By Magnus Rudolph Hestenes (auth.)

Shortly after the tip of worldwide warfare II high-speed electronic computing machines have been being built. It used to be transparent that the mathematical facets of com putation had to be reexamined with the intention to make effective use of high-speed electronic pcs for mathematical computations. for this reason, below the management of Min a Rees, John Curtiss, and others, an Institute for Numerical research was once manage on the college of California at la below the sponsorship of the nationwide Bureau of criteria. an identical institute was once shaped on the nationwide Bureau of criteria in Washington, D. C. In 1949 J. Barkeley Rosser turned Director of the crowd at UCLA for a interval of 2 years. in this interval we prepared a seminar at the examine of solu tions of simultaneous linear equations and at the choice of eigen values. G. Forsythe, W. Karush, C. Lanczos, T. Motzkin, L. J. Paige, and others attended this seminar. We chanced on, for instance, that even Gaus sian removal was once no longer good understood from a laptop viewpoint and that no powerful computer orientated removal set of rules have been built. in this interval Lanczos built his three-term courting and that i had the great fortune of suggesting the tactic of conjugate gradients. We dis lined later on that the elemental rules underlying the 2 techniques are basically an analogous. the concept that of conjugacy was once no longer new to me. In a joint paper with G. D.

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38 I Newton's Method and the Gradient Method We assume further that the Jacobian matrix G(x) of g(x) has rank n on our domain of search for a solution of g(x) = O. 8) S'(x) = G(x)*g(x), S"(x) = G(x)*G(x) + N(x), where N(x) has g(x) as a factor so that N(x) = 0 when g(x) = O. 9b) and Q(x) has g(x) as a factor so that Q(x) = 0 when g(x) = O. 11) without significantly altering the rate of convergence when g(x) = 0 possesses a solution. 13) G(X)-l = H(x)G(x)* = [G(x)*G(X)]-lG(X)* is the inverse of G(x) when m = n and is the pseudo inverse of G(x) when m > n.

Let m and M be, respectively, the smallest and largest eigenvalues of HA. For 0 < b S 1 we have L = 1 - b(2 - b) 4Mm (M + m)2 < 1. For a given point x set r = -F'(x), Then for b s {J s p = Hr, p*r r*Hr a=--=--. p*Ap p*Ap 2 - b we have + {Jap) - F(xo) s L[F(x) - F(x o)] U so that H = UU*. Set x = Uy and G(y) = F(Uy). 1 to G(y) and interpret the result in terms of F and H. 51 6 Gradient Methods-The Quadratic Case 12. Show that the conclusions in Exercise 11 hold when m and M are chosen to be any pair of positive numbers such that the inequality mq*H-Iq :s; q*Aq :s; Mq*H-Iq holds for every vector q.

5. Show that Newton's algorithm is invariant under a nonsingular linear transformation x = Uy. '(Xk) for f is transformed into the Newton algorithm Yk+ 1 = Yk for g. Under this transformation Xk IIU - I II IX k I. - g"(Yk)-lg'(Yk) = UYk so that IXkl ~ IIUlllYkl and IYkl ~ 6. Find a nonsingular linear transformation in function 2f = (x + 10y)2 + 5(z - W)2 tff4 which transforms the Powell + (y - 2Z)4 + 10(x - W)4 40 I Newton's Method and the Gradient Method into the function Show that the Newton algorithm for g is Yk+1 = 0, Consequently a Newton sequence for J and for g converges linearly with constant L = f.