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  1.11.2 (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 maxnorm (Theorem 4.1 and 4.2). Convergence of Newtons method (Theorem 4.4).  4.14.5.  Project 1: (05.0916.09)  
37  Numerical linear algebra Naiv Gauss elimination LU factorization Gausselminiation with partial pivoting.  2.12.6. Note on linear algebra sec. 1 and 3.  
Vector and matrix norms, subordinate 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, positivedefinite, diagonally dominant, sparse, tridiagonal, band. Cholesky factorization, LUfactorization for tridiagonal matrix.  3.13.3 in S&M Section 5 in the note on linear algebra.  
Gershgorin's theorem. Iterative methods for linear systems, Jacobi method, GaussSeidel 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.  6.16.3  
The maxnorm of function spaces. Weierstrass approximation theorem. Minimax polynomials (existence, uniqueness, properties). Chebyshev poynomials, their properties, why they are useful in the interpolation context.  8.18.3 (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.  7.17.5 (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  9.12 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 EulerMaclauring 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  11.14. 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 12.  
Existence and uniqueness of solutions of ODEs, Examples of RungeKutta methods General RungeKutta methods  Lecture notes 24 (up to 4.1) rk2.m, LotkaVolterra.m, ordertest.m  Exercise 9: Problem set 7: Task 1 and 2. 

44  Order conditions for RungeKutta methods Error estimates and stepsize selection Embedded RungeKutta methods  Note, section 4.  
Linear multistep methods  Note, section 7.17.3. 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 7.47.5.  Exercise 10 Solution of problem 2. 
Summary 