Common sources of error in numerical computations can be categorized into two main types: round-off error and truncation error.

  1. Round-off Error: This type of error occurs due to the limitations of representing real numbers with finite precision in computers. Since most calculations involve a large number of floating-point operations, the accumulation of round-off errors can lead to significant discrepancies between the computed values and the actual values.

  2. Truncation Error: This error arises from approximating mathematical processes or functions by using finite methods, such as numerical integration or differentiation. Truncation error occurs when an infinite process is replaced by a finite one, resulting in an incomplete or truncated representation. The error is proportional to the step size used in the approximation and decreases as the step size decreases.

  Other common sources of errors in numerical computations include:

  3. Input Data Error: Errors can arise from inaccurate or imprecise input data. This can be due to measurement errors, data corruption during storage or transmission, rounding errors in manually entered data, or errors in data preprocessing.

  4. Algorithmic Error: Mistakes or inconsistencies in the formulation or implementation of algorithms can introduce errors. These errors can stem from incorrect assumptions, approximation methods, or faulty logic in the algorithm design.

  5. Convergence Error: Numerical methods that rely on iterative procedures, such as root finding or optimization algorithms, may not achieve exact convergence due to chosen termination criteria or convergence thresholds. The obtained solutions may only be approximate.

  6. Programming Error: Errors in coding or programming can lead to incorrect calculations. These errors can include syntax errors, logic errors, or data handling mistakes in the implementation of numerical algorithms.

  To mitigate these errors, it is important to use appropriate numerical methods, select algorithms with desired properties, validate input data, implement robust error-handling mechanisms, and thoroughly test the code. Additionally, using higher precision arithmetic or implementing improved error analysis techniques can help reduce the impact of round-off and truncation errors.