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An iterative algorithm for determining a class of solutions of the dispersionful 2-Toda hierarchy characterized by string equations is developed. This class includes the solution which underlies the large N-limit of the Hermitian matrix model in the one-cut case. It is also shown how the double scaling limit can be naturally formulated in this schemedoi:10.1088/1751-8113/40/47/013 fatcat:c53ikmivnfcxpihy4cfbm7e4wa