The objective of this study is to develop an algorithmic approach for solving problems associated with the convergence to the local minima in adaptive IIR filtering. The approach is based on a numerical method called the homotopy continuation method;The homotopy continuation method is a "solution exhaustive method" for calculating all solutions of a set of nonlinear equations. The globally optimum filter coefficients correspond to the solutions with minimum mean square error. In order to apply the technique to the adaptive IIR filtering problem, the homotopy continuation method is modified to handle a set of nonlinear polynomials with time-varying coefficients. Then, the adaptive IIR filtering problem is formulated in terms of a set of nonlinear polynomials using the mean square output error minimization approach. The adaptive homotopy continuation method (AHCM) for the case of time-varying coefficients is then applied to solve the IIR filtering problem. After demonstrating the feasibility of the approach, problems encountered in the basic AHCM algorithm are discussed and alternative structures of the filter are proposed. In the development of the proposed algorithm and its variations, the instability problem which is a second disadvantage of IIR filters is also considered;Simulation results for a system identification example validate the proposed algorithm by determining the filter coefficients at the global minimum position. For further validation, the AHCM algorithm is then applied to an adaptive noise cancellation application in ultrasonic nondestructive evaluation. Ultrasonic inspection signal reflections from defects and material grain boundaries are considered. The AHCM algorithm is applied to the noise cancellation mode to filter out the material noise. The experimental results show that the proposed algorithm shows considerable promise for real as well as for simulated data.