i) Start from a known solution at given (simple) parameter
values.

ii) Use a solution computed at a nearby set of parameter values to generate an initial guess to feed to an iterative  scheme to compute the solution at a new set of parameter values.

iii) Repeat step ii) as often as desired.

Thus (hiding all of the many substantial difficulties underlying an efficient implementation) we can describe a parameter continuation algorithm as a numerical implementation of a homotopy between solutions of different problems.

 

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