What does the floating point arithmetic contribute to the rounding error in Mathematica?

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I am a theoretical mathematician who works on the iterative method to find some generalized inverse of my interest. When I have tried some numerical example in Mathematica $11.0$, this does not converge to what I have proved theoretically. I have convinced myself to some extend that this is because of the rounding error when you use floating point arithmetic.

I need a proper explanation with the proper reference to justify this.

How does one theoretical mathematician should look into this and stop the discussion in his paper?

Thank you in advance.