In the assessment of the minimal eigenvalue of large sparse positive definite matrices the shift method easily fails to give successful results since its rate of convergence may prove unacceptably slow. A very efficient alternative approach is provided by a double iterative scheme wherein external and internal iterations are performed by the reverse power and the modified conjugate gradients, respectively. The wider the spectral interval, the greater the success of this technique whose efficiency in engineering practice therefore increases with the problem size N. The results obtained with large (800 ⩽ N ⩽ 2200) space positive definite matrices, arising from the finite element integration of flow and elasticity equations, show that an accurate estimate of the minimal eigenvalue is achieved after very few (<10) external iterations and a number of internal iterations of the order of √ N.