Lecture plan

Will be updated throughout the semester.

Week Themes Curriculum Notes
2 Introduction to MATLAB, absolute and relative error, Taylor series, O(h) notation, floating point numbers, computer artithmetic, and roundoff errors. Ch. 1.1, 1.2, 1.3
3 Loss of significance (cancellation errors), linear systems, Gaussian elimination, Matrix and vector norms, ill-conditioned systems Ch. 1.4, 2.1, 8.4 (part about norms and the condition number)
4 Row exchanges and scaled partial pivoting in Gaussian elimination, numerical complexity analysis, Gaussian elimination for tridiagonal linear systems, matrix factorizations, connection between LU factorization and Gaussian elimination Ch. 2.2, 2.3 (not pentadiagonal systems), 8.1
5 LDLT factorization for symmetric matrices, Cholesky factorization. Non-linear equations: bisection method. Ch. 8.1, 3.1
6 Non-linear equations: Method of false position, fixed-point iterations, Newton's method. Ch 3.1, 3.2 . note
7 Newton's method for nonlinear systems, the secant method. Eigenvalues. Ch. 3.2, 3.3, 8.2 (up to SVD)
8 Power method (including inverse and shifted power methods), iterative solvers for linear systems (meaning: (damped) Richardson, Jacobi, Gauss-Seidel, and SOR), a short discussion on their convergence properties. Polynomial interpolation (Lagrangian interpolation) Ch. 8.3, 8.4 (not conjugate gradient), 4.1
9 Newton interpolation and divided differences, Project work Ch. 4.1
10 Interpolation errors, minimization of interpolation errors: Chebyshev points, estimation of derivatives, Richardson extrapolation. Ch. 4.2, 4.3
11 Numerical integration: Trapezoid rule, Simpson rule, Newton-Cotes rules, approximation errors and convergence rates. Ch. 5.1, 5.2, 5.3.
12 Numerical integration: Romberg integration, adaptive quadrature, Gaussian quadrature & Legendre polynomials, singularities. Ch. 5.3, 5.4.
13 Splines: Linear splines, natural cubic splines. Ch. 6.1, 6.2. (Not specific material on quadratic splines).
15 Least squares, linear regression, and normal equations. Initial value problems: Euler's method, Runge-Kutta methods, Ch. 9.1, 9.3, 7.1, 7.2
16 Initial value problems: Multi-step methods, implicit methods and stiff ODEs. Ch. 7.3, 7.4, 7.5

There will also be a double lecture with repetition of desired topics and questions (we might also go through some relevant former exam problems), closer to the final exam.

2016-04-25, Asif Mushtaq