Explain principles and methods on numerical computation techniques with computational machinery and introduce examples of application to simulations and analyses of systems; including simultaneous linear equations, large scale matrix handlings, methods for solving nonlinear equation,
computation of matrix eigenvalues, numerical integration methods and numerical solution methods for differential equations.
Students will be able to obtain a perspective on numeric computation that is playing a fundamental role in various engineering areas.
By the end of this course, students will be able to:
1) Explain numerical solutions for linear, nonlinear, and differential equations and apply them to new problems.
2) Explain numerical solutions for eigenvalue problems and gradient methods and apply them to new problems.
3) Explain how to formulate numerical errors, and perform error analysis.
4) Explain interpolation and approximation.
linear systems, least squares solutions, error analysis, eigenvalue problems, interpolation and approximation, non-linear equations, numerical integration, ordinary differential equation, partial differential equation
|Intercultural skills||Communication skills||Specialist skills||Critical thinking skills||Practical and/or problem-solving skills|
At the beginning of each class, answers are given for the exercises given in the previous class. At the end of each class, students will work on exercises related to the lecture of that date. Students are expected to prepare for the next class checking the course schedule. Reviewing the lecture is also very important.
|Course schedule||Required learning|
|Class 1||Introduction to numeric computation||Understand basic nature of numerical computation.|
|Class 2||Direct methods for linear systems||Understand direct methods for solving simultaneous linear equations.|
|Class 3||Iterative methods for sparse linear systems||Understand iterative methods for solving simultaneous linear equations.|
|Class 4||Least squares solutions to overdetermined systems||Understand least squares solutions to overdetermined systems.|
|Class 5||Error analysis of linear transformation||Understand error analysis of linear transformation.|
|Class 6||Numerical solution of eigenvalue problems||Understand numerical solution of eigenvalue problems.|
|Class 7||Test level of understanding with exercise problems and summary of the first part of the course - Solve exercise problems covering the contents of classes 1–6.||Test level of understanding and evaluate achievement for classes 1–6.|
|Class 8||Interpolation and approximation||Understand interpolation and function approximation|
|Class 9||Solutions of nonlinear equations (bisection method, Newton's method, secant method)||Understand methods to solve nonlinear equations (bisection method, Newton's method, secant method).|
|Class 10||Solutions of nonlinear equations (solutions of algebraic equations)||Understand solutions of nonlinear equations (solutions of algebraic equations).|
|Class 11||Gradient method||Understand steepest descent method and conjugate gradient method|
|Class 12||Numerical integration (interpolatory integration rules)||Understand interpolatory integration rules|
|Class 13||Numerical integration (periodic function, rapidly decreasing function,double exponential formula)||Understand numeric integration for periodic function and rapidly decreasing function. Understand double exponential formula|
|Class 14||Numerical solution of ordinary differential equation||Understand numerical solution of ordinary differential equation|
|Class 15||Numerical solution of partial differential equation||Understand numerical solution of partial differential equation|
Handouts are distributed.
Numerical Computing with MATLAB, by Cleve Moler, http://jp.mathworks.com/moler/chapters.html
Students' knowledge of numeric solutions to equations, error analysis, numeric integration, optimization and so on that are explained in the lecture, and their ability to apply them to new problems will be assessed.
Final exam 70%, exercises 30%.
Basic concepts of linear algebra and mathematical analysis, computational machinery.