Problem with NIntegrate over a piecewise function

Question

The following code seems to generate wrong results
q2 = 1/y^0.9;
G2 = 1.1 x^0.045;
b2bar = x /. Solve[G2 == 1, x][[1]]
ff = FullSimplify[PiecewiseExpand[D[Min[G2, 1], x]*x*q2]]
NIntegrate[ff, {y, 0, 1}, {x, y, b2bar}]
NIntegrate[ff, {y, 0, 1}, {x, y, 1}]

Since ff=0 for x>=b2bar, one would expect NIntegrate[ff, {y, 0, 1}, {x, y, b2bar}] and NIntegrate[ff, {y, 0, 1}, {x, y, 1}] to generate the same results. But NIntegrate[ff, {y, 0, 1}, {x, y, b2bar}] gives 0.038246 and NIntegrate[ff, {y, 0, 1}, {x, y, 1}] gives 0.022375.
Is there an error with NIntegrate when integrating piecewise functions? How can I avoid this?

Michael E2 · Accepted Answer

As I mentioned in a comment, NIntegrate does solve the condition 1.1 x^0.045 < 1 for the singularity at x == b2bar and this causes a problem with the integration, which is itself an issue. But that issue can be avoided by reducing the condition to something NIntegrate can handle. If we throw in the domain restriction 0 <= x <= 1 && 0 <= y <= 1 from the integral, or just the x component 0 <= x <= 1, then Reduce will solve the condition. ff = PiecewiseExpand[D[Min[G2, 1], x]*x*q2, Method -> {"ConditionSimplifier" -> (Quiet[ Reduce[# && 0 <= x <= 1, x, Reals], Reduce::ratnz] &)}] NIntegrate[ff, {y, 0, 1}, {x, y, 1}]

Anton Antonov · Answer

If your second numerical integration uses appropriate options nearly the same results are obtained.
(Those options can be figured out from the messages NIntegrate issues while evaluating your second integral.)
One such way is:
NIntegrate[ff, {y, 0, 1}, {x, y, 1}, 
 Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 10^6}, 
 MaxRecursion -> 1000]

Here is another:
NIntegrate[ff, {y, 0, 1}, {x, y, 1}, MinRecursion -> 3]

(See res3 and res4 in the attached screenshot.)

Problem with NIntegrate over a piecewise function

2 Answers

Add your own answers!

Ask a Question