Theorem 22 (Round-off error)

By decreasing the step size, one needs more calculations to reach a given value of , and hence the effects of round-off error become important. The effect of round-off error is estimated in the following theorem, given without proof:

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Assume is continuous and satisfies a Lipschitz Condition in the variable on the domain

.

Assume a constant exists such that

for all in ,

and let be the unique solution to the initial value problem

,

and call the successive approximations generated by Euler's method for some positive integer , for the perturbed problem

,

with and .

Then for each , we have

.

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From this result, it is clear that the total error will increase for sufficiently small values of .

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