Analysis and applications of the weighted central direction method
Abstract
Description
A thesis Submitted in Partial Ful llment of the Requirements for the Degree of MASTER OF SCIENCE in MATHEMATICS from Texas A&M University-Corpus Christi in Corpus Christi, Texas.
Iterative methods yield an approximated solution to a given problem by producing a sequence of points that converges to the exact solution. Due to the effectiveness of these methods, they become one of the core mathematical procedures widely used in many major mathematical areas such as Differential Equations, Linear Algebra, and Matrix Analysis. The Methods of Alternating Projection, which we will use in this thesis, form a class of iterative methods based on the relevant projection algorithm introduced by John von Neumann. This outstanding algorithm received considerable attention by mathematicians which contributed to a number of different algorithms to solve several problems. Recently, two former graduate students at Texas A&M University-Corpus Christi, Melina Wijaya and Zulema Cervantes, introduced a couple of new algorithms in this area. Wijaya presented the Weighted Direction algorithm, and Cervantes introduced the Weighted Central Direction algorithm by combining the Weighted Direction and the Central Direction. The algorithm with the combination of two directions achieved a faster convergence than the algorithm with the Weighted Direction only. However, the Weighted Central Direction algorithm needs a parameter which depends on the size of the problems. This thesis carries two objectives. Firstly, study the role of the wide angle condition which guarantees the convergence in the Weighted Direction algorithm in order to analyze if this condition is necessary for the convergence of the algorithm. Secondly, improve the algorithms developed by Wijaya and Cervantes in order to obtain a faster convergence by finding an adequate ratio between the Weighted Direction and the Central Direction that is independent of the size of the problems.
Mathematics and Statistics
College of Science and Engineering
Iterative methods yield an approximated solution to a given problem by producing a sequence of points that converges to the exact solution. Due to the effectiveness of these methods, they become one of the core mathematical procedures widely used in many major mathematical areas such as Differential Equations, Linear Algebra, and Matrix Analysis. The Methods of Alternating Projection, which we will use in this thesis, form a class of iterative methods based on the relevant projection algorithm introduced by John von Neumann. This outstanding algorithm received considerable attention by mathematicians which contributed to a number of different algorithms to solve several problems. Recently, two former graduate students at Texas A&M University-Corpus Christi, Melina Wijaya and Zulema Cervantes, introduced a couple of new algorithms in this area. Wijaya presented the Weighted Direction algorithm, and Cervantes introduced the Weighted Central Direction algorithm by combining the Weighted Direction and the Central Direction. The algorithm with the combination of two directions achieved a faster convergence than the algorithm with the Weighted Direction only. However, the Weighted Central Direction algorithm needs a parameter which depends on the size of the problems. This thesis carries two objectives. Firstly, study the role of the wide angle condition which guarantees the convergence in the Weighted Direction algorithm in order to analyze if this condition is necessary for the convergence of the algorithm. Secondly, improve the algorithms developed by Wijaya and Cervantes in order to obtain a faster convergence by finding an adequate ratio between the Weighted Direction and the Central Direction that is independent of the size of the problems.
Mathematics and Statistics
College of Science and Engineering