Backward error

Definition

Backward error refers to the difference between the exact solution of a problem and the approximate solution derived from numerical computations, typically measuring how much the input data would need to be perturbed to achieve the computed output. It helps in understanding the stability and accuracy of numerical algorithms, revealing how errors propagate through calculations. This concept is particularly relevant when analyzing methods for solving linear systems, eigenvalue problems, and in assessing the reliability of numerical approximations.

5 Must Know Facts For Your Next Test

1. Backward error gives insight into how small changes in input can affect the output, allowing for a deeper understanding of algorithm performance.
2. In LU factorization, analyzing backward error helps determine how accurately we can solve linear systems based on matrix properties.
3. The backward error framework aids in assessing convergence rates for iterative methods like power and inverse power methods.
4. By comparing backward error with forward error, we can better evaluate the effectiveness of different numerical methods and their impact on solutions.
5. Backward error analysis is crucial for determining whether an approximate solution is sufficiently close to the true solution to be considered valid.

Review Questions

• How does backward error help evaluate the performance of numerical algorithms?
  • Backward error analysis is essential for evaluating numerical algorithms because it quantifies how much the input data would need to change for the computed solution to be exact. This provides insights into the stability and robustness of an algorithm under small perturbations. By analyzing backward errors, we can determine if the algorithm is suitable for practical applications where precision is critical.
• Discuss the relationship between backward error and conditioning when solving linear systems.
  • The relationship between backward error and conditioning is fundamental when solving linear systems. If a system is well-conditioned, small perturbations in the input data will lead to small changes in the solution, resulting in minimal backward error. Conversely, poorly conditioned systems may exhibit significant backward error even with minor changes in input, indicating that solutions obtained may not be reliable. Thus, understanding conditioning helps anticipate potential issues in computational accuracy.
• Evaluate how backward error analysis can influence the choice of numerical methods for eigenvalue problems.
  • Backward error analysis can significantly influence the choice of numerical methods for eigenvalue problems by highlighting which algorithms maintain accuracy under varying conditions. For example, certain iterative methods may exhibit better backward error characteristics compared to direct methods depending on the nature of the matrix involved. By evaluating these errors, one can select algorithms that not only yield accurate eigenvalues but also demonstrate stability against input perturbations, ultimately leading to more reliable computations in practice.

Related terms

forward error:

Forward error measures the difference between the computed solution and the exact solution of a problem, directly assessing the accuracy of the result.

conditioning: Conditioning refers to how sensitive a function or problem is to changes in input data; well-conditioned problems have small backward errors.

numerical stability: Numerical stability describes how errors are controlled during computations, ensuring that small perturbations in input do not lead to large errors in output.