Computing Derivatives of Eigenvalues and Elgenvectors by Simultaneous Iteration

Computation of mixed partial derivatives of eigenvalues and eigenvectors by simultaneous iteration

“…That method is analysed, improved and generalized in Reference 16, which also includes some useful references and a discussion of problems associated with both repeated and closely clustered eigenvalues. In the more common case in which all the eigenvalues are distinct, no additional calculation is required to obtain X, and, as previously shown, 10 the remaining quantities required in Step 1 of our new algorithm (Algorithm 1 below) may be computed using Algorithm 0 below. Algorithm 1 places no restriction on the choice of method used for the initial computation of eigenvalues and eigenvectors.…”

“…We showed 10 that, if jl i À sj 4 jl r1 À sj, then the ith columns of M(k) and U(k) converge at the rate jl r1 À sjajl i À sj k , and we also derived and analysed the following algorithm.…”

“…This method, which follows our earlier approach 10 for ®rst derivatives, includes the method of Tan et al…”

“…For this example we used a 100 Â 100 randomly generated matrix with elements from a uniform distribution on [0, 1]. It is known 10 that one eigenvalue of matrices obtained from this distribution is usually of much greater magnitude than the others. This was indeed true with our example, which had l 1 50 .…”

Iterative computation of derivatives of repeated eigenvalues and the corresponding eigenvectors

“…See also Reference [4]. Earlier [10], we developed a simultaneous iteration algorithm for numerical computation of the local values of λ i,j (t) and x i,j (t) for simple eigenvalues, and mentioned that Algorithm 1 of Reference [10] could also be used to compute the derivatives of repeated eigenvalues (though not the corresponding eigenvectors) of non-defective matrices. That statement, though correct, is potentially misleading.…”