數值分析

This textbook is intended to introduce advanced undergraduate and early-career graduate students to the field of numerical analysis. This field pertains to the design, analysis, and implementation of algorithms for the approximate solution of mathematical problems that arise in applications spanning science and engineering, and are not practical to solve using analytical techniques such as those taught in courses in calculus, linear algebra or differential equations.

Topics covered include computer arithmetic, error analysis, solution of systems of linear equations, least squares problems, eigenvalue problems, nonlinear equations, optimization, polynomial interpolation and approximation, numerical differentiation and integration, ordinary differential equations, and partial differential equations. For each problem considered, the presentation includes the derivation of solution techniques, analysis of their efficiency, accuracy and robustness, and details of their implementation, illustrated through the Python programming language.

This text is suitable for a year-long sequence in numerical analysis, and can also be used for a one-semester course in numerical linear algebra.

James V Lambers is a Professor in the School of Mathematics and Natural Sciences at The University of Southern Mississippi, and an ACUE Distinguished Teaching Scholar. His research interests span numerical methods for partial differential equations and numerical linear algebra.
Amber C Sumner is an Assistant Professor of Mathematics at William Carey University. She was a PhD student of Lambers at USM. Her research interests include deep learning and numerical linear algebra.
Vivian A Montiforte is a PhD student of Lambers at USM. Her research pertains to spectral methods and meshless methods for partial differential equations.

Chapter 1: What is Numerical Analysis?
Chapter 2: Understanding Error
Chapter 3: Direct Methods for Linear Systems
Chapter 4: Least Squares Problems
Chapter 5: Iterative Methods for Linear Systems
Chapter 6: Eigenvalue Problems
Chapter 7: Polynomial Interpolation
Chapter 8: Approximation of Functions
Chapter 9: Differentiation and Integration
Chapter 10: Zeros of Nonlinear Functions
Chapter 11: Optimization
Chapter 12: Initial Value Problems
Chapter 13: Two-Point Boundary Value Problems
Chapter 14: Partial Differential Equations
Appendices