Computation of Derivatives of Repeated Eigenvalues and the Corresponding Eigenvectors of Symmetric Matrix Pencils

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

“…The assumption of dierentiability is more restrictive in the case of repeated eigenvalues 8,17,19 then for simple eigenvalues. Diculties arise with the numerous notorious pathological examples such as Rellich's 20 matrix A 1 given by A 1 (0) 0 and…”

Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem

“…Unfortunately, C 31 and C 13 cannot be computed since Γ 3 is not yet determined. However, (3.7c) and (3.7b) give expressions for C 13 and C 31 in terms of Γ 1 and Γ 3 .…”

“…In [4], the theory is extended by assuming that there are still eigenvalue derivatives which are repeated, but the second order derivatives of the eigenvalues have to be distinct. In [3], this theory is generalized to complex-valued, Hermitian matrices. Here it is assumed that for an eigenvalue with multiplicity r, all its eigenvalue derivatives up to kth order are also equal and the (k + 1)st order derivatives are distinct again.…”

Iterative computation of derivatives of repeated eigenvalues and the corresponding eigenvectors

“…, t ρ ) ʦ ‫ޒ‬ ρ . In most, though certainly not all, applications, these functions are differentiable, and much effort (see [2,[4][5][6] for some references) has been devoted to the development of numerical methods for computing the local values of the partial derivatives λ i,j (t) and x i,j (t). (Here, and throughout this paper, the subscripts ', j' and ', jj' denote the first-and second-order partial derivatives with respect to the j th argument, t j .)…”

“…(Here, and throughout this paper, the subscripts ', j' and ', jj' denote the first-and second-order partial derivatives with respect to the j th argument, t j .) Until recently [4,7], most of this work was restricted to the case of simple eigenvalues, although it is known that eigenvalues often coalesce as a design structure approaches an optimum [1,8], and, even before optimizing, repeated eigenvalues may occur when a structure has certain symmetry properties [9]. Moreover, in the presence of uncertain data or roundoff errors from numerical computation, the distinction between repeated eigenvalues and very close eigenvalues is blurred.…”