Graduate Seminar

Location:  CL 305 and Live Stream

Speaker: Kenneth Lie

MSc Student supervised by Chun-Hua Guo

Title:  Iterative Methods for a Quadratic Matrix Equation

 Zoom Link:  https://uregina-ca.zoom.us/j/92805435386?pwd=42pEMT2QL4b8W8bqIOr5Y39hln8bSI.1

Abstract:

We consider the quadratic matrix equation (QME) \(AX^2 - BX + C = 0\), where B and \(B-A-C\) are nonsingular M-matrices, and \(B^{-1}A \geq 0, B^{-1}C \geq 0\). This class of QMEs is more general than some classes of QMEs studied earlier by others. We show that the more general QME always has a minimal nonnegative solution \(X_*\) and establish some theoretical properties about \(X_*\). To approximate the solution \(X_*\), we consider Bernoulli iteration, Newton’s method, and the structure-preserving doubling algorithm. With \(X_0 = 0\), we show that Bernoulli iteration and Newton’s method will each generate a sequence \(\{X_k\}\) that converges to \(X_*\) monotonically. The convergence of Bernoulli iteration is shown to be linear and the convergence of Newton’s method is quadratic. We also show that we can stop the Bernoulli iteration with any iterate \(X_{k_0}\) and use it as an initial guess for Newton’s method. The monotonic convergence of Newton’s method is still guaranteed. So Newton’s method can be used as a correction method following Bernoulli iteration, which is much less expensive each iteration. The convergence of the structure-preserving doubling algorithm is also shown to be quadratic.