Local error

Definition

Local error refers to the error that occurs in a numerical method at a single step of the computation, measuring how far the approximation deviates from the exact value at that specific point. It highlights how well a numerical method can reproduce the true solution locally, and understanding it is essential for assessing the overall accuracy of algorithms. Local error is particularly relevant when analyzing iterative methods or approximations, as it helps determine convergence and stability.

5 Must Know Facts For Your Next Test

1. Local error is often related to the step size used in numerical approximations, where smaller step sizes typically reduce local error.
2. In finite difference methods, local error can arise from approximating derivatives using differences between function values at discrete points.
3. The order of local error indicates how quickly it decreases as the step size approaches zero; higher-order methods usually have smaller local errors.
4. In integration methods like Trapezoidal and Simpson's Rules, local error provides insight into how accurately each segment of the integral contributes to the total approximation.
5. Local error is crucial for iterative methods like Euler's Method because it determines how accurately each new approximation builds upon the previous one.

Review Questions

• How does local error impact the accuracy of finite difference approximations?
  • Local error plays a significant role in finite difference approximations since it measures how accurately a numerical derivative represents the actual derivative at a specific point. By understanding local error, one can assess the effect of discretization on derivative estimates. For example, a smaller step size reduces local error, leading to more accurate results, which is critical when solving differential equations numerically.
• Discuss how local error is evaluated in methods like Trapezoidal and Simpson's Rules and its implications on convergence.
  • In methods like Trapezoidal and Simpson's Rules, local error is evaluated based on how well these rules approximate the area under a curve over small segments. The local error can be derived from Taylor series expansions, revealing that it depends on higher-order derivatives of the function being integrated. This evaluation not only indicates the accuracy of each segment but also influences overall convergence; smaller local errors lead to better global accuracy as more segments are added.
• Evaluate the relationship between local error and stability in Euler's Method and how this affects long-term approximations.
  • Local error in Euler's Method directly affects its stability and reliability for long-term approximations of differential equations. If the local error remains small with appropriately chosen step sizes, the method tends to produce stable results. However, if local errors accumulate due to instability or inappropriate step sizes, they can lead to significant divergence from the true solution over time. This highlights the need for careful consideration of both local error and stability to ensure accurate long-term predictions in dynamic systems.

Related terms

global error:

The cumulative error across all steps of a numerical method, reflecting the total deviation from the exact solution over an entire interval.

convergence: The property of a numerical method where the results approach the exact solution as the number of steps increases or as step size decreases.

truncation error:

The error introduced when a mathematical procedure is approximated by truncating an infinite series or by neglecting higher-order terms in calculations.