Linearly implicit Runge–Kutta methods with approximate matrix factorization can solve efficiently large systems of differential equations that have a stiff linear part, e.g. reaction–diffusion systems. However, the use of approximate factorization usually leads to loss of accuracy, which makes it attractive only for low order time integration schemes. This paper discusses the application of approximate matrix factorization with high order methods; an inexpensive correction procedure applied to each stage allows to retain the high order of the underlying linearly implicit Runge–Kutta scheme. The accuracy and stability of the methods are studied. Numerical experiments on reaction–diffusion type problems of different sizes and with different degrees of stiffness illustrate the efficiency of the proposed approach.