Steffensen method for A^(1/2)

Matrix functions are very important in various fields such as science, engineering and applied mathematics. Matrix functions are used in transformations in computer graphics, data processing in data analysis and machine learning algorithms, and data analysis in statistics. In this thesis study, we obtained a new approach to calculate the matrix square root function using the Steffensen method. We also investigated an alternative approach using the relationship between the matrix sign function and the matrix square root function and used an iterative method specifically derived for the matrix sign function based on the Steffensen method. We compared these two approaches with the most common methods in the literature, the Newton and Denman-Beavers methods. In the comparison, we also observed the construction errors using orthogonal, symplectic and perplex matrices from specially structured matrices. Finally, we examined the condition number for the matrix square root and supported it with the numerical tests. We examined all these in detail in four different sections throughout the thesis. The contents of these sections are as follows: In the introduction section, we gave the three most common general definitions from the general definitions used in the literature for matrix functions. The three definitions given are Jordan canonical form, interpolation polynomial and Cauchy integral theorem. After these definitions, we introduced the matrix square root function and presented its various usage areas. In the second section, we introduced the Steffensen method and obtained a new iterative approach by arranging it in a way suitable for the matrix square root function. However, we introduced a second approach using the relationship between the matrix sign function and the matrix square root function. In order to compare these approaches, we explained the most common methods in the literature, the Newton and Denman-Beavers methods. Finally, we introduced special structured matrices and used the orthogonal, symplectic, perplectic matrices as test matrices. In the third section, we examined the condition numbers for matrix functions and explained how to calculated the matrix square root function. In the last part of the thesis, we compared the Newton method, Steffensen method, the third method obtained from the relationship between the matrix sign function and the matrix square root function and the Denman-Beavers methods with numerical tests to provide an approximation to the matrix square root function of the orthogonal, symplectic and perplectic special structured matrices that we used. Finally, we supported the condition number studies for the matrix square root with numerical tests and presented them in tables.