Truncation error
In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process.In reality, we can only use a finite number of these terms as it would take an infinite amount of computational time to make use of all of them.So let's suppose we use only three terms of the series, thenIn this case, the truncation error isExample A: Given the following infinite series, find the truncation error for x = 0.75 if only the first three terms of the series are used.Solution Using only first three terms of the series givesThe sum of an infinite geometrical seriesFor our series, a = 1 and r = 0.75, to giveThe truncation error hence isThe definition of the exact first derivative of the function is given byHowever, if we are calculating the derivative numerically,The error caused by choosingto be finite is a truncation error in the mathematical process of differentiation.Example A: Find the truncation in calculating the first derivative ofusing a step size ofThe truncation error hence isThe definition of the exact integral of a functionbe a function defined on a closed intervalThis implies that we are finding the area under the curve using infinite rectangles.However, if we are calculating the integral numerically, we can only use a finite number of rectangles.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.find the truncation error if a two-segment left-hand Riemann sum is used with equal width of segments.Using two rectangles of equal width to approximate the area (see Figure 2) under the curve, the approximate value of the integralTruncation Error{\displaystyle {\begin{aligned}{\text{Truncation Error}}&={\text{Exact Value}}-{\text{Approximate Value}}\\&=234-135\\&=99.\end{aligned}}}Occasionally, by mistake, round-off error (the consequence of using finite precision floating point numbers on computers), is also called truncation error, especially if the number is rounded by chopping.That is not the correct use of "truncation error"; however calling it truncating a number may be acceptable.Truncation error can causehas a truncation error equal to 1.This truncation error occurs because computers do not store the least significant digits of an extremely large integer.