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:
“To become a token woman—whether you win the Nobel Prize or merely get tenure at the cost of denying your sisters—is 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)
“History is strewn with the wrecks of nations which have gained a little progressiveness at the cost of a great deal of hard manliness, and have thus prepared themselves for destruction as soon as the the movements of the world gave a chance for it.”
—Walter Bagehot (1826–1877)
“I knew that the wall was the main thing in Quebec, and had cost a great deal of money.... In fact, these are the only remarkable walls we have in North America, though we have a good deal of Virginia fence, it is true.”
—Henry David Thoreau (1817–1862)