A new family of modal methods for the calculation of eigenvector derivatives in non-self-adjoint systems with a singular coefficient matrix is developed. The family contains the modal and modified modal methods as a subset. In the family, the component of the mth eigenvector in the expansion of the derivative of the jth eigenvector is multiplied by various powers of the eigenvalue ratio lambda(i)/lambda(m), thereby accelerating convergence. The family of methods is applied to a self-adjoint example problem, namely, a cantilever beam whose root depth Is used as the design variable to which the sensitivity of the first four mode shapes is sought. Two different numbers of elements are used to model the beam in two cases in an effort to investigate the effect of the system size on the performance of the new methods. Central processor time and number of modes needed for convergence are determined. For a given problem, one of the methods in the family takes the smallest number of modes and shortest time to converge. The method is applied to a non-self-adjoint system with zero eigenvalues as well. The family is compared with Nelson's method and the modified Rudisill and Chu method on the basis of operation counts and is expected to perform better than the two when more than one eigenvector derivative is of interest.