#### Abstract

There are several different methods for computing a square root of a matrix. Previous research has been focused on Newton's method and improving its speed and stability by application of Schur decomposition called Schur-Newton.

In this thesis, we propose a new method for finding a square root of a matrix called the exponential method. The exponential method is an iterative method based on the matrix equation (X - I)^(2) = C, for C an n x n matrix, that finds an inverse matrix at the final step as opposed to every step like Newton's method. We set up the matrix equation to form a 2n x 2n companion block matrix and then select the initial matrix C as a seed. With the seed, we run the power method for a given number of iterations to obtain a 2n x n matrix whose top block multiplied by the inverse of the bottom block is C^(1/2) + I. We will use techniques in linear algebra to prove that the exponential method converges to a specific square root of a matrix when it converges while numerical analysis techniques will show the rate of convergence. We will compare the outcomes of the exponential method versus Schur-Newton, and discuss further research and modifications to improve its versatility.

#### Library of Congress Subject Headings

Matrices; Square root; Algorithms

#### Publication Date

9-13-2016

#### Document Type

Thesis

#### Student Type

Graduate

#### Degree Name

Applied and Computational Mathematics (MS)

#### Department, Program, or Center

School of Mathematical Sciences (COS)

#### Advisor

Matthew J. Hoffman

#### Advisor/Committee Member

Manuel Lopez

#### Advisor/Committee Member

James Marengo

#### Recommended Citation

Nichols, John, "A New Algorithm for Computing the Square Root of a Matrix" (2016). Thesis. Rochester Institute of Technology. Accessed from

https://scholarworks.rit.edu/theses/9265

#### Campus

RIT – Main Campus

## Comments

Physical copy available from RIT's Wallace Library at QA188 .N43 2016