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The local truncation error is the error that our increment function, , causes during a single iteration, assuming perfect knowledge of the true solution at the previous iteration. Unlike truncation error which only indicates that error is due to taking small step sizes, local truncation error give us error in terms of complexity theory The local truncation error (defined as the error made in one step) of the backward euler method is , using the big o notation

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The error at a specific time is. These can be derived from the definition of the truncation error itself. The local truncation error of the euler method is the error made in a single step

It is the difference between the numerical solution after one step, , and the exact solution at time.

The error caused by choosing a finite number of rectangles as opposed to an infinite number of them is a truncation error in the mathematical process of integration. An expression of general interest is the local truncation error of a method That is, it is the quantity if refers to the exact value and to the numerical approximation. There are also accompanying requirements if one requires the method to have a certain order p, meaning that the local truncation error is o (hp+1)