- Methods for a single IVP (continued)
- Conditioning and stability
- 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 f
_{x}<= 0, where f_{x}is the partial of f with respect to x.

- 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).
- 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_error*=_{k+1}*magnification_factor***global_error*+_{k}*local_error*_{k+1}*magnification_factor*= (1 +*h f*_{x})From this we can conclude that

- If
*f*> 0 we are in serious trouble._{x} - If
*f*<= 0, then we still need_{x}*h*= -2 /*f*_{x}

- If

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

- Conditioning
- A couple of left-over issues
- Systems of 1st-order IVPs. Most methods for a single IVP
generalize easily to systems of equations.
- 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).

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

- Conditioning and stability