Cost
Each Jacobi rotation can be done in n steps when the pivot element p is known. However the search for p requires inspection of all N ≈ ½ n2 off-diag elements. We can reduce this to n steps too if we introduce an additional index array with the property that is the index of the largest element in row i, (i = 1, …, n − 1) of the current S. Then (k, l) must be one of the pairs . Since only columns k and l change, only must be updated, which again can be done in n steps. Thus each rotation has O(n) cost and one sweep has O(n3) cost which is equivalent to one matrix multiplication. Additionally the must be initialized before the process starts, this can be done in n2 steps.
Typically the Jacobi method converges within numerical precision after a small number of sweeps. Note that multiple eigenvalues reduce the number of iterations since .
Read more about this topic: Jacobi Eigenvalue Algorithm
Famous quotes containing the word cost:
“The one regret I have about my own abortions is that they cost money that might otherwise have been spent on something more pleasurable, like taking the kids to movies and theme parks.”
—Barbara Ehrenreich (b. 1941)
“To become a token womanwhether you win the Nobel Prize or merely get tenure at the cost of denying your sistersis to become something less than a man ... since men are loyal at least to their own world-view, their laws of brotherhood and self-interest.”
—Adrienne Rich (b. 1929)
“Every new stroke of civilization has cost the lives of countless brave men, who have fallen defeated by the dragon, in their efforts to win the apples of the Hesperides, or the fleece of gold. Fallen in their efforts to overcome the old, half sordid savagery of the lower stages of creation, and win the next stage.”
—D.H. (David Herbert)