Applied Mathematics |
College | College of Engineering |
Department | Department of Information Science and Engineering |
Year | 2nd grade |
Semester | Spring Semester |
Credit | 2 |
1. | Understanding the least-square method and being able to approximate given sequences of data or functions by linear functions or quadratic functions |
2. | Understanding orthogonal functions and being able to do the orthogonal function expansion for given functions by some given set of orthogonal functions |
3. | Understanding Gram-Schmidt orthogonalisation, which is a method (algorithm) for orthogonalise a set of vectors in an inner product space, and being able to construct an orthogonal set of functions from a given set of functions. |
4. | Being able to do Fourier series expansion, which is an important instance of the orthogonal function expansion. |
Class schedule | HW assignments (Including preparation and review of the class.) | |
1. | Introduction and the least-square method (1) - Approximation of sequences of data in linear functions | Read Section 20.5 of the reference book. |
2. | The least-square method (2) - Approximation of sequences of data in quadratic functions | Example 2 in Section 20.5 of the reference book |
3. | The least-square method (3) - Approximation of sequences of data in linear combination of some fixed set of functions | It is not treated in the reference book. |
4. | The least-square method (4) - Approximation of functions in linear combination of some fixed set of functions | Confer Problem 16 in Section 20.5 of the reference book. |
5. | The least-square method (5) and the orthogonal function expansion(1) - Approximation of column vectors - Approximation of functions in linear combination of some fixed set of orthogonal functions - An orthogonal set of functions --- Legendre polynomials | Read Section 5.7 and 5.8 of the reference book. Confer Example 2 in Section 5.8 for Legendre polynomials. Confer Section 7.1 for column vectors. |
6. | The orthogonal function expansion (2) - An orthogonal set of functions --- Trigonometric functions - The orthogonal function expansion | Read Section 5.8 of the reference book. Confer Appendix 3 for formulae about trigonometric functions. |
7. | The orthogonal function expansion (3) - An example of the orthogonal function expansion --- Fourier series expansion - Orthogonal set of functions with a weight function - An example --- Chebyshev polynomials | Read Section 11.1 for Fourier series expansion. Confer Problem 20 in Section 5.7 for Chebyshev polynomials |
8. | Mid-term examination and explanation of the answers - Pencil-and-paper test for checking the understanding of the contents of the lectures from the first to the eighth | Review the contents of all the lectures until the last one. |
9. | The orghogonal function expansion (4) - Examples --- Hermite polynomials and Laguerre polynomials | Confer Problem 18 in Section 5.8 for Hermite polynomials. Confer Example 2 in Section 5.8 for Legendre polynomials. |
10. | The orthogonal function expansion (5) - The orthogonal function expansion in Chebyshev, Hermite, and Laguerre polynomials - Inner product spaces - An inner product space --- n-dimensional Euclidean space | Read Section 5.7 and 5.8 of the reference book for the orthogonal function expansion. Read Section 7.9 for the inner product spaces. Confer Example 3 in Section 7.9 for the n-dimensional Euclidean space. |
11. | The orthogonal function expansion (6) - Cauchy–Schwarz inequality - Triangle inequality - Orthonormal basis | Read Section 7.9 for Cauchy-Schwarz inequality and Triangle inequality. Read Section 5.7 for the definition of orthonormality Confer Section 7.4 for basis. |
12. | The orthogonal function expansion (7) - Orthogonal projection - Orthogonal basis - Gram-Schmidt orthogonalisation | Read Section 9.2 for projections. Gram-Schmidt orthogonalisation is not treated in the reference book. Consult some linear algebra textbook. |
13. | The orthogonal function expansion (8) - An example of Gram-Schmidt orthogonalisation | Gram-Schmidt orthogonalisation is not treated in the referece book. Consult some linear algebra textbook. |
14. | The orthogonal function expansion (9) - Obtaining Legendre polynomials by Gram-Schmidt orthogonalisation | Gram-Schmidt orthogonalisation is not treated in the reference book. Consult some linear algebra textbook. |
15. | Final examination and explanation of the answers - Paper-and-pencil test for checking the understanding of the contents of the lectures from the first to the fourteenth | Review the contents of all lectures |
・ | Before and after each lecture or any time agreed on by email |
・ | Course that cultivates a basic problem-solving skills |