1 Fundamentals

1.1 Sets and Mappings

1.2 Numbers

1.3 Peano Axioms

1.4 Absolute Value

1.5 Some Useful Equalities and Inequalities

1.6 Important Functions

2 Induction and Relations

2.1 Induction

2.2 Relations and Orderings

2.3 Countable and Uncountable

3 Groups and Construction of Integers

3.1 Groups

3.2 Construction of Integers

4 Numbers and Floating Point Numbers

4.1 Representation of Numbers

4.2 Floating Point Representation and Machine Accuracy

5 On Stability and Error Propagation

5.1 Propagation of Errors

5.2 Absolute and Relative Error

5.3 Condition Number and Well Posed Problems

5.4 Condition of Elementary Operations

5.5 Landau Symbols

6 Polynomial Interpolation

6.1 Interpolation Conditions

6.2 Properties of the Interpolation Process

6.3 Condition of Polynomial Interpolation

6.4 Divided Differences

6.5 The Algorithm of Aitken-Neville

6.6 Interpolation Error

6.7 Piecewise linear Interpolation

7 Direct Solution of Linear Systems

7.1 Linear Systems

7.2 Triangular Systems and Forward/Backward Substitution

7.3 Gaussian Elimination

7.4 Matrices and Vectors

7.5 Elementary Matrix-Transformations

7.6 LU-Decomposition

7.7 Practical Considerations

7.8 LU-decomposition in Detail

8 Condition Number of Matrices

8.1 Euclidean Norm

8.2 Condition Number of Matrices

8.3 Stability Analysis of Gaussian Elimination

9 Symmetry, Cholesky Decomposition, and Rank

9.1 Scalar Product

9.2 Cholesky-Decomposition

9.3 Rank of a Matrix

9.4 Linear Independence

10 Interpolation Revisited - Splines

10.1 Piecewise Linear Interpolation

10.2 Cubic Spline Interpolation

10.3 A Mechanical Motivation

10.4 Computing a cubic spline

11 Least Square Problems and the Normal Equations

11.1 General Setting and Choice of the Approximation

11.2 Least Squares

11.3 Normal Equations

11.4 Condition Number of the Normal Equations