Initial Value Problems - Numerical Methods in Engineering with MATLAB - page 296

These two methods, as well as all the methods describedinin this topic, belong
to agroup known as single-step methods . The name stemsfrom the fact that the
informationat a single pointon the solution curve issufficienttocompute the next
point. There are also multistep methods that utilize several points on the curveto
extrapolate the solutionat the next step. These methods were popular once, buthave
lost someoftheir lusterinthe last fewyears. Multistepmethodshavetwo shortcomings
thatcomplicate their implementation:
The methods are not self-starting, but must be providedwith the solutionat the
first fewpoints by a single-stepmethod.
The integration formulas assumeequally spaced steps, which makes it makes it
difficult to change the step size.

Both of these hurdles canbeovercome, but the price iscomplexity of the algorithm
that increases with sophistication of the method. The benefits of multistepmethods
are minimal—the best of them can outperform their single-step counterparts in cer-
tain problems, but these occasions are rare. MATLAB provides one general-purpose
multistepmethod:

[xSol,ySol] = ode113(dEqs,[xStart xStop],yStart) uses the variable-

order Adams-Bashforth-Moultonmethod.

MATLAB has also severalfunctionsfor solving stiff problems. These are ode15s
(this is the first method to try whenastiff problemisencountered), ode23s , ode23t
and ode23tb .