This is a programming project for Calculus 3 for CS majors @ Georgia Tech. I'm using Matlab to program a random 5x5 matrix, A, to be run through the Jacobi Algorithm until the Off of the diagonalized matrix, B, is < 10^(-9). I've got the algorithm programmed, but I don't understand this next part of the assignment.
"If bk denote the value of ln(Off(B)) after the kth partial diagonalization, we will have the theoretical bound bk =< k ln (9/10) + ln (Off(A)) : (The value 9/10 comes from 1−2/(n^2 −2n) with n = 5). Plot your points (k; bk) together with a graph of the line y = x ln (9/10) + ln (Off(A)) : do this for 10 randomly generated matrices. How does the actual data compare with the theoretical bound?"
So I understand that for the theoretical bound, bk is to be substituted for ln(Off(B)), which refers to the diagonalized matrix, and k is the number of iterations necessary to achieve an Off(B) value < 10^(-9).
For the second equation, what do Y and X correspond to??? It seems that they've just replaced bk and k, respectively, but that wouldn't make sense. Also, do you think I would have to recalculate the theoretical bounds for each of those 10 matrices?
I'm so confused, and I'm stuck at a hotel for my cousin's wedding. Everyone expects me to attend the rehearsal dinner... ugh.
"If bk denote the value of ln(Off(B)) after the kth partial diagonalization, we will have the theoretical bound bk =< k ln (9/10) + ln (Off(A)) : (The value 9/10 comes from 1−2/(n^2 −2n) with n = 5). Plot your points (k; bk) together with a graph of the line y = x ln (9/10) + ln (Off(A)) : do this for 10 randomly generated matrices. How does the actual data compare with the theoretical bound?"
So I understand that for the theoretical bound, bk is to be substituted for ln(Off(B)), which refers to the diagonalized matrix, and k is the number of iterations necessary to achieve an Off(B) value < 10^(-9).
For the second equation, what do Y and X correspond to??? It seems that they've just replaced bk and k, respectively, but that wouldn't make sense. Also, do you think I would have to recalculate the theoretical bounds for each of those 10 matrices?
I'm so confused, and I'm stuck at a hotel for my cousin's wedding. Everyone expects me to attend the rehearsal dinner... ugh.