數值方法

臺北科技大學 114-2 學期課程，課號 353237。

課程大綱

先修課程：3002011或負責教授同意。基本觀念及其領域、非線性代數之數值解法、聯立方程式之數值解、多項式內外插、微分、積分、常微分方程、起始與邊界值問題之數值解法。 1. Introduction to the basic concepts of the numerical method 2. Numerical methods of solving nonlinear algebraic equations. 3. Numerical methods of solving systems of linear equations. 4. Numerical interpolation and extrapolation. 5. Numerical differentiation and integration.6. Numerical methods of ordinary differential equations with initial-value problems and boundary value problems.

課程進度

Week 1: Course Introduction and Overview Introduction to Numerical Methods Overview of the course, expectations, and resources Introduction to mathematical modeling and problem-solving with computers Week 2: Foundations of Numerical Computing Introduction to numerical computing languages Basics of floating-point arithmetic and representation on computers Week 3: Mathematical Modeling Techniques Translating real-world problems into mathematical models Introduction to simple numerical algorithms Week 4: Error Analysis Understanding numerical accuracy, stability, and round-off errors Significance and impact of errors in computational methods Week 5: Linear Algebra - Basic Concepts Overview of matrix operations and their computational aspects Introduction to systems of linear equations Week 6: Linear Algebra - Advanced Concepts Direct and iterative methods for solving linear systems Eigenvalue problems and decompositions Week 7: Interpolation and Approximation Polynomial and spline interpolation Approximation theory and practice Week 8: Numerical Integration and Differentiation Techniques and methods for numerical integration Numerical differentiation and its applications Week 9: Solving Ordinary Differential Equations (ODEs) Overview of ODEs and numerical methods for solving them Initial value problems and boundary value problems Week 10: Solving Partial Differential Equations (PDEs) Introduction to PDEs and numerical methods for solving them Finite difference and finite element methods Week 11: Monte Carlo Methods Introduction to Monte Carlo simulations Applications in physical sciences and engineering Week 12: Numerical Optimization Basic concepts in optimization Gradient-based and gradient-free optimization methods Week 13: Advanced Topics in Numerical Linear Algebra Sparse matrix techniques Advanced decomposition methods and their applications Week 14: Numerical Methods in Data Science Numerical techniques in data fitting and regression Introduction to machine learning algorithms Week 15: High-Performance Numerical Computing Understanding parallel computing and its use in numerical methods Introduction to GPUs and distributed computing Week 16: Review and Project Work Comprehensive review of course material Guidance and consultation for final projects Note: Week 16(2026.06.12): Online class and accessment 線上預錄課程(please go to microsoft teams) Week 17: Final Project Presentations Presentation of final projects by students Peer review and feedback sessions Note: Online and remote class please record and present online.(線上課程) Week 18: Course Wrap-up and Examination Final examination covering the entire course content Feedback and course closure Each week will include lectures, readings from the recommended texts, and practical assignments or problem sets to reinforce the concepts taught.

評量方式

Homework/In class assignment 40% Midterm exam 30% Final (oral) report 30%

教材

【遵守智慧財產權觀念，請使用正版教科書，不得使用非法影印教科書】 使用外文原文書：是 1.self-made materials 2.Applied Numerical Methods with MATLAB for Engineers and Scientists 作者 Steven C. Chapra (Author)