Least-mean-square (LMS) algorithms for adaptive infinite impulse response (IIR) filters are proposed which sequentially adjust the parameters of the feedforward and feedback paths so as to minimise the equation error. Both the feed-forward and feedback paths are implemented in cascade second-order sections. However, instead of adapting the coefficients of the second-order sections, the actual root locations are adjusted either in cartesian or in polar co-ordinates. Prequantising the movements of the roots eliminates the need for the arbitrary convergence factors used in normal LMS algorithms. Hardware implementations of the adaptive filters have been investigated, and new filter structures for second-order filters are also proposed. An efficient algorithm for adapting the roots of second-order filters in cascade is given. This algorithm combines a simple gradient approximation together with the prequantisation of root movements. Simulations are conducted to assess the performance of the proposed algorithms.