Thesis

Iterative methods for symmetric positive definite matrices with applications to covariance matrices

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Awarding institution
  • University of Strathclyde
Date of award
  • 2026
Thesis identifier
  • T18033
Person Identifier (Local)
  • 202287929
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Department, School or Faculty
Abstract
  • Obtaining the inverse of, and solving linear systems involving, a large symmetric positive definite matrix A ∈ R p×p is a continual challenge across many mathematical disciplines. Direct methods of obtaining the inverse of a symmetric positive definite matrix can be computationally expensive, making it infeasible to calculate the entire inverse for large matrices. In this thesis we present a novel iterative block matrix inversion (IBMI) algorithm, which is designed to approximate the inverse of a large, dense, symmetric positive definite matrix. We partition the matrix A into blocks and use an iterative process involving block matrix inversion to update our approximation until it reaches a satisfactory level of accuracy. For the two-block non-overlapping approach, we show that the IBMI algorithm will always converge given any symmetric positive definite matrix. Additionally, in this thesis, we present two novel preconditioners which can be used with the preconditioned conjugate gradient (PCG) method to solve large positive definite linear systems. Both preconditioners are described in detail, beginning with the block diagonal IBMI preconditioner. After preliminary numerical experiments, we then introduce the IBMI hierarchically off-diagonal low rank (HODLR) preconditioner. We show that the IBMI HODLR preconditioner can perform better than state-of-the-art preconditioners, such as Block Jacobi and incomplete Cholesky, for various problems. Finally, we describe how both the IBMI algorithm, and the preconditioners derived from it, can be applied to any large symmetric positive definite matrix.
Advisor / supervisor
  • Pestana, Jennifer
  • Dolean, Victorita
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DOI
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