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The reasons a class of codes for solving ordinary differential equations might want to use an extremely small step size are investigated. For this class the likelihood of precision difficulties is evaluated and remedies examined. The investigation suggests a way of selecting automatically an initial step size which should be reliably on scale.doi:10.1090/s0025-5718-1978-0501936-x fatcat:5tjwn6uhmvdm7dpuawne76abka