1 Introduction
2 Background
2.1 Types of Linear Systems
2.2 Solving Linear Systems
3 Variations of the Kaczmarz Method
3.1 The Kaczmarz Method
3.2 Randomized Kaczmarz Method
3.3 Simple Randomized Kaczmarz Method
3.4 Sampling Rows Using Quasi-random Numbers
3.5 Randomized Block Kaczmarz Method
3.6 Randomized Coordinate Descent Method
3.7 Randomized Extended Kaczmarz Method
3.8 Randomized Double Block Kaczmarz Method
3.9 Greedy Randomized Kaczmarz Method
3.10 Selectable Set Randomized Kaczmarz Method
3.11 Randomized Kaczmarz with Averaging Method
3.12 Summary
3.13 Parallel Implementations of the Kaczmarz Method
4 Applications of the Kaczmarz Method
4.1 Relationship with Other Iterative Methods
4.2 Adaptations for Linear System of Inequalities
4.3 Application in Image Reconstruction in Computed Tomographies
5 Implementation and Results for the Variations of the Kaczmarz Method
5.1 Implemented Methods
5.2 Stopping Criterion
5.3 Data Sets
5.4 Effect of Randomization
5.5 Results
6 Conclusion