Računanje i analiza matrične funkcije drugi korijen

Stopić, Paula (2016) Računanje i analiza matrične funkcije drugi korijen. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

General matrix function is defined via Jordan canonical form. If function is defined on the spectrum of some matrix, matrix function has some very good properties which are useful for matrix square root like commutation of matrix function and matrix, block triangular matrix is mapped in block triangular matrix, block diagonal matrix is mapped in block diagonal matrix, eigenvalues of matrix are mapped in eigenvalues of function of that matrix and so on. Every solution of equation $X^2 = A$ is called a square root of A. If matrix A is nonsingular or singular with a semisimple zero eigenvalue, the matrix square root is defined on the spectrum of A. In other cases matrix square root doesn’t have to exist at all. The square roots of a nonsingular matrix fall into two classes. The first class comprises finitely many primary square roots and second class may be empty or it comprises a finite number of square roots which are sharing the same spectrum. The principal square root is defined for matrices which have no eigenvalues on $R^-$. It is a unique square root all of whose eigenvalues lie in the open right half-plane. I gave the algorithms for computing the square root via a Schur decomposition especially when a matrix is nonsingular where complexity is $O(n^3)$, and especially when a matrix is singular with a semisimple zero eigenvalue where computing the square root can be solved by solving a linear system. Through Newton’s method I used some properties of matrix $sign$ function. Newton’s method is also described for a nonsingular matrix where iteration converges quadratically to $A^{\frac{1}{2}} sign(A^{-\frac{1}{2}} X_0)$; while for a singular matrix is proven that the square root exists if matrix is under some conditions. The iterations which I described have different stability properties. The Newton iteration is unstable at $A^{\frac{1}{2}}$ unless the eigenvalues of A are very closely clustered. The other versions of Newton’s iteration are all stable at $A^{\frac{1}{2}}$. Further I showed few special matrices which all have good properties to compute matrix square root. Although the binomial expansion which I described with $(I-C)^{\frac{1}{2}}$ does not converge for $\rho(C) > 1$, the iteration nevertheless continues to converge when the eigenvalues of C lie outside the unit disk but within the cardioid. Except the binomial iteration, I explained what are M-matricces and Hmatrices and I showed that for that kind of regular matrices the principal square root exists and it is also M or H-matrix. At the end we can see also how Hermitian positive definite matrix is good for computing the square root via Cholesky factorization and polar decomposition.

Item Type: Thesis (Diploma thesis)
Supervisor: Bosner, Nela
Date: 2016
Number of Pages: 64
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 26 Aug 2016 08:17
Last Modified: 26 Aug 2016 08:17
URI: http://digre.pmf.unizg.hr/id/eprint/5009

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