Nearness Problems in Numerical Linear Algebra

We consider the theoretical and the computational aspects of some nearness problems in numerical linear algebra. Given a matrix $A$, a matrix norm and a matrix property P, we wish to find the distance from $A$ to the class of matrices having property P, and to compute a nearest matrix from this class. It is well-known that nearness to singularity is measured by the reciprocal of the matrix condition number. We survey and compare a wide variety of techniques for estimating the condition number and make recommendations concerning the use of the estimates in applications. We express the solution to the nearness to unitary and nearness to Hermitian positive (semi-) definiteness problems in terms of the polar decomposition. A quadratically convergent Newton iteration for computing the unitary polar factor is presented and analysed, and the iteration is developed into a practical algorithm for computing the polar decomposition. Applications of the algorithm to factor analysis, aerospace computations and optimisation are described; and the algorithm is used to derive a new method for computing the square root of a symmetric positive definite matrix. This leads us, in the remainder of the thesis, to consider the theory and computation of matrix square roots. We analyse the convergence properties and the numerical stability of several well-known Newton methods for computing the matrix square root. By means of a perturbation analysis and supportive numerical examples it is shown that two of these Newton iterations are numerically unstable. The polar decomposition algorithm, and a further Newton square root iteration are shown not to suffer from this numerical instability. For a nonsingular real matrix $A$ we derive conditions for the existence of a real square root, and for the existence of a real square root which is a polynomial in $A$; the number of square roots of the latter type is determined. We show how a Schur method recently proposed by Bj\"orck and Hammarling can be extended so as to compute a real square root of a real matrix in real arithmetic. Finally, we investigate the conditioning of matrix square roots and derive an algorithm for the computation of a well-conditioned square root