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The modeling of High Temperature Superconductor (HTS) devices has presented considerable challenges over the years. One of the main difficulties arises due to the high degree of non-linearity associated with the HTS materials. Iterative techniques are applied to deal with such nonlinearities, and construct computational prototypes and solvers. The conventional approaches such as the Newton-Raphson (NR) method, and the Successive Substitution (SS) method have been applied in the past but they have shortcomings of being unstable in certain scenarios, and at times, are very slow to converge. This paper presents a new algorithm which attempts to address these issues based on the ideas of the Aitken technique.