What is the Saddle Point Method and Its Relation to Asymptotic Expansions?

[Jaggis]

Hi!

Could someone please explain the saddle point method i.e. the method of deepest descends to me in layman's terms? All I need is a rough idea of what it is. I've tried to read explanation from various sources but perhaps I have a hard time following the examples due to their mathematical complexity.

If you could also tell me what the saddle point method has to do with asymptotic expansions, again as simply and roughly as possible, I'd be grateful.

Thanks in advance.

 

[Mandelbroth]

Hi!

Could someone please explain the saddle point method i.e. the method of deepest descends to me in layman's terms? All I need is a rough idea of what it is. I've tried to read explanation from various sources but perhaps I have a hard time following the examples due to their mathematical complexity.

If you could also tell me what the saddle point method has to do with asymptotic expansions, again as simply and roughly as possible, I'd be grateful.

Thanks in advance.

Are you familiar with Laplace's method? If so, an explanation is a little easier.

 

[Jaggis]

Are you familiar with Laplace's method?

I'm afraid not.

 

[Mandelbroth]

I'm afraid not.

Suppose we have an integral of the form $$\int_a^be^{nf(x)}~dx$$ where $f$ is a twice differentiable function and $n$ is large. Suppose $f(x_0)$ is the unique global maximum of $f$. Then, by definition of global maximum, $f(x_0)>f(x)$ for all $x\neq x_0$. Significant contributions to the value of the integral will then be from some neighborhood of $x_0$. This is the basic idea of Laplace's method.

The saddle-point method basically deforms a contour of integration to one in which Laplace's method can be used. There's a little more to it, but you asked for a rough explanation, so I'll spare you the details.