This paper describes several enhancements to accelerate the direct determination of the periodic states of fixed-bed adsorption cycles. The new methods make it possible to calculate the periodic states up to 40 times faster than our original formulation (Croft & LeVan, Chem. Eng. Sci. 49 (1994) 1821) and in many cases 100-1000s of times faster than by just running cycle after cycle. We have tested the methods on several examples and have obtained fast, stable, robust convergence. The enhancements are (i) the use of a hybrid Newton-Broyden method to reduce some of the Jacobian matrix calculation processes required by Newton's method. (ii) the initial application of a novel iterative-secant approach to increase stability and to avoid the calculation of the first Jacobian matrix for the hybrid Newton-Broyden method, (iii) the implementation of a sensitivity interpolation technique with dynamic grid allocation, and (iv) the dynamic specification of integration error tolerances.