#### CS3414 Afterclass Notes --- 10 June, 2002

Ordinary Differential Equations (parts of Chapters 8 and 12)
1. Methods for a single IVP (continued)

1. Conditioning and stability

1. Conditioning

• Warning: ``stability'' is often used to refer to what we've called conditioning, i.e., to a property of the IVP problem (as opposed to a property of a numerical method).

• Intuition. Changing the initial condition slightly corresponds to stepping onto a different solution curve, one of a whole family of solution curves. So the question is --- what are the ramifications of stepping onto a different curve? We looked at some pictures (e.g., Figures 8.2 and 8.3 in our text) which give an intuitive answer to this question.

• A more formal answer says that the problem is well-conditioned (i.e., the cost of stepping off the curve is not catastrophic) if fx <= 0, where fx is the partial of f with respect to x.

2. Stability

• Question: do small errors made at each step accumulate catastrophically?

• Very high level summary:
Explicit methods: must keep h small
Implicit methods: often independent of h (``unconditionally'' stable)

• Example: stability analysis for Euler's method.

Can show that

global_errork+1 = magnification_factor * global_errork + local_errork+1

where magnification_factor = (1 + h fx)

From this we can conclude that

• If fx > 0 we are in serious trouble.
• If fx <= 0, then we still need h = -2 / fx

2. A couple of left-over issues

1. Systems of 1st-order IVPs. Most methods for a single IVP generalize easily to systems of equations.

2. Higher-order IVPs. Can transform an ODE

x(m) = f(t, x, x(1), x(1), ... x(m-1))

into a system of m 1st-order equations (see Section 9.2 in our text).