The authors explore the properties of the Arnoldi process when applied to a large matrix \(A\in\mathbb{R}^{n\times n}\) with a skew-symmetric part
\[ A-A^*=\sum_{k=1}^sf_kg_k^*,\quad f_k,g_k\in\mathbb{R}^{n} \tag{1} \]
of low rank \(s.\)
The first part presents the main features of the Arnoldi process and the general minimal residual (GMRES) method for iterative solving of a large system of linear equations with a nonsymmetric matrix. Thus the application of \(j\) steps of the Arnoldi process to the matrix \(A\) with initial vector \(r_0\neq 0\) yields the decomposition
\[ AV_j=V_jH_j+h_je_j^* \tag{2} \]
where \(V_j=\{v_1,v_2,\dots,v_j\}\in\mathbb{R}^{n\times j}\) and \(h_j\in\mathbb{R}^n\) satisfy \(V_j^*V_j=I_j,\,V_j^*h_j=0\) and \(v_1=r_0/\| r_0\|.\) \(H_j\in\mathbb{R}^{j\times j}\) is an upper Hessenberg matrix, \(I_j\) denotes the identity matrix of order \(j,\) \(e_k\) is the \(k\)th column of an identity matrix of appropriate order and \(\|\cdot\|\) denotes the Euclidian vector norm. When \(h_j\neq 0,\) one can express ({2}) in the form
\[ AV_j=V_{j+1}\overline H_j \tag{3} \]
where \(v_{j+1}=h_j/\| h_j\|,\) \(V_{j+1}=[V_j,v_{j+1}]\in\mathbb R^{n\times(j+1)},\) \(\overline H_j=\Big[\begin{matrix} H_j\\ \| h_j\| e_j^*\end{matrix} \Big]\in\mathbb R^{(j+1)\times j}\).
In the second section the authors show that the property ({1}) of \(A\) makes it possible to determine the columns \(v_k\) of \(V_j\) with a short recursion formula, the number of terms of which depends on \(s\) in ({1}) but can be bounded independently of \(k.\)
The third section discusses the structure of the matrices \(H_j\) and \(\overline H_j\) in ({2}) and ({3}), respectively, and presents a fast algorithm for determining the Arnoldi decomposition ({3}), assuming that the decomposition exists. The algorithm can be applied to compute approximations of a few extreme eigenvalues and associated eigenvectors of \(A\) similarly to the standard implementation of the Arnoldi process.
The short recursion formula for the columns of \(V_j\) and the structure of \(H_j\) make it possible to derive a progressive GMRES method for the solution of linear systems \(Ax=b\) with a matrix that satisfies ({1}). Such a method is described in the fourth section. The storage requirement of the method, as well a s the computational effort per iteration, are bounded independently of the number of iterations \(j.\)
The fifth section contains some computed examples.
Concluding remarks are presented in the last section.