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As was pointed out above, this is a good summary of Mathematica's constrained optimization methods. Read through this if you want to know a lot more. A quick answer is below: The answer to your question is strongly dependent on the function you want to maximize. Convex functions can be maximized quite easily, with the error controlled by the PrecisionGoal ...


A way to work with RealDigits, if desired: Drop @@ RealDigits[Round[Sqrt[2], 1*^-7]] // FromDigits 4142136


FractionalPart@N[Sqrt[2], 7] 10^7 // Round (* 4142136*)


I would say that the answer to the first question is, no, if Mathematica does not give any errors when evaluating an expression, then we cannot be absolutely certain that its answer is reliable up to a decent precision. This is shown in the section Examples of Pathological Behavior in the tutorial NIntegrate Integration Rules. Given that the previous ...


In V10 there has been added some symbolic processing of integrands containing an InterpolatingFunction. In particular if the interpolation grid divides the domain of integration into a number of subintervals, the number being at most the value of the option "MaxSubregions", the integrand will automatically be integrated over each subinterval. In V9, this is ...

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