error caused due to floating point arithmetic limit in MATLAB?

I am trying to do an error analysis of RK4 in MATLAB. But after the step size reaches a particular value the error starts increasing with decrease in step size. I know that these occurs due to the round off error generated in MATLAB. What can be the solution of this?

[Stephen23]

"Can I eliminate roundoff error caused due to floating point arithmetic limit in MATLAB?"

The very general answer to your very general question is no.

If there was a general solution to "eliminate roundoff error caused due to floating point arithmetic" then everyone would use it.

In some circumstances you can change an algorithm to make it more numerically robust, but as you did not provide any details of the arithmetic that you are performing we cannot help you know if that can be applied to your code.

[Ankhiparna Guha]

I am solving a second order ODE with RK4 algorithm. I am reducing the step size and observing the error is reducing. I saw that upto step size = 10e-4 the order of error is coming right. But for step size less than that, the order changes.

Answer 1

[Star Strider]

Floating-point approximation error is simply inherent in binary representations of floating-point numbers. The best way to deal with it is to use a tolerance, for example —

flag = abs(x - target) < 1E-4;

or something similar, to test for the equality of ‘x’ and ‘target’ unless all the values are integers, since in that situation exact representations and comparisons are possible. Use whatever tolerance value (here 1E-4) is appropriate.

.

[Ankhiparna Guha]

I have used

format long;

I think I am getting correct result upto 10e-15. After that, the expected order of error is not coming.

[John D'Errico]

@Ankhiparna Guha - You CANNOT go beyond roughly 15-16 significant digits in floating point arithmetic. So no. You cannot eliminate floating point errors when working in floating point arithmetic. At best you can use good numerical practices, the things taught in numerical analysis and numerical methods courses, to avoid issues. These are the kinds of things I assume you will learn in the classes you are taking, since you are trying to solve this problem. One such idea is to use tolerances, as Star suggests. As for what is exactly happening in your code, we cannot know, since we see no code from you, nor any idea what exactly you did.

[Ankhiparna Guha]

Okay .. I understood.