There are several common techniques for error analysis in numerical computations. These techniques help to identify and quantify the errors that arise in numerical calculations. Some of the most common techniques include:

  1. Error Propagation: This technique involves analyzing how errors in the input data propagate through the computational process to affect the final result. Various mathematical methods, such as Taylor series expansion, can be employed to estimate the propagated errors.

  2. Order of Accuracy: Order of accuracy analysis determines the rate at which the errors decrease as the step size or grid size used in numerical methods decreases. It allows us to assess the convergence properties of numerical algorithms and compare their accuracy.

  3. Stability Analysis: Stability analysis investigates whether a numerical algorithm will produce meaningful and convergent results or if it will lead to excessive errors or even divergence. This analysis is particularly relevant for iterative numerical methods and can be performed using various techniques like von Neumann analysis or Lyapunov stability analysis.

  4. Consistency Analysis: Consistency analysis examines how well a numerical method approximates the underlying mathematical problem. It involves comparing the truncation errors in the numerical method with the errors introduced due to the discretization or approximation of the original problem.

  5. Residual Analysis: Residual analysis is often used in the context of iterative numerical methods, such as solving systems of linear equations or optimization problems. It measures the discrepancy between the solution obtained and the original problem by evaluating the residual or the difference between the computed and predicted values.

  6. Sensitivity Analysis: Sensitivity analysis assesses the impact of variations or uncertainties in the input data on the computed result. It involves perturbing the input data within specified ranges and observing how the output changes. Sensitivity analysis helps identify which input variables have the most significant influence on the output and assists in understanding the robustness of the results.

  These techniques, applied individually or in combination, provide a systematic way to analyze errors in numerical computations and allow for the identification and improvement of accuracy in numerical algorithms. However, it is essential to note that error analysis is a complex discipline and often requires expertise to obtain meaningful and reliable results.