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Integration warning: Merely an error estimate, or an upper bound?

Posted 1 year ago

This integral

i1 = NIntegrate[ Sqrt[1 + (x^6 + y^6)/(1 - x^4 - y^4)^(3/2)] ,
  {x, 0, 1/3^(1/4)}, {y, 0, x}, WorkingPrecision -> 100, PrecisionGoal -> 40 ]

returns, roughly, 0.30155... along with the warning "NIntegrate obtained 0.30155... and 3.20396...*10^-16 for the integral and error estimates"

If the estimate 3.20396...*10^-16 is an upper bound, this is valuable information.

But if the actual error could be larger than this, then this number is not useful (to me).

Which is it? A mere estimate of the error, or an upper bound on the error?

POSTED BY: Robert Baillie
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