The authors state that secant methods have had a bad reputation for solving linear systems in practice, although these in exact arithmetic may give an exact solution after a finite number of steps. Their aim is to improve this reputation.
To this end they consider the linear system \(Ax=b\) in terms of a generalized Broyden’s method. Here A is a non-Hermitian non-singular \(n\times n\) matrix and \(b\in {\mathbb{C}}^ n\). The method introduced is defined by a family of algorithms with two family parameters, denoted by \(v_ k\) and \(t_ k\) at the \((k+1)st\) step. \(v_ k\) is an n-dimensional vector and \(t_ k\in {\mathbb{C}}\). The latter appears in the formulae \(x_{k+1}=x_ k+t_ k\Delta_ k\) and \(r_{k+1}=r_ k+t_ kq_ k\) with \(x_ k\) the kth iterate of x and \(r_ k\) the residual vector \(r_ k=b-Ax_ k.\) The original Broyden’s method is the special case where \(t_ k=1\) for all k.
The problem is to find pairs \(v_ k\), \(t_ k\) which make the corresponding algorithm competitive with the favoured GMRES method. In principle, \(t_ k\) is chosen in such a way that \(\| e_{k+1}(t_ k)\| =\min_{t\in {\mathbb{C}}}\| e_{k+1}(t)\|\) where \(e_{k+1}(t)=e_ k-t\Delta_ k=x-x_{k+1}.\) In practice, only an approximative condition \(\| C_ kr_{k+1}(t_ k)\| =\min_{t\in {\mathbb{C}}}\| C_ kr_{k+1}(t)\|\) is available. There are 3 alternatives: (a) \(C_ k=H_{k+1}\), (b) \(C_ k=I\), (c) \(C_ k=H_ k.\)
H\({}_ k\) is the “preconditioning” matrix at the \((k+1)st\) step.
The family parameter \(v_ k\) appears in the rank-1 update formula of \(H_{k+1}\). There are two useful alternatives: \((A)\quad v_ k=\Delta_ k=H_ kr_{k+1}\) or \((B)\quad H^*_ kv_ k=A\Delta_ k=q_ k.\) It turns out that from the six possible combinations (Aa), (Ab), (Ac), (Ba), (Bb), (Bc) the two (Aa) and (Bb) really are competitive with GMRES.
Two numerical examples are given which indicate that the combination (Aa) may be even better then GMRES, and proves to be the more competitive the larger the system dimension is. The authors emphasize that this observation is backed not only by the given examples, but also by further more extensive tests.