Iterative computation of second‐order derivatives of eigenvalues and eigenvectors

Abstract

SUMMARYAn iterative method is introduced for computing second-order partial derivatives (sensitivities) of eigenvalues and eigenvectors of matrices which depend on a number of real design parameters. Numerical tests confirm the viability of the method and support our theoretical analysis. Alternative methods are reviewed briefly and compared with the one proposed here. . INTRODUCTIONThe optimum design of structures often involves the matrix eigenvalue problem A x ; = X i x i , i = 1 , 2 , . . , nwhere the (real or complex) n x n matrix A (and consequently the eigenvalues X i and eigenvectors x i ) are functions of a number of real design parameters P I , Pz, ..., P,,,. We are concerned here with numerical computation of Xi,;[ and x;,;r for a small number of X i and x i , where the subscript 'J' denotes partial differentiation with respect to Pj and the subscript ', j/' denotes the second-order partial derivative with respect to P; and PI. The importance of these second-order derivatives (also called 'sensitivities') has been demonstrated by Brandon, for example, who suggested computing X i , j t and x i , j / when X i is a simple (i.e. unrepeated) eigenvalue using what Murthy and Haftka2 called 'adjoint methods'. Unfortunately adjoint methods have two important drawbacks for computing derivatives of eigenvectors: 394 (i) accurate knowledge of a// eigenvalues and right and left eigenvectors is required to compute x i , ; and xi,;/ for a single i; (ii) if any eigenvalues of A are ill-conditioned then, for aN i , computation of xi,; and (especially) X i , j l involves computation of the difference between two nearly equal quantities, sometimes resulting in the loss of all significant figures, even if the particular eigenvector whose derivatives are required is well conditioned. In this paper we describe a new iterative method for computation of A;, ;[ and xi,;/. The method is introduced, its rate of convergence established and its efficient implementation discussed in Section 2. In Section 3 we describe our numerical experience with the iterative 0748-8025/94/010001-09$09.50

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

“…Although our earlier paper 14 on the case r 1 did consider mixed partial derivatives, the numerical examples tested there all involved only the case j l. Here we describe some numerical tests involving examples with r 4 1 and j T l.…”

“…The case r 1 treated earlier 14 would never have been competitive with direct methods had it not been for the fact that its performance can be dramatically improved by the use of certain etype extrapolation 22 methods discussed in our earlier papers. 10,23±25 As well as substantially reducing the amount of computation required for the dominant eigenvalue, these methods also make it possible to obtain useful results for the derivatives of subdominant eigenvalues and the corresponding eigenvectors, though the stability properties of these methods are often unsatisfactory when subdominant eigenvalues are involved, as the extrapolation methods must then be applied to divergent sequences.…”

“…Like the earlier method, 14 the method described here is readily extended to compute partial derivatives of order higher than two. Details are omitted, as the competitiveness of both iterative methods relative to direct methods decreases somewhat as the order of the derivative required increases.…”

“…By using a good initial approximation we can usually reduce the number of iterations required. As noted before, 14 changing the component of x i,jl in the x i direction is equivalent to using a dierent normalizing condition for x i .…”

“…Previous methods include modal expansion methods 4 (which, unlike the method proposed here, require accurate knowledge of all eigenvalues and right and left eigenvectors for the accurate calculation of a single x i,jl ), and various direct methods, 5,6 one of which 7 is valid for problems with non-linear dependence on the eigenvalue parameter. For ®rst derivatives, several iterative methods have been proposed 2,8±12 which oer a number of advantages, 2,13,14 at least for the large sparse matrices that arise in ®nite element computations, but the solitary iterative method 14 that has been proposed for computing l i,jl and x i,jl is really suitable only for the dominant eigenvalue, or one which can be made dominant by a suitable origin shift 14,15 or for a dominant complex conjugate pair.…”