By Charles Hutton

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The spaces generated in this manner are known as the Krylov subspaces. The algorithm, as described would rapidly become unworkable since the search space increases in size on each iteration. However, due to a truly amazing result which we do not have time to prove, it turns out that for quadratics, it is possible to achieve the same result as searching over all the space spanned by all the previous search directions by searching …rstly along ¡rQ (x0 ) and subsequently over only a two-dimensional space on each iteration.

On the next iteration the value of Q is recalculated at this point and it is a good check to see that the actual value agrees with the predicted value. M. Tan and Colin Fox, The University of Auckland 3-12 S(:,1) = 2*res; % Search direction along negative gradient [xnew,Qpred] = search1(x0,res,Hfunc,Qnow,S); S(:,2) = xnew - x0; % Second search direction x0 = xnew; keyboard % Type ’’return’’ to continue to next iteration end Notice how the matrix of search directions S is set up. On the …rst iteration, it consists of a single column containing 2 (G ¡ Hx0 ) : This is the negative gradient at the starting guess.

Now AA∗ = (1) which has one eigenvalue of 1 with eigenvector (1) . , the constant function with value 1. The left-hand side is the matrix A ∗ acting on the data vector (1) ,and since A = a∗1 (x) = 1∗ , in this case, then A∗ = a1 (x) = 1 (x) and the function 1 (x) acting on the number 1 simply results in the function 1 (x) which is the constant function with value 1. M. 7 The three right singular vectors       0. 82706 0. 54744 0. 12766 which has the normalized eigenvectors: u1 =  0. 46039 , u2 =  −0.