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
Applied and Computational Mathematics (MS)
Department, Program, or Center
School of Mathematical Sciences (COS)
Matthew J. Hoffman
Nichols, John, "A New Algorithm for Computing the Square Root of a Matrix" (2016). Thesis. Rochester Institute of Technology. Accessed from
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