Multi-dimensional Integral by Change of Variables

[junt]

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?

 

[dft5]

In $\sum_{i=1}^{4}((x_{i}-x_{i-1})^2+k_ix_i)$ you have $x_0$ term, so I assume that indices are cyclic and $x_0=x_4$. You have to expand $\sum_{i=1}^{4}(x_{i}-x_{i-1})^2$ and then write it as (Einstein convention used) $A_{ij}x_i x_j$. Notice that as $x_i x_j=x_j x_i$, matrix $\mathbb{A}$ is not unique. Defining $\mathbb{x}=[x_1,x_2,x_3,x_4]^T$ you can write $\sum_{i=1}^{4}(x_{i}-x_{i-1})^2=\mathbb{x}^T \mathbb{A}\mathbb{x}$. Now you have to make diagonalization: $\mathbb{A}=\mathbb{M}^T\mathbb{D}\mathbb{M}$. Your new coordinates that will allow you to write your integral as a product of four one-dimensional integrals are $\mathbb{y}=\mathbb{M}\mathbb{x}$. However you can do it faster as there is general formula for $n$-dimensional gaussian integrals for which you only have to know matrix $\mathbb{A}$ and its determinant. You can find derivation of that formula for example here:
http://www.weylmann.com/gaussian.pdf