The problem of convergence in modal synthesis is solved by a minimization approach. Substructure modes corresponding to fixed-interface boundaries are considered. The Rayleigh quotient of the structure is defined in a coordinate space which includes the initial selection of substructure modes augmented by some of the unused modes. It is scaled and projected onto a subspace orthogonal to the lower order modes. An unconstrained minimization problem is formulated for each mode using this function. Starting points correspond to eigenvectors obtained from the initial solution. The scaling transformation tends to point the vector of steepest descent toward the minimum. An optimum step in this direction provides a lower bound on the eigenvalue error and a corresponding estimate of eigenvector error. The magnitude of the gradient vector provides another measure of the truncation error. These are compared formally and by numerical example. For small errors, the Rayleigh quotient is minimized to first-order approximation by a single step consistent with small perturbation theory. For larger errors, the Conjugate Gradient algorithm is used to drive each mode to convergence in an iterative manner.