On Producing Multiple Solutions Using Repeated Trials

The number of repeated trials that is required by an algorithm to produce a given fraction of the problem solutions with a speciied level of conndence is analyzed. The analysis indicates that the number of trials required to nd a large fraction of the solutions rapidly decreases as the number of solutions obtained on each trial by an algorithm increases. In applications where multiple solutions are sought, this decrease in the number of trials could potentially ooset the additional computational cost of algorithms that produce multiple solutions on a single trial. The analysis framework presented is used to compare the eeciency of a homotopy algorithm to that of a Newton method by measuring both the number of trials and the number of calculations required to obtain a speciied fraction of the solutions.