The concepts of stability of algorithms for solving the symmetric and generalized symmetric-definite eigenproblems are discussed. An algorithm for solving the symmetric eigenproblem $Ax={lambda}x$ is stable if the computed solution z is the exact solution of some slightly perturbed system $(A+E)z={lambda}z$. We use both normwise approach and componentwise way of measuring the size of the perturbations in data. If E preserves symmetry we say that an algorithm is strongly stable (in a normwise or componentwise sense, respectively). The relations between the stability and strong stability are investigated for some classes of matrices.