Numerical calculation is the basis for numerically analyzing and simulating the real world using computers. In this class, students learn how to make mathematical models of the real world using differential equations, and 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 analytical and numerical solutions of differential equations
・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
Intercultural skills | Communication skills | ✔ Specialist skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
The class style is basically the lecture style. Student are required to promote a better understanding of the lectures by performing report assignments.
Course schedule | Required learning | |
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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 | Analytical solution of ordinary differential equations, Numerical solution of ordinary differential equations (1) | Analytical solution of ordinary differential equations (non-computational solution), 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 | Numerical solution of simultaneous linear equations (1) | Direct methods: Gaussian elimination, LU decomposition |
Class 9 | Numerical solution of simultaneous linear equations (2) | Iteration methods: Jacobian iteration method, Gauss-Seidel method, Successive over-relaxation (SOR) method |
Class 10 | Estimation methods of curves (1) | Interpolation and approximation of functions, Lagrange interpolation, Spline interpolation |
Class 11 | Estimation methods of curves (2) | Least-square method, data fitting |
Class 12 | Analytical and numerical solution of partial differential equations (1) | Basics of partial differential equations, Analytical solution of partial differential equations (non-computational solution) |
Class 13 | Analytical and numerical solution of partial differential equations (2) | Finite-difference method, Gauss-Seidel method, Successive over-relaxation (SOR) method |
Class 14 | Introduction to Monte Carlo method and statistical analysis of experimental data | Numerical calculation using stochastic processes, Statistical analysis of experimental data |
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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.
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