Iterative methods for the Drazin inverse of a matrix with a complex spectrum

Invariant Subspaces, Derivative Arrays, and the Computation of the Drazin Inverse

“…converges to E D provided the eigenvalues of E are real, see [19], or satisfy some angle condition, see [17]. If E κ is large in norm, the parameter α must be chosen very small.…”

A Higher Order Iterative Method for Computing the Drazin Inverse

“…In 2004, Li and Wei in [16] proved that the matrix method of Schulz (1) can be used for finding the Drazin inverse of square matrices both possessing real or complex spectra. They proposed the following initial matrix: $\begin{array}{c}{V}_{\mathrm{0}}=W_{\mathrm{0}}=\alpha A^l,\hspace{1em}l\ge \text{ind}\left(A\right)=k,\end{array}$ where the parameter α must be chosen so that the condition || I − AV 0 || < 1 is satisfied.…”

“…The aim of this example is to apply the discussions of Section 4, for finding the Drazin inverse of the following square matrix (taken from [16]): …”

Drazin inverse based numerical methods for singular linear differential systems

“…We have used a Newton method proposed in [11] and [18] that computes an approximation for the Drazin inverse of a matrix A with index k, using the recurrence…”

“…The convergence of the method is quadratic [11] and to use the method we have selected the value α = 0.01.…”