Sensitivity analysis of the largest dependent eigenvalue functions of eigensystems (Q1294264)

Local behavior of the largest eigenvalues of a symmetric tridiagonal matrix is assessed by a numerical procedure. Let \(T(\lambda)\) be a real \((n,n)\) symmetric tridiagonal matrix with continuous differentiable elements. Assume that at \(\lambda\mu_j(\lambda)\), \(j= 1,\dots,k< n\) are the \(k\) largest eigenvalues of \(T(\lambda)\). Choose \(\lambda_1\), \(\lambda_2\), \(\lambda_3\) and generate numerically \(\mu_j(\lambda_1)\), \(\mu_j(\lambda_2)\), \(\mu_j(\lambda_3)\). An approximation to \(\mu_j(\lambda)\) is obtained by setting \[ \mu_j(\lambda)= (a_j+ b_j\lambda)/(c_j\lambda+ 1)\tag{i} \] and choosing \(a_j\), \(b_j\), \(c_j\) so that \(\mu_j(\lambda)\) agrees with the computed \(\mu_j(\lambda_i)\), \(i= 1,2,3\). The directional derivative of \(\mu_j(\lambda)\) in positive and negative directions \(b=\pm 1\) is using (i): \[ \mu_j'(\lambda)= (b_j- a_jc_j)/(c_j\lambda+ 1)b\tag{ii} \] and the first-order sensitivity of \(\mu_j(\lambda)\) is established by \[ \mu_j(\lambda+ \delta b)- \mu_j(\lambda)= \delta\mu_j'(\lambda, b),\quad \delta\ll 1.\tag{iii} \] The analysis may be applied to the study of sensitivity of nonlinear symmetric matrix eigenvalue problems \[ (K-\lambda M(\lambda))x= 0,\quad x\neq 0\tag{iv} \] which may be reduced by the Lanczos method to a system of equations for \(\lambda\) which involve a tridiagonal matrix \(T(\lambda)\) and it's corresponding eigenvalues \(\mu_j(\lambda)\).