Henrici's transformation and its application to the computation of derivatives of eigensystems (Q1899964)

Application of Henrici's transformation to the computation of derivatives \(\lambda_{i,j}\) and \(x_{i,j}\) of eigenvalues \(\lambda_i\) and corresponding eigenvectors \(x_i\), respectively, of an \(n \times n\) matrix which depends smoothly on real parameters is studied. An \(H\)- algorithm involving 4 steps is described for this computation. This algorithm gives exact values of \(\lambda_{i, j}\) and \(x_{i, j}\) in the absence of roundoff errors. It requires less iterations than the \(\varepsilon\)-algorithm. Numerical comparisons suggest that it is the best for sufficiently large sparse matrices.