104 Ordinary Differential Equations
per step; (3) the maximum number of iterations, K , to be taken per step; and
( 4) what to do if K is reached before the error requirement E or E is satisfied.
And, of course, if the predictor-corrector algorithm being utilized involves
a multistep formula , one must obtain starting values using some single-step
method. In selecting the stepsize, the maximum iteration error per step, and
the maximum number of iterations per step, one must keep in m ind that these
are not independent but are related through the algorithm local error formula
which in turn depends upon the differential equation. Usually, the maximum
iteration error desired per step is chosen, the maximum number of iterations
per step is set at a small number- often two or three, and the stepsize is then
determined so that it is consistent with the maximum iteration error, the
maximum number of iterations per step, the algorithm, and the differential
equation.
Using the predictor-corrector formulas (24), (25), and (26), we generated
numerical approximations to the solution of the IVP (7) y' = y + x; y(O) = 1
on the interval [O, 1] using a constant stepsize h = .1. Where n ecessary (for
equations (25) and (26)), we used the exact solution values as the starting
values. We set the maximum absolute iteration error, E , equal to 5 x 10-^6
and recorded the number of iterations per step , k, required to achieve this
accuracy. The results of our calculations are shown in Table 2.8. Observe
that the single-step predictor-corrector formula (24) is not able to maintain
accuracy with four iterations per step while the multistep predictor-corrector
formulas (25) and (26) a re able to do so with only two iterations per step.
Table 2.8 Predictor-corrector approximations to the solution of the IVP
(7) y' = y + x; y(O) = 1 on [O, 1] with stepsize h = .1Equations (24) Equations (25) Equations (26) Exact
Xn Yn k Yn k Yn k solution.0 1.00000
.1 1.11053 4 1.11034
.2 1.24321 4 1.24281
.3 1.40039 4 1.39972.4 1.58464 (^4) 1.58365 2 1.58365 2 1.58365
.5 1.79882 4 1.7974 (^4 2) 1.79744 2 1.79744
.6 2.04606 4 2.04424 2 2.04424 2 2.04424
.7 2.32985 4 2.32751 2 2.32751 (^2) 2.32751
.8 2.65405 4 2.65108 2 2.65108 (^2) 2.65108
.9 3.02290 4 3.01921 2 3 .01921 (^2) 3.01921
1.0 3.44109 4 3.43656 (^2) 3.43657 (^2) 3.43656