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Algebraic topology by Edwin H. Spanier

By Edwin H. Spanier

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Sin : current stepsize. • r: step adjustment factor, between 0 and 1, exclusive. Output: • Pout : new precision. • sout : new stepsize. Algorithm: Initialize Pout := Pin and sout := r ∗ sin . While sout < ε(Pout ) Increment Pout to the next available precision. The safety error and step success algorithms both rely upon an algorithm, called minimize cost, to adjust the precision and stepsize to avoid stalling on the path, satisfy Eq. 4, and reduce the cost. The precision selected by this algorithm is between the current precision and the maximum precision, inclusively, and the stepsize selected is not larger than the current stepsize.

We write ε(L; x) = ε(X, L; x) etc. There are another three interesting numbers which can be defined taking infimums over various spaces of parameters. 12 (Seshadri constants of a line bundle, a point and a variety). (a) The number ε(X, L) := inf ε(X, L; x) x∈X is the Seshadri constant of the line bundle L. (b) The number ε(X; x) := inf ε(X, L; x) L ample is the Seshadri constant of the point x ∈ X. (c) The number ε(X) := inf L ample ε(X, L) = inf ε(X; x) x∈X is the Seshadri constant of the variety X.

STEPSIZE CONTROL FOR PATH TRACKING 25 5 The algorithm is outlined in the flowchart presented in Figure 1. Some key parts presented in the diagram require additional explanation. As in [1], there is a maximum number of steps and a maximum precision, Pmax , allowed. These limits prevent computational waste and guarantee the termination of the algorithm. “Setup f (z, t)” constructs f (z, t) described by Eq. 3 combining the prediction step and correction steps. If the step in t would go beyond tf , then the stepsize is adjusted to land exactly on tf .

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