Lecture plan

This schedule is tentative, changes will appear.

Textbook: E. Süli and D. Mayers, An introduction to Numerical Analysis, Cambridge University Press (2003).

Week	Topics	Lecturer	Reading	Exercises
34	Introduction to the course. Taylor's theorem, big O-notation, rounding errors	TK	Introduction
	Numerical solution of nonlinear equations Existence of solutions Simple iterations Fixed point theorem Rate of convergence Contraction mapping theorem	TK	1.1-1.2 in S&M (up to Theorem 1.4) Scalar equations: Convergence of Newton's method (Theorem 1.8) and max number of fixed point iterations (Theorem 1.4) with proofs.
35	Rate of convergence, asymptotic error constant, order of convergence Convergence of fixed point iterations. Bisection and Newton's method, Convergence analysis Secant method	TK	1.2 (from Theorem 1.4) - 1.8 in S&M Slides: Solving non-linear equations	Exercise 2 is posted here.
	Multi-variate Taylor's expension. Newtons method for system of equations. Convergence of Newtons method (Theorem 4.4) (self study). Fixed point iterations for systems of equations. The contraction mapping theorem in max-norm (Theorem 4.1 and 4.2).	TK	Notes: - Solution of systems of nonlinear equations - Newton's method for systems of non-linear equations - Examples of nonlinear iteration
36	Numerical linear algebra Naiv Gauss elimination LU factorization	TK	2.1-2.2 in S&M	Project 1 is available here.
	Gauss-elminiation with partial pivoting, Vector and matrix norms, sub-ordinate matrix norms,Stability of linear system, Condition number, Gershgorin’s theorem	TK	2.2-2.6 in S&M Notes: Numerical methods for linear algebra
37	Special matrices: Symmetric, positive-definite, diagonally dominant, tridiagonal Cholesky factorization, Iterative methods for linear systems, Jacobi method, Gauss-Seidel method, spectral radius.	TK	3.1-3.3 in S&M Section 5 in the note on Linear Algebra.	Exercise 3 is posted here.
	Numerical interpolation Lagrange interpolation. Existence and uniqueness of interpolation polynomials. Error formula. All with proofs.	TK	6.1-6.3 in S&M
38	Hermite interpolation with Lagrange polynomials. Numerical differentiation.	TK	6.4 and 6.5 in S&M	Exercise 4 is posted here.
39	The max-norm of function spaces. Weierstrass approximation theorem. Minimax polynomials (existence, uniqueness, properties).	TK	8.1-8.3 (no proofs required.)
	Chebyshev poynomials, their properties, why they are useful in the interpolation context.	TK	8.4 and 8.5 (with proofs)	Exercise 5 is posted here.
40	Numerical integration How Lagrange interpolation polynomials can be used to construct numerical quadrature. Error estimates.	TK	7.1-7.3
	Composite formulas. The Euler-Maclaurin expansion. Extrapolation methods	TK	7.4-7.7	Exercise 6 is posted here.
41	Polynomial expansion in the 2-norm Inner product space. Best approximation in the 2-norm.	TK	9.1-9.3
	Newton interpolation polynomial and divided differences. Error for Lagrange interpolation.	TK	Slides: Divided differences and Newton interpolation polynomial Reading material: Newton form	No new exercises this week.
42	Adaptive Simpson. Orthogonal polynomial. Comparisons.	TK	Notes: Adaptive Simpson with Matlab code 9.4-9.5
43	Gauss Quadrature	TK	Chapter 10.1-10.5	Exercise 7 is posted here.
44	Splines	TK	Chapter 11
	Ordinary differential equations Eulers method: implementation, convergence proof, how to measure the order of a method.	TK	Notes: Numerical solution of ordinary differential equations Notes, section 1 and 2
45	Numerical solution of ODE's	TK	Notes, section 3.
	Order conditions for Runge-Kutta methods Error estimates and stepsize selection Embedded Runge-Kutta methods	TK	Notes, section 4.	Exercise 8 is posted here.
46	Stiff ordinary differential equations Linear Multistep Methods	TK	Notes, section 5 and 7
47	No lectures. Finishing the project work

Lecturers:

TK: Trond Kvamsdal