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