# Lecture 12 (Round Off Error Stability and ODEs)

======= Today we will be talking about round-effor error stability and other quadrature rules.

## Round off Error Stability

Suppose that we make an error while evaluating \(f(x_k)\) on \(k = 0,…,n\).

In this case, \(\tilde{f}(x_k)\) is our approximation, and \(e_k\) is our error. Let’s say:

Hopefully you followed all of that. We are now going to make one more logical leap, which is that \(|e_k| \leq \epsilon \). Once we’ve accepted this, you can see that the following holds:

Epsilon at this point is just a constant, so we can simply multiply it \(\frac{n}{2}\) times:

Now recall that \(h=\frac{b-a}{n}\), so if we replace the h, and cancel out the twos, we get:

Our final result is that our error does not depend on \(h\), so it is is Stable!

## Initial Value Problems for ODEs

Well-posed Initial Value Problems are of the following form:

### example

This is a really terrible problem to solve, so who cares about the exact solution. Let’s just figure out a numerical solution!
Our first questions to ask before doing any solving:

- Does the IVP have a solution? (Existence)
- Is this solution uniqueness (Uniqueness)
- Is the solution STABLE? (Stability) We will talk about stability in many different ways, so be careful about which test for stability you are applying in the future. Note: Stability implies that a small perturbation of the initial condtion and/or equation only imply small perturbations of the solution.