Exercise 6.6.7 additional info on next page

Exercise 6.6.7 Let (ug) be a Ritz pair of A from some subspace. Show that i = q* Aq/q*q. Thus w is the Rayleigh quotient of q. Show that if s is determined by the Jacobi- Davidson equation (6.6.2), then q+s = v(A-ul)-'q. Therefore q+s is the result of one step of Rayleigh quotient iteration with starting vector q. AND RELATED ALGORITHMS 467 A second possibility is to take (D-u)r, where is the diagonal matrix span{919). Then ex+1 is obtained by Orthonormalizing s against 91. by the Gram-Schmidt process. The methods differ in how they obtains from The simplest possibility is to take ser. This leads to a method that is equivalent the Amoldi process. See Exercise 6.6.6. that has the same main-diagonal entries as A. This leads to Davidson's method, which dus heen used extensively in quantum chemistry calculations. In these applications the matrices are symmetric and extremely large. They are also strongly diagonally want, which means that the main-diagonal entries are much larger than the entries Notice that the computation of 8 is quite inexpensive, as (D-AT-1 is a diagonal w the main diagonal. This property is crucial to the success of Davidson's method. as follows. If q is close to an eigenvector of A, then a small corrections can make A third way of choosing a leads to the Jacobi-Davidson method, which we motivate an exact eigenvector. Thus mr. A(q + 5) = (+ 1)(+3), (6.6.1) currection to be orthogonal to g.1.e. *8 = 0. The Jacobi-Davidson method chooses where it is a small correction to the Ritz value . Furthermore, we may take the approximation to g. Suppose= O(6) and V = O(e), where