Numerical Evaluation of Problematic Situations: Error Analysis and Solution Methods
Numerical Evaluation of Problematic Situations
Error Analysis
In numerical evaluation, errors can arise due to:
- Rounding error: Limited precision of computers leads to data being represented with a finite number of decimal places, causing arithmetic operations to produce errors.
- Truncation error: Infinite processes are replaced by finite processes, leading to errors.
- Inherent error: Approximation of reality by mathematical models and use of error-affected data.
- Modeling or programming error: Errors in the algorithm.
- Error propagation: Errors spread through arithmetic operations, with subtraction producing the greatest rounding error.
Theory of Error
The theory of error provides criteria for determining the correctness of results.
Iterative Techniques for Rounding Error
Iterative techniques can be used to reduce rounding error, but only in the last iteration to avoid chaining operations and ensure true input values.
Bairtow Method
The Bairtow method approximates complex roots by decomposing the polynomial into quadratic factors, allowing it to work with real arithmetic.
Bisection Method
The bisection method ensures that the calculated error is greater than the round error, making it suitable for approximating roots of nonlinear equations.
Methods for Solving Linear Algebraic Equations
Finite Methods
- Gauss Jordan
- Crouttes
- Gaussian Elimination
Infinite Methods
- Jacobi
- Gauss Seidel
Infinite methods are suitable for large or ill-conditioned systems because they can continue until the error converges within a specified tolerance, controlling the desired error level.
Methods of Approximation for Nonlinear Algebraic Equations
Open Methods
- Secant method
- Von Misses method
- Newton-Raphson method
- Method of fixed point iteration
Open methods may not always converge, but when they do, they converge faster.
Closed Methods
- Bisection rule
- False position method
Closed methods always converge.
Convergence Speed
The order of a method determines its convergence speed. Higher-order methods have a faster convergence rate.
Finite and Infinite Methods
Infinite methods are iterative and do not propagate error, making them suitable for ill-conditioned systems. Finite methods are used for smaller systems and require more significant figures.
Difference between Bairtow and Newton-Raphson
| Bairtow | Newton-Raphson |
|---|---|
| Finds roots using quadratic factors | Finds roots directly by factors |
| Works with real arithmetic | Works with complex arithmetic |
| More stable | Less stable |
| Requires two applications of Ruffini | Requires one application of Ruffini |