Discrete Math , Eleventh Problem Set ( July 14 )

Recall that the “coefficient reduction” process reduces each μij to |μij | ≤ 1/2 without affecting the orthogonalized sequence and therefore without affecting the potential function. Once “coefficient reduction” has been performed, we check the second property. If there is some i with ‖bi+1‖ < 1 2‖b ∗ i ‖, we simply swap these two elements. This will change the orthogonalized vectors bi and b ∗ i+1, but the potential is only affected by: Pnew Pold = ‖(bi )new‖ ‖(bi )old‖