Introduction to Numerical Analysis (MAGIC015)
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Autumn 2009 (Monday, October 5 to Friday, December 11)
Undergraduate analysis and linear algebra.
This is a 20 lecture course. The aim of the course is to introduce students to a number of key ideas and methods in numerical analysis and for the students to learn to implement algorithms in Matlab. Syllabus Lecture 1: Introduction and prerequisites. Description of the ideas to be covered and the assessment activities. Lecture 2: Stable and unstable computation, relative and absolute error, floating point computation and round off errors. Lecture 3: Finding roots of nonlinear equations. Bisection, secant and Newton's methods. Lecture 4: Approximation of functions I. Polynomial interpolation, Lagrange and Newton forms: divided differences. Lecture 5: Approximation of function II. Piecewise polynomial approximation. Splines and their generalisations into higher dimensions. Lecture 6: Approximation of functions III. Least squares and orthogonal polynomials. Lecture 7: Numerical integration. Newton-Cotes and Gauss formulae. Integration of periodic functions. Romburg integration. Lecture 8: The Fast Fourier transform. Lecture 9: Wavelets I. Lecture 10: Wavelets II. Lecture 11: Solving systems of linear equations I. Gauss elimination, pivoting. Cholesky factorisation. Lecture 12: Solving systems of linear equations II. Conditioning and error analysis. Lecture 13: Solving systems of linear equations II. Iterative methods: Jacobi, Gauss-Seidel, SOR. Lecture 14: Least squares solution, Schur decomposition, the QR and QZ algorithms. Lecture 15: Power method and singluar value decomposition. Lecture 16: Krylov subspace methods: Arnoldi algorithm. Lecture 17: Conjugate gradient method and GMres. Lecture 18: Functions of a matrix. Lecture 19: This lecture will be set aside for expansion of topics in the course previously. Lecture 20: Summarising and finishing course. This lecture also allows some time if other topics take longer than expected.
Reading list and references
There are a number of excellent books on numerical analysis and you are encouraged to consult these books for alternative and often better accounts of what you have heard in lectures. In the main I have followed Kincaid and Cheney  and Higham .
Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will see links to places where you can buy the book. There is also link marked 'Find this book in a library'. This sometimes works well, but not always. (You will need to enter your location, but it will be saved after you do that for the first time.)
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No assignments have been set for this course.
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