Numerical Analysis
A comprehensive textbook on the theory and application of numerical approximation techniques. It covers mathematical preliminaries, error analysis, solutions of equations, interpolation, and numerical solutions to differential equations.
Course Overview
Content Summary
A comprehensive textbook on the theory and application of numerical approximation techniques. It covers mathematical preliminaries, error analysis, solutions of equations, interpolation, and numerical solutions to differential equations.
Master the art and science of modern numerical approximation techniques.
Author: Richard L. Burden, J. Douglas Faires
Acknowledgments: Supported by Youngstown State University and contributors including John Carroll (Dublin City University) and various student assistants like Mario Sracic.
Learning Objectives
- Apply the Intermediate Value Theorem and Rolle’s Theorem to prove the existence and uniqueness of solutions.
- Construct Taylor polynomials and use their remainder terms to establish rigorous error bounds for numerical approximations.
- Differentiate between rounding and chopping arithmetic and calculate absolute and relative errors in floating-point systems.
- Apply the Bisection, Fixed-Point, Newton, Secant, and False Position methods to approximate roots.
- Analyze the order of convergence and error bounds for various iterative methods.
- Utilize Aitken’s \Delta^2 and Steffensen’s methods to accelerate the convergence of linear sequences.
- State and explain the Weierstrass Approximation Theorem and its implications for function approximation.
- Construct Lagrange, Newton’s Divided-Difference, and Hermite Interpolating Polynomials for given data sets.
- Apply Neville’s Method to iteratively generate polynomial approximations.
- Derivation and application of numerical differentiation formulas (Three-Point) and error estimation.