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Magnetic fields render a unique ability to control magnetic objects without a direct mechanical contact. To exploit this potential for a broad range of medical, microrobotics, and microfluidics applications, noncontact magnetic manipulators have been designed using both electromagnets and permanent magnets. By feedback control of these manipulators, magnetic objects can be precisely driven in the directions required by an application of interest. The feedback design process for these manipulators is normally complicated by their highly nonlinear nature, particularly for those utilizing permanent magnets. Yet, feedback linearization techniques can be applied to compensate for the nonlinear nature of most magnetic manipulators. This goal can be achieved by solving an underdetermined system of nonlinear algebraic equations. This paper adopts a homotopy continuation approach to solve this system of equations. It is shown by simulations that the proposed feedback linearization scheme drastically improves the control performance compared to the alternative control design methods used in prior work.