There are many recent studies concerning reduced-order computational methods, especially reductions by projection on a small-sized basis. But it is difficult to control the quality of the solutions if the basis is fixed once and for all. This is why we attempt to define efficient and low-cost strategies for correction and updating of the basis. These correction steps re-use previously computed quantities such as: vectors and triangulated matrices. The proposed algorithms use alternately full and reduced-size steps, allowing a strong reduction in the number of full-size tangent matrices. Two classes of applications are discussed. First, we consider an algorithm for determining Hopf bifurcation points in 2D Navier-Stokes equations, but which requires time-consuming preliminary frequency-dependent calculations. New reduction-correction procedures are applied to reduce these preliminary computations. The second application concerns the response curves of viscoelastic structures. A key point is the definition of the reduced basis. Vector Taylor series are computed within the asymptotic numerical method and the relevance of this set of vectors is analysed.