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Discrete Gronwall inequality

If nonnegative quantities satisfy

e_{n + 1} \leq (1 + h L) e_{n} + h b_{n},

then iteration gives

e_{n} \leq (1 + h L)^{n} e_{0} + h \sum_{j = 0}^{n - 1} (1 + h L)^{n - 1 - j} b_{j} .

For

nh \leq T

, the growth factor is bounded by

e^{L T}

.

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