Fitting Linear Models: An Application of Conjugate Gradient by Allen Mclntosh (auth.)

By Allen Mclntosh (auth.)

The expanding energy and lowering fee of smalI desktops, especialIy "personal" pcs, has made them more and more well known in statistical research. The day is probably not too far flung whilst each statistician has on his or her computer computing strength on a par with the big mainframe pcs of 15 or two decades in the past. those related components make it really effortless to obtain and manage huge amounts of information, and statisticians can anticipate a corresponding elevate within the dimension of the datasets that they have to study. regrettably, due to constraints imposed via structure, dimension or fee, those smalI desktops don't own the most reminiscence in their huge cousins. therefore, there's a starting to be desire for algorithms which are sufficiently low-budget of area to allow statistical research on smalI pcs. One sector of study the place there's a desire for algorithms which are cost-efficient of house is within the becoming of linear models.

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See for instance Ortega and Rheinboldt (1970) chapter 14. - 40- lion independent the algorithm converges to a minimizer of cJ>(P) regardless of the parameterization. A useful convergence criterion might involve the change in cJ>, or in u(kl'u(kl = u(kl'q(kl. Golub and Nash (I980) discuss this in the context of the precoriditioned conjugate gradient algorithm discussed below. It is possible to apply Hemmerle's preconditioning scheme to the conjugate gradient algorithms dis- cussed above. 5 of appendix A.

3. For the Hestenes and Stiefel method, 13(H \} = Qk (G )g(O) where Qk(') is a polynomial in G of degree k. p over all such polynomials. The result now follows because G-1 can be written as a polynomial in G of degree m - 1. Unfortunately, the proof does not seem to generalize to Beale's algorithm when pro) is arbitrary. 3 given above can be generalized, but the upper bound is no longer m. 6, we obtain - 32 - p(k)=-g(O)-a(O)Gp(O)- k-I a(j)Gp(j) +a(k)p(k-l)+d(k)p(O) j=1 Since g(O) and pro) need not have any eigenvectors in common, the best that we can do is assert that the search directions lie in a space of dimension 2m.

7 and d(k) - - ! ) _ g s, ) . otherWIse We will refer to this as the Beale algorithm. 2, and we will refer to this as the Hestenes-Stiefel algorithm. The mutual orthogonality of the search directions has a number of useful consequences which we present in the following theorems. The results apply to quadratic restarting is done. q,. p on the linear manifold where pro) ... 3, and an exact line search has been done to find a(O) ••• a(k- I). 3) with equality if and only if (I i = a (j). Summing this for i = 0, I, ...

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