# NIntegrate Which Includes a Definite Integral

I need to calculate a numerical integral which consists of the following functions:

f7 := 1/
(Pi^8*(
(1/8)*(y0 + y1 + y2 + y3 - y4 - y5 - y6 - y7)^2 + 1)
*
((1/8)*(y0 + y1 - y2 - y3 + y4 + y5 - y6 - y7)^2 + 1)
*
((1/8)*(y0 - y1 + y2 - y3 + y4 - y5 + y6 - y7)^2 + 1)
*
((1/8)*(y0 - y1 - y2 + y3 - y4 + y5 + y6 - y7)^2 + 1)
*
((1/8)*(y0 - y1 - y2 + y3 + y4 - y5 - y6 + y7)^2 + 1)
*
((1/8)*(y0 - y1 + y2 - y3 - y4 + y5 - y6 + y7)^2 + 1)
*
((1/8)*(y0 + y1 - y2 - y3 - y4 - y5 + y6 + y7)^2 + 1)
*
((1/8)*(y0 + y1 + y2 + y3 + y4 + y5 + y6 + y7)^2 + 1))

f6 := Integrate[f7, {y7, -Infinity, Infinity},
Assumptions -> {Element[{y0, y1, y2,  y3, y4, y5, y6},Reals]}]

h7 = NIntegrate[f7*Log[f7/f6], {y0, -Infinity, Infinity}, {y1,
-Infinity, Infinity}, {y2, -Infinity, Infinity},
{y3, -Infinity, Infinity}, {y4, -Infinity, Infinity},
{y5, -Infinity, Infinity}, {y6, -Infinity, Infinity},
{y7, -Infinity, Infinity}]


My question here is, f6 is hard to calculate symbolically. However, I only need h7. Is there any way to calculate h7 using NIntegrate without trying to calculate f6 symbolically?

• People here generally like users to post code as Mathematica code instead of images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find this this meta Q&A helpful Oct 1 '16 at 17:01
• Don't use subscripts. Subscript[y, 1] doesn't create a variable $y_1$, it takes a variable y that is inside a two-argument function Subscript. Compute the following to see what I mean: D[Subscript[y, 1], y]. Use simply y1, y2 etc. Oct 1 '16 at 18:07

You can do the multiple Integral completeley with NIntegrate. Try this code:

    In[31]:= f7[y0_, y1_, y2_, y3_, y4_, y5_, y6_, y7_] =
1/(Pi^8*((1/8)*(y0 + y1 + y2 + y3 - y4 - y5 - y6 - y7)^2 +
1)*((1/8)*(y0 + y1 - y2 - y3 + y4 + y5 - y6 - y7)^2 +
1)*((1/8)*(y0 - y1 + y2 - y3 + y4 - y5 + y6 - y7)^2 +
1)*((1/8)*(y0 - y1 - y2 + y3 - y4 + y5 + y6 - y7)^2 +
1)*((1/8)*(y0 - y1 - y2 + y3 + y4 - y5 - y6 + y7)^2 +
1)*((1/8)*(y0 - y1 + y2 - y3 - y4 + y5 - y6 + y7)^2 +
1)*((1/8)*(y0 + y1 - y2 - y3 - y4 - y5 + y6 + y7)^2 +
1)*((1/8)*(y0 + y1 + y2 + y3 + y4 + y5 + y6 + y7)^2 + 1));

In[32]:= f6[y0_?NumericQ, y1_?NumericQ, y2_?NumericQ, y3_?NumericQ,
y4_?NumericQ, y5_?NumericQ, y6_?NumericQ] :=
NIntegrate[
f7[y0, y1, y2, y3, y4, y5, y6, y7], {y7, -Infinity, Infinity},
MaxRecursion -> 100]

(h7 = NIntegrate[
f7[y0, y1, y2, y3, y4, y5, y6, y7]*
Log[f7[y0, y1, y2, y3, y4, y5, y6, y7]/
f6[y0, y1, y2, y3, y4, y5, y6]], {y0, -Infinity,
Infinity}, {y1, -Infinity, Infinity}, {y2, -Infinity,
Infinity}, {y3, -Infinity, Infinity}, {y4, -Infinity,
Infinity}, {y5, -Infinity, Infinity}, {y6, -Infinity,
Infinity}, {y7, -Infinity, Infinity},
MaxRecursion -> 20]) // Timing

NIntegrate::eincr: The global error of the strategy
GlobalAdaptive has increased more than 2000 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 -7.47918 and 4.45574566213166 for the integral  and
error estimates. >>

{50349.7, -7.47918}

In[33]:= (h7 =
NIntegrate[
f7[y0, y1, y2, y3, y4, y5, y6, y7]*
Log[f7[y0, y1, y2, y3, y4, y5, y6, y7]/
f6[y0, y1, y2, y3, y4, y5, y6]], {y0, -Infinity,
Infinity}, {y1, -Infinity, Infinity}, {y2, -Infinity,
Infinity}, {y3, -Infinity, Infinity}, {y4, -Infinity,
Infinity}, {y5, -Infinity, Infinity}, {y6, -Infinity,
Infinity}, {y7, -Infinity, Infinity}, MaxRecursion -> 100,
Method -> {"GlobalAdaptive", "MaxErrorIncreases" -> 10000,
Method -> "GaussKronrodRule"}]) // Timing

Out[33]= \$Aborted
`

Use ?NumericQ for dhe definition of f6 in order to force it to be evaluated only when it has got numerical values from the h7 - Integration.

I first got a result with a very large error after one day of calculation on a intel i7 4790 Processor. To get a better result, you should try the code wiht higher MaxErrorIncreases. Attention:This may take a few days of calculation.