Inverse iteration method with a complex parameter (Q1201482)

Let \(A\) be a real symmetric \(n\times n\)-matrix with the pairs \((\lambda_ k,\phi_ k)\), \(k=1,\dots,n\) of eigenvalues and the corresponding normalized real eigenvectors. Let \(\inf_{k\neq j}| \lambda_ j-\lambda_ k| > 2c\), \(|\lambda_ j-\lambda| < \varepsilon\) and \(0<\tau < \varepsilon < c/2\). Using the \(l_ 2\)-norm, consider the iteration procedure \((A- (\lambda^{(m)}+i\tau)I)w^{(m)}=z^{(m)}\) with \(\lambda^{(0)}=\lambda\), \(w^{(m)}=u^{(m)}+iv^{(m)}\) and \(z^{(m+1)}=v^{(m)}/\| v^{(m)}\|\) as well as \(\lambda^{(m+1)}=(Az^{(m+1)},z^{(m+1)})\) if \(\| v^{(m)}\| > \| u^{(m)}\|\) and \(\lambda^{(m+1)}=\lambda^{(m)}\) else. Then \(z^{(m)}\to\pm\phi_ j\). An error estimate and numerical examples show that the method is more efficient than the standard one with \(\tau=0\).