Numerical calculation is the basis for numerically analyzing and simulating the real world using computers. In this class, we learn the essential knowledge of numerical calculation and some famous methods and algorithms in order to apply them to the real world analysis.
・Learn how to model the real world and how to numerically analyze the model using computers
・Learn the important topics when you perform numerical analysis (e.g., errors, loss of digits)
・Learn numerical solution of simultaneous linear equations
・Learn numerical solution of nonlinear equations
・Learn numerical differentiation and numerical integral, and apply them to numerical solution of ordinary/partial differential equations
・Learn interpolation and data fitting based on least-square method
Simultaneous linear equations, Ordinary differential equations, Partial differential equations, Numerical integral, Nonlinear equations, Interpolation, Least-square method, Monte-Carlo method, Errors, Dynamical systems, System modeling
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||✔ Practical and/or problem-solving skills|
Mainly, lectures. Sometimes, practice for promoting a better understanding of the lectures.
|Course schedule||Required learning|
|Class 1||Introduction to numerical calculation methods (1)||Modeling of the real world and numerical analysis, Units and dimensions, Expression of the real number (floating number), Types of errors|
|Class 2||Introduction to numerical calculation methods (2)||Analysis of errors, Loss of significant digits, Amount of calculation, Step size， Numerical analysis libraries and applications|
|Class 3||Numerical differentiation and numerical integral||Difference, Trapezoidal rule, Simpson's rule|
|Class 4||Numerical solution of nonlinear equations||Bisection method, Newton method|
|Class 5||Numerical solution of ordinary differential equations (1)||Initial value problem of ordinary differential equations, Explicit methods, Euler method, Runge-Kutta method|
|Class 6||Numerical solution of ordinary differential equations (2)||Stiff equations, Implicit methods, Boundary value problem of ordinary differential equations|
|Class 7||Numerical solution of ordinary differential equations (3)||Second-order ordinary differential equations, Modeling of dynamical systems, Example of dynamical system (damped oscillation, van der Pol equation)|
|Class 8||Practice (1)||Practice of classes 1-7|
|Class 9||Numerical solution of simultaneous linear equations (1)||Direct methods: Gaussian elimination, LU decomposition|
|Class 10||Numerical solution of simultaneous linear equations (2)||Iteration methods: Jacobian iteration method, Gauss-Seidel method, Successive over-relaxation (SOR) method|
|Class 11||Estimation methods of curves (1)||Interpolation and approximation of functions, Lagrange interpolation, Spline interpolation|
|Class 12||Estimation methods of curves (1)||Least-square method, data fitting|
|Class 13||Numerical solution of partial differential equations||Finite-difference method, Gauss-Seidel method, Successive over-relaxation (SOR) method|
|Class 14||Introduction to Monte Carlo method||Numerical calculation using stochastic processes|
|Class 15||Practice (2)||Practice of classes 10-14|
Lecture slides (Japanese) will be uploaded on OCW-i.
Reference books: 数値計算（高橋大輔，岩波書店），数値計算の常識（伊理正夫・藤野和建，共立出版），Numerical Recipes in C (W. H. Press et al., Cambridge University Press).
It will be explained in the first class.