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Because of its stable solution and despite its slow convergence, the fixed-point technique is commonly used for solving hysteretic field problems. In this paper, we propose a new method for accelerating the convergence of the fixed-point technique in solving time-stepping nonlinear field problems. The method ensures locally convergent iteration in an interval that contains the initial value and the fixed-point solution. We provide a thorough discussion and geometric interpretation to clarify and highlight the principle of the method. We also use a finite-element formulation to test the method by computing the magnetic field of an electric machine. Finally, we assess the efficiency and applicability of the method by a comparative investigation. The method proves to be simple and remarkably fast.