Random Jacobi Java Applet  Andrew Trusty (gtg877q) 



Questions & Answers Q. How does the actual data compare with the theoretical bound? A. For 5x5 matrices the actual data stays relatively close under the theoretical bound for the first step or two but then progressively descends steeper than the theoretical bound with each step until it has reached the diagnolized form at which point the theoretical bound hovers near where the actual data was in its 4th to 6th step. For each increase in matrix size the actual data hovers longer close to the theoretical bound but eventually always breaks away in a progressively steeper descent. Q. What happens if you simply do not bother with the sorting step, but just "sweep through" the upper right entries in a systematic fashion? A. Without the sorting step the resulting data follows closer to the theoretical bound for more of the beginning steps. For some matrices the results even break the theoretical bound going above it for a few steps. The rate of descent of this data is also differs from that of the sorting data in that the nonsorting data is characterized by smooth periods of almost level descent followed by great leaps toward the diagnolized form which repeat until it reaches the diagnolized form whereas the sorting data shows a smooth constantly increasing descent. The nonsorting algorithm also takes more steps to reach the diagnolized form, ranging anywhere from 511 extra steps for a given matrix. For larger matrices the effects worsen with much longer periods spent above the theoretical bound. Q. What do you conclude about the importance of sorting versus its expense? A. Sorting is an important part of the Jacobi Algorithm that allows it to choose a better step for each iteration than a systematic method. The cost of sorting inside the algorithm is small as in a symmetric matrix only the offdiagnol elements on either the uppper or lower triangular area need to be searched for the largest element which eliminates over half the elements in a given matrix. Sorting keeps the algorithm performing approximately 25% fewer steps than the systematic approach. 