What method does NIntegrate use by default?

Question

There is a variety of algorithms for performing numerical integration (See wiki). What method does NIntegrate use by default?

I looked at the documentation page and I saw that the function NIntegrate can use a ton of different algorithms, but I couldn't find which one is used by default. Where can I find this info?

Answer 1

the answer is in the links -- just to demonstrate or validate what it says:

 First@Last@
    Reap@NIntegrate[Sin[x], {x, 0, 1}, EvaluationMonitor :> Sow[x], 
          MaxRecursion -> 0, Method -> "GaussKronrodRule"]
 xi=First@Last@
    Reap@NIntegrate[Sin[x], {x, 0, 1}, EvaluationMonitor :> Sow[x], 
          MaxRecursion -> 0]

{0.00795732, 0.0469101, 0.122917, 0.230765, 0.360185, 0.5, 0.639815, 0.769235, 0.877083, 0.95309, 0.992043}

{0.00795732, 0.0469101, 0.122917, 0.230765, 0.360185, 0.5, 0.639815, 0.769235, 0.877083, 0.95309, 0.992043}

the default uses the same evaluation points as the "GaussKronrodRule" method.

You can then pull out the weights like this:

 w=(f[x_?NumericQ] := Boole[x == #];
    Quiet@NIntegrate[f[x], {x, 0, 1}, MaxRecursion -> 0]) & /@ xi

{0.021291, 0.0576167, 0.0934004, 0.12052, 0.136425, 0.141494, 0.136425, 0.12052, 0.0934004, 0.0576167, 0.021291}

and show they are of course the same whether you specify "GaussKronrodRule" or nothing.

for completeness, just to show the weight extraction is correct:

 {w.Sin[xi] , NIntegrate[Sin[x], {x, 0, 1}]}
 SameQ @@ %

{0.459698, 0.459698}

True

Finally note, even with MaxRecursions->0 the automatic method can change, for example if it detects an oscilatory function:

 First@Last@Reap@NIntegrate[Sin[4 x], {x, 0, 1}, EvaluationMonitor :> Sow[x], 
      MaxRecursion -> 0]

{1., 0.996057, 0.984292, 0.964888, 0.938153, 0.904508, 0.864484, 0.818712, 0.767913, 0.71289, 0.654508, 0.593691, 0.531395, 0.468605, 0.406309, 0.345492, 0.28711, 0.232087, 0.181288, 0.135516, 0.0954915, 0.0618467, 0.0351118, 0.0157084, 0.00394265, 0.}