Truncation error

In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process. A summation series for is given by an infinite series such as 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, then In this case, the truncation error is Example 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 gives The sum of an infinite geometrical series is given by For our series, a = 1 and r = 0.75, to give The truncation error hence is The definition of the exact first derivative of the function is given by However, if we are calculating the derivative numerically, has to be finite. The error caused by choosing to be finite is a truncation error in the mathematical process of differentiation. Example A: Find the truncation in calculating the first derivative of at using a step size of Solution: The first derivative of is and at , The approximate value is given by The truncation error hence is The definition of the exact integral of a function from to is given as follows. Let be a function defined on a closed interval of the real numbers, , and be a partition of I, where where and . This 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. Example A. For the integral find the truncation error if a two-segment left-hand Riemann sum is used with equal width of segments.