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Calculating Integral in Mathematica

  1. Jul 4, 2008 #1
    I have a function f(x) which is defined as

    [tex]f(x) = \int_{x}^{c} \int_{x}^{c} f(x_1,x_2) dx_1 dx_2 [/tex]

    where c is a known constant and f(x1,x2) is a multivariate Gaussian. Unfortunetaly there is no closed form solution for f(x).
    My problem is I want to numerically calculate

    [tex] \int_{c_1}^{c} f(x) dx [/tex]

    where again c_1, and c are known constants.
    How do I numerically evaluate such an integral in Mathematica? I get errors every time saying "x is not a valid limit of integration".
    Any ideas, how I would input the above into Mathematica to get a numerical solution?
  2. jcsd
  3. Jul 4, 2008 #2


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    NIntegrate[ f[x], {x, c1, c} ]
    where c1 and c are numerical values or constants (like Pi, 2*E, 33.401 etc).
  4. Jul 4, 2008 #3
    Unfortunately f(x) has no closed form solution (so the expression for f(x) still has the symbols x1, x2) and thus evaluating "NIntegrate[ f[x], {x, c1, c} ]" just gives the error the "Integrand ..... is not numerical at...".

    I've also tried "NIntegrate[ f[x], {x1, x, c}, {x2, x, c}, {x, c1, c} ]", but this gives the error "x1 = x is not a valid limit of integration. "

    I'm sure there's some way to do it, but just can't figure it out?
  5. Jul 4, 2008 #4


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    Can you post some code, in particular be a little more specific about f(x1, x2) ?
  6. Jul 4, 2008 #5
    Basically, f(x1,x2) is a multivariate Normal pdf. An example in Mathematica code would be:

    f(x) = "Integrate[PDF[MultinormalDistribution[{5,6}, {{1,1}, {1,2}}], {x1,x2}], {x1,x,c},{x2,x,c}]"

    For any particular value of x and c (c is a known constant) I can numerically evaluate the above expression, however, I want to evaluate the above expression for all possible values of x, where x is a countinous number, and hence I need to integrate the above expression for all possible values of x (say, x ranges from 0-10 in the above example)
  7. Jul 4, 2008 #6


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    You are right that
    Code (Text):
    NIntegrate[ f[x], {x1, x, c}, {x2, x, c}, {x, c1, c} ]
    doesn't work, but I think that
    Code (Text):
    NIntegrate[ f[x], {x, c1, c}, {x1, x, c}, {x2, x, c} ]
    does. At least it gives an answer... apparently the integration is in a different order than you'd expect.

    Code (Text):

    NIntegrate[PDF[MultinormalDistribution[{5, 6}, {{1, 1}, {1, 2}}], {x1, x2}], {x, 5, 10}, {x1, x, 10}, {x2, x, 10}]

  8. Jul 4, 2008 #7
    Of course, make sure you load the package with the MultinormalDistribution:
    Code (Text):

    Then turn your code into a function:

    Code (Text):
    f[x_, c_] :=  NIntegrate[PDF[MultinormalDistribution[{5, 6}, {{1, 1}, {1, 2}}], {x1,
        x2}], {x1, x, c}, {x2, x, c}]
    Now you say that you want to evaluate this repeatedly for different values of x, you could use a table:

    Code (Text):
  9. Jul 4, 2008 #8
    Cool, thanks guys! I've used your suggestions and I think I've got it now, just need to test the result a bit more. Anyway, cheers for the help!
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