Lecture 01 Introduction & course outline, Iterations in 1D, Finding roots of nonlinear problems in 1D: here bisection method and its realization in MATLAB. (Read Sec 1.1)

Lecture 02 Fixed Point, Fixed Point Iteration, Contractivity, Fixed Point Theorem, Examples

Lecture 03 Fixed Point Iterations: Examples, A priori and a posteriori error estimates

Lecture 04 - Newton iteration, variants of Newton iteration, order and rate of convergence, Start with linear systems, matrix and vector norms. Read Sec. 1.4 and Sec. 2.3

Lecture 05 Linear Equations Systems, LU factorization (p. 101 ff), forward and backward substition, row pivoting, condition Number (just motivation)

Complement to Lecture 5 Hand-written notes on Direct Methods: Gaussian elimination, back substitution, LU factorization. Discussion about computational costs, ill-conditioning & Pivoting.

Lecture 06 Linear Space (vector space), vector norms, condition number

Complement to Lecture 6 Vector and matrix norms, eigenvalues, spectral radius, condition number.

Lecture 07 Matrix norms, condition number, start with iterative methods and fixed point iteration in n dimensions. (among others Sec. 2.3.1, 2.5)

Complement to Lecture 7 Jacobi and Gauss-Seidel iterative schemes - notes from class

Lecture 8 Fixed point iteration in R^n, Newton's method & Jacobian in R^n

Complement to Lecture 8 More on iterative schemes, Newton's method in R^n.

Lecture 9 Polynomial interpolation, monomial, Vandermonde approach, polyfit, polyval

Complement to Lecture 9 Finishes up on Newton's method in R^n. Then on to Lagrange + Vandermonde interpolation.

Lecture 10 Polynomial Space, Lagrange basis, inner products in function spaces

Complement to Lecture 10 Interpolation error, Chebyshev polynomials and nodes

Lecture 11 Inner products and norms, uniqueness of interpolation polynomials, error formula for interpolation polynomials, extrapolation error

Complement to Lecture 11 Chebyshev interpolation & Least Squares method

Lecture 12 Tschebyschev polynomials and the use of Tschebyschev points for interpolation.

Complement to Lecture 12 Nonlinear Least Squares, Gram-Schmidt ortho-normalization and QR factorization

Lecture 13/14 Introduction to cubic interpolatory Splines, some additional comments on the homework.

Lecture 15 Natural spline (cont.), minimality property of cubic splines, linear space of splines and its basis, B-splines

Complement to Lecture 15 Review of vector spaces and inner products

Lecture 16 & 17 Quadrature (Numerical Integration, Introduction)

Complement to Lecture 16 Quadratures (Newton-Cotes Formulas)

Complement to Lecture 17 Composite Newton-Cotes Formulas and Romberg Integration

Lecture 18 Quadrature (Repetition), Gauss Quadrature

Complement to Lecture 18 Gaussian Quadratures and Legendre Polynomials

Lecture 19 ODEs and initial value problems: Repetition of basic facts such as directional field, stability, linear ODEs, autonomous ODEs. Start with explicit Euler method

Complement to Lecture 19 Gaussian Quadratures. Starting on ODEs. Euler's Method.

Lecture 20 Reduction of Order, Theoretical Results for IVP

Lecture 21 Explicit and Implicit Euler: Stability, predictor/corrector way to implement, Newton- and Fixed point corrector iteration, linear test equation

Lecture 22 notes Consistency, Convergence, Stability and Stiffness of Numerical Methods for ODEs

Lecture 23 Multi-step methods, stability, convergence, consistency and derivation

Lecture 24 Multistep methods: derivation of Adams Bashforth, Adams-Moulton and BDF methods

Complement to Lecture 24 Backward Difference Formulas, Introduction to Boundary Value Problems - Shooting Method

Lecture 25 Finite Difference Methods for (linear/nonlinear) BVP

Review 1