2025 (Current Year) Faculty Courses School of Computing Undergraduate major in Mathematical and Computing Science
Numerical Analysis
- Academic unit or major
- Undergraduate major in Mathematical and Computing Science
- Instructor(s)
- Kenichiro Tanaka
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MCS.T335
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 4Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This course introduces numerical methods for solving mathematical problems encountered in science and engineering. Many of these problems require concrete solutions, but obtaining exact mathematical solutions is often difficult. Therefore, numerical methods using computers are essential, especially in today's world where data science is increasingly important. This course provides an overview of numerical computation methods and their mathematical foundations.
The primary objective of this course is to understand and explain numerical methods for solving fundamental mathematical problems, including their underlying mathematical principles. Additionally, students will gain hands-on experience executing numerical computations using a computer.
Course description and aims
By completing this course, students will be able to:
(1) Explain numerical methods for fundamental mathematical problems, including their mathematical foundations.
(2) Execute numerical computations using a computer.
Keywords
numerical analysis, numerical computation, algorithms, approximation theory, computers
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This course will cover numerical computation methods and their mathematical principles while introducing a wide range of applications. Several small programming assignments will be given to implement these numerical methods on a computer.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Introduction | Understand examples where numerical computation is necessary. |
| Class 2 | Representation of numbers in Computers | Explain how numbers are represented in computers and associated issues. |
| Class 3 | Solving Linear Systems (1) - Direct Methods | Explain direct methods for solving systems of linear equations numerically. |
| Class 4 | Solving Linear Systems (2) - Iterative Methods | Explain iterative methods for solving systems of linear equations numerically. |
| Class 5 | Eigenvalue Problems (1) - Power Method | Explain the power method for solving eigenvalue problems numerically. |
| Class 6 | Eigenvalue Problems (2) - QR Method and Singular Value Decomposition | Explain the QR method for solving eigenvalue problems numerically and describe the relationship between eigenvalue problems and singular value decomposition. |
| Class 7 | Function Interpolation and Approximation (1) - Lagrange Interpolation, Spline Interpolation | Explain function interpolation and approximation methods, specifically Lagrange interpolation polynomials and spline interpolation. |
| Class 8 | Function Interpolation and Approximation (2) - Advanced Methods | Explain advanced function approximation methods, including neural network-based function approximation. |
| Class 9 | Numerical Integration (1) - Newton-Cotes Formula, Gauss Quadrature | Explain numerical integration methods, particularly Newton-Cotes formulas and Gauss quadrature. |
| Class 10 | Numerical Integration (2) - Advanced Methods | Explain advanced numerical integration methods, particularly variable transformation-based methods. |
| Class 11 | Numerical Methods for Solving Ordinary Differential Equations (1) - Euler Method, Runge-Kutta Methods | Explain numerical methods for solving ordinary differential equations, particularly the Euler method and Runge-Kutta methods. |
| Class 12 | Numerical Methods for Solving Ordinary Differential Equations (2) - Convergence and Stability Analysis | Perform convergence and stability analysis of numerical methods learned in the previous lecture. |
| Class 13 | Optimization Methods (1) - Gradient Descent, Newton's Method, Conjugate Gradient Method | Explain numerical optimization methods, particularly gradient descent, Newton's method, and the conjugate gradient method. |
| Class 14 | Optimization Methods (2) - Relationship with Numerical Methods for Ordinary Differential Equations | Explain the relationship between numerical optimization methods and numerical methods for ordinary differential equations. |
| Class 15 | No Lecture | No lecture scheduled. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class. They should do so by referring to textbooks and other course material.
Textbook(s)
No designated textbook. Lecture materials will be provided as needed, and reference books will be recommended.
Reference books, course materials, etc.
Tetsuro Yamamoto: Introduction to Numerical Analysis [Revised Edition], SAIENSU-SHA (Science Publishing), Tokyo, 2003. (in Japanese)
Norikazu Saito: Introduction to Numerical Analysis, University of Tokyo Press, Tokyo, 2012. (in Japanese)
Evaluation methods and criteria
Small assignments given during the course: 10%
Final exam: 90%
Related courses
- MCS.T302 ： Mathematical Optimization
Prerequisites
Students should have prior knowledge of basic calculus and linear algebra. Knowledge of optimization methods is also recommended.