# NIntegrate Error-Estimate Question

When I do some complicated 4-dimensional integral numerically, Mathematica tells me its estimate for the value of the integral, and its estimate for the value of the error in the calculation.

The integrand:

Integrand[x_, y_, z_, Ekb_] := Sqrt[-4.14195*^-29 +1.38065*^-29 Ekb + (1.7055100000000003*^-29 2.71828^(-((2.76817*^8 ((0.986286 x + 0.165048 y)^2 + z^2))/(1. + 4707.27 (-0.165048 x + 0.986286 y)^2))))/(1. + 4707.27 (-0.165048 x +0.986286 y)^2) + (2.4364400000000005*^-29 2.71828^(-((2.76817*^8 ((0.986286 x -0.165048 y)^2 + z^2))/(1. + 4707.27 (0.165048 x + 0.986286 y)^2))))/(1. + 4707.27(0.165048 x + 0.986286 y)^2)]/(1. + 2.71828^(7.2429699999999995*^28 (-2.7613*^-29 + 1.38065*^-29 Ekb)))


The Integration:

Re@NIntegrate[Integrand[x,y,z,Ekb],{x,-9.53853 10^(-5),9.53853 10^(-5)},{y,-57 10^(-5),57 10^(-5)},{z,-4.37824 10^(-5),4.37824 10^(-5)},{Ekb,0,7*3/17},Method ->     {"GlobalAdaptive", Method -> "MultiDimensionalRule", "MaxErrorIncreases" -> 100}]


It puts out the error message:

NIntegrate::eincr: The global error of the strategy GlobalAdaptive has increased more than 100 times. The global error is expected to decrease monotonically after a number of integrand evaluations. Suspect one of the following: the working precision is insufficient for the specified precision goal; the integrand is highly oscillatory or it is not a (piecewise) smooth function; or the true value of the integral is 0. Increasing the value of the GlobalAdaptive option MaxErrorIncreases might lead to a convergent numerical integration. NIntegrate obtained 1.51764*10^-27+7.31069*10^-26 I and 1.0039003041591675*^-28 for the integral and error estimates. >>

Should this be interpreted as (1.51764*10^-27+7.31069*10^-26 I) ± 1.0039003041591675*^-28,
or
(1.51764*10^-27+7.31069*10^-26 I) ± 5.0195*10^-29 ?

• I have never seen Mathematica output like that. Are you sure you are actually using that? If so, please provide the code you are using. – Sjoerd C. de Vries Feb 15 '13 at 11:41
• Are you perhaps referring to error messages/warnings like this: NIntegrate::maxp: The integral failed to converge after 33 integrand evaluations. NIntegrate obtained 1.9558072180392028 and 0.06781302015519788 for the integral and error estimates.`? – Sjoerd C. de Vries Feb 15 '13 at 11:47
• Yes indeed! I'm sorry for being unclear. was Just trying to get that message in the typing box :) – Henk Spaan Feb 15 '13 at 11:50
• @SjoerdC.deVries So in your example, should i interpret the answer as: 1.95581 ([PlusMinus]0.067813) or 1.95581 ([PlusMinus]0.0339065) Thanks again! – Henk Spaan Feb 15 '13 at 11:56
• Boundary values? – Sjoerd C. de Vries Feb 15 '13 at 12:26