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I Multi-dimensional Integral by Change of Variables

  1. Feb 12, 2017 #1
    Hi All,

    $$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$
    I can perform the integration of the integral above easily by changing the variable
    $$u=x_2+x_1\\
    v=x_2-x_1$$
    Of course first computing the Jacobian, and integrating over ##u## and ##v##

    I am wondering how you perform the change of variable for 4-dimensional integral like below:

    $$\int{\exp(\sum_{i=1}^{4}((x_{i}-x_{i-1})^2+k_ix_i))dx_1dx_2dx_3dx_4}$$

    Is it something like:
    $$x_2-x_1=u \\
    x_2+x_1=v \\
    x_4-x_3=p \\
    x_4+x_3=q$$
    Should this be enough? I think one only needs 4 new variables right? Because I was thinking the integral would be easier if I could do something like:
    $$x_2-x_1=u \\
    x_3-x_2=v \\
    x_4-x_3=p \\
    x_1-x_4=q$$
    Or in general, how do you perform change of variable in multi-dimensional case. Should you generally perform it by doing:
    $$u=a_1x_1+a_2x_2+a_3x_3+a_4x_4 \\
    v=b_1x_1+b_2x_2+b_3x_3+b_4x_4 \\
    p=c_1x_1+c_2x_2+c_3x_3+c_4x_4 \\
    q=d_1x_1+d_2x_2+d_3x_3+d_4x_4$$

    And what kind of change of variable does one need to perform to perform the integration of such integration easily?
     
    Last edited by a moderator: Feb 13, 2017
  2. jcsd
  3. Feb 19, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
  4. Feb 20, 2017 #3
    In [itex]\sum_{i=1}^{4}((x_{i}-x_{i-1})^2+k_ix_i) [/itex] you have [itex]x_0[/itex] term, so I assume that indices are cyclic and [itex]x_0=x_4[/itex]. You have to expand [itex]\sum_{i=1}^{4}(x_{i}-x_{i-1})^2 [/itex] and then write it as (Einstein convention used) [itex]A_{ij}x_i x_j[/itex]. Notice that as [itex]x_i x_j=x_j x_i[/itex], matrix [itex]\mathbb{A}[/itex] is not unique. Defining [itex]\mathbb{x}=[x_1,x_2,x_3,x_4]^T[/itex] you can write [itex]\sum_{i=1}^{4}(x_{i}-x_{i-1})^2=\mathbb{x}^T \mathbb{A}\mathbb{x}[/itex]. Now you have to make diagonalization: [itex]\mathbb{A}=\mathbb{M}^T\mathbb{D}\mathbb{M}[/itex]. Your new coordinates that will allow you to write your integral as a product of four one-dimensional integrals are [itex]\mathbb{y}=\mathbb{M}\mathbb{x}[/itex]. However you can do it faster as there is general formula for [itex]n[/itex]-dimensional gaussian integrals for which you only have to know matrix [itex]\mathbb{A}[/itex] and its determinant. You can find derivation of that formula for example here:
    http://www.weylmann.com/gaussian.pdf
     
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