# Seeking a precedent – two-stage Gaussian integration?

Sometimes, by iteration, linear algebra can be used to solve non-linear equations. For example, consider the system $$Ax=a \qquad B(x)y=b(x),$$ where $$a$$ is a vector with scalar entries, $$A$$ is a matrix with scalar entries, $$b(x)$$ is a vector whose entries are functions of $$x$$, and $$B(x)$$ is a matrix whose entries are functions of $$x$$. This system can be solved by first solving $$Ax=b$$, then substituting the solution into the second equation $$By=b$$, and then solving the second equation. The system can also be solved by first solving $$By=b$$ over the ring of functions of $$x$$, and then solving the first equation.

Similarly, formal$$^*$$ Gaussian integration techniques can sometimes be used iteratively to compute the exact integrals of non-Gaussian integrands. Here's a 3D example in the variables $$a,b,x$$; it is easy to raise this example to higher dimensions by replacing scalars with vectors and matrices.

Let $$L=\lambda ab+\frac12 q(b)x^2+\alpha a+\beta b+\xi x$$, where all the letters represent scalars except for $$q(b)$$ which is a function of $$b$$. We wish to compute $$I := \int e^Lda\,db\,dx$$. This is not a Gaussian integral because the $$q(b)x^2$$ term is not quadratic in the integration variables.

Yet first computing the $$ab$$ integral we get $$I(x) := \int e^Lda\,db = e^{\xi x}e^{q(\partial_\beta)x^2/2}\int e^{\lambda ab+\alpha a+\beta b}da\,db$$ $$= \frac{2\pi}{\lambda}e^{\xi x}e^{q(\partial_\beta)x^2/2}e^{-\alpha\beta/\lambda} = \frac{2\pi}{\lambda}e^{-\alpha\beta/\lambda + \xi x + q(-\alpha/\lambda)x^2/2}.$$

Thus $$I(x)$$ is a Gaussian with respect to $$x$$, so we can (formally) compute $$I = \int I(x)dx = \frac{(2\pi)^{3/2}}{\lambda\sqrt{q(-\alpha/\lambda)}} e^{-\alpha\beta/\lambda - q(-\alpha/\lambda)^{-1}\xi^2/2}.$$

We could have arrived at the same result by first computing the $$x$$ integral as a formal Gaussian over the ring of functions of $$b$$ and then computing the $$ab$$ integral.

Question. Is there a precedent for this procedure? A name? Is there a place where people routinely iterate Gaussian integration to integrate non-Gaussians?

$$^*$$Meaning, applying standard formulas without worrying about convergence. Add conditions if you must, or think that we're really imitating some QFT-like context in which convergence is not an issue.

• I did something similar in this old paper of mine mat.univie.ac.at/~slc/s/s49abdess.html Jun 22, 2021 at 14:57
• Nice paper! And closely relevant to work I'm doing with Roland van der Veen (see drorbn.net/la19 and the in-progress drorbn.net/dpg/DPG.pdf; I'll cite your paper within Comment 2.9). But I could find no "two-stage" stuff in your paper. If it's there, can you be specific about where? Jun 22, 2021 at 15:32
• The closest would be section II.1 of my paper where I iterate the use of formal Gaussian integration to get a formula of the composition of two functions. That part is heuristic but then in is made rigorous at the level of formal power series in section III.1. As I said I think it is similar or in the same spirit of you are considering, but I wouldn't claim it is exactly the kind of computation you described. BTW, please say Hi to Roland for me. Jun 22, 2021 at 15:43
• Also perhaps of interest to you is the so called intermediate field formalism (a.k.a Hubbard-Stratonovich transformation in the physics literature). A paper where this iterated is arxiv.org/abs/1601.02805 There the goal is not exact computation but rather proving higher (Borel Leroy) summability. Jun 22, 2021 at 15:47
• Can't edit - the earlier links are fixed here: drorbn.net/la19 and drorbn.net/dpg/DPG.pdf. Jun 22, 2021 at 15:57