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3 years ago in Applied Mathematics , Mathematical Analysis By Keerthi Gupta

How can Bernoulli’s inequality be approximated effectively?

I'm working on a computational model where I need to handle expressions of the form (1+x)^n frequently. The standard Bernoulli inequality gives a bound, but for my iterative algorithms, I need a tighter, computationally efficient approximation. I'm looking for guidance on robust simplification techniques that maintain accuracy without over-complicating the code.

Inequality Numerical Stability

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By Raj Shravan Answered 1 year ago

In my work on optimization algorithms, I've often needed to move beyond the basic Bernoulli bound. For practical approximation, I would recommend a two-pronged approach. First, for a quick and reliable estimate, use the first two terms of the binomial expansion, (1 + nx), as your linear approximation; it's surprisingly robust near zero. For tighter bounds in iterative schemes, consider a Padé approximant instead of a pure Taylor series it often gives better accuracy over a wider interval with similar computational cost. Always validate the error against your specific parameter ranges.

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