# Homework Help: Pade Approximation and it's Applications

1. Jun 6, 2006

### LLT

Pade Approximation states that a power series can be written as a rational function. Which is a series divided by another series.
(An easy example of this will be the geometric series with mod'r' < 1)

I've read books about the abstract bit of this. But I am completely stuck when it goes onto applications.

How does pade approximation help solving PDE and ODE? And sometimes, people optain a power series solution for PDE and ODE, (of polynomials of x) how did they do that?

I would also very much like to understand the Navier-Stokes equations.. but this can come later.

2. Jun 8, 2006

### LeBrad

Pade approximations are very useful in the area of model order reduction. If you have a large linear dynamical system, e.g.
$$\frac{d x(t)}{dt} = Ax(t) + bu(t) \;\;\;\; , \;\;\;\;\;y(t) = c^Tx(t)$$
where A is a big matrix, b is a vector, c is a vector, x is a vector of unknowns, and y is the output, you can use this Pade approximation of the system's transfer function to create a low-order transfer function of some different system which matches some number of derivatives of the original system's transfer function.

The end result of this is that if you give me a system of 10,000 ODEs, I can return to you a system of 20 ODEs which are a very good approximation to the original system over some range of frequencies.

For more information, search for pade approximation along with 'model order reduction'.

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