Lecture plan

This schedule is tentative, changes will appear.

Textbook: Süli and Mayers, An introduction to Numerical Analysis

Week	Topics	Reading	Exercises
34	Introduction to the course. Taylor's theorem, big O -notation, rounding errors	slides
	Introduction to MATLAB	slides	Exercise 1 Euler
35	Numerical solution of nonlinear equations Existence of solutions Simple iterations Rate of convergence Contraction mapping theorem	1.1-1.2 (up to Theorem 1.4)
	Rate of convergence, asymptotic error constant, order of convergence Convergence of fixed point iterations. Secant and bisection method (self study)	1.2 (from Theorem 1.4) - 1.8. slides	Exercise 2
36	Scalar equations: Convergence of Newton's method (Theorem 1.8) and max number of fixed point iterations (Theorem 1.4) with proofs. Newtons method for system of equations.	A comment on an example from the lecture. Note on systems of equations.
	Fixed point iterations for systems of equations. The contraction mapping theorem in max-norm (Theorem 4.1 and 4.2). Convergence of Newtons method (Theorem 4.4).	4.1-4.5.	Project 1: (05.09-16.09)
37	Numerical linear algebra Naiv Gauss elimination LU factorization Gauss-elminiation with partial pivoting.	2.1-2.6. Note on linear algebra sec. 1 and 3.
	Vector and matrix norms, sub-ordinate matrix norms Stability of linear system, Condition number	2.6 in S&M Section 2 the note on linear algebra.	Exercise 3
38	Special matrices: Symmetric, positive-definite, diagonally dominant, sparse, tridiagonal, band. Cholesky factorization, LU-factorization for tridiagonal matrix.	3.1-3.3 in S&M Section 5 in the note on linear algebra.
	Gershgorin's theorem. Iterative methods for linear systems, Jacobi method, Gauss-Seidel method, spectral radius.	Section 6 and 7 in the note on linear algebra.	Exercise 4 (corrected 03.10) gs.m rhoSOR.m Solution task 2: ex4t2.m and sor.m.
39	Numerical interpolation Lagrange interpolation. Existence and uniqueness of interpolation polynomials. Error formula. All with proofs.	6.1-6.3
	The max-norm of function spaces. Weierstrass approximation theorem. Minimax polynomials (existence, uniqueness, properties). Chebyshev poynomials, their properties, why they are useful in the interpolation context.	8.1-8.3 (no proofs required.) 8.4 and 8.5 (with proofs)	Exercise 5: Set 3 lagrange.m
40	Divided differences (including Newton forward- and backward difference formula). Hermite interpolation, both with Lagrange polynomials and divided differences.	Chapter 3.2 and 3.3 in Burden and Faires, "Numerical Analysis", on it's learning.
	Numerical integration How Lagrange interpolation polynomials to construct numerical quadrature. Degree of accuracy. Composite formulas. Error formulas for the Trapezoidal rule.	7.1-7.5 (the proof of error formula for Simpson will be given on Tuesday)	Exercise 6 solution
41	Proof of the error estimate of Simpsons formula. Adaptive Simpson	Note on Adaptive Simpsons method with Matlab code
	Inner product space Orthogonal polynomials	9.1-2 and 9.4 (to Theorem 9.2) The curriculum is covered by the note on Gauss quadrature and orthoganal polynomials (see below).	Exercise 7
42	Gauss quadrature. Composite Gauss The Euler-Maclauring expansion Romberg integration	Note on orthogonal plynomials and Gauss quadrature 10.5 Theorem 7.4 (no proof required) 7.7.
	Splines Definition of a spline of degree k Linear and cubic spline	11.1-4. The note on splines cover the curriculum.	Exercise 8
43	Ordinary differential equations Eulers method: \\implementation, convergence proof, how to measure the order of a method.	Note on numerical solutions of ODEs. Most of the lectured material can be found in this note. Lecture note 1-2.
	Existence and uniqueness of solutions of ODEs, Examples of Runge-Kutta methods General Runge-Kutta methods	Lecture notes 2-4 (up to 4.1) rk2.m, LotkaVolterra.m, ordertest.m	Exercise 9: Problem set 7: Task 1 and 2.
44	Order conditions for Runge-Kutta methods Error estimates and stepsize selection Embedded Runge-Kutta methods	Note, section 4.
	Linear multistep methods	Note, section 7.1-7.3. lmm.m	Project 2
45	No lectures
46	No lectures
47	Leftovers Stiff ordinary differential equations Adams methods Predictor -corrector methods	Note, section 5 and 7.4-7.5.	Exercise 10 Solution of problem 2.
	Summary