# Lecture plan

This schedule is tentative, changes will appear.

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

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
No new exercises this week.
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