By Magnus Rudolph Hestenes (auth.)

Shortly after the top of worldwide battle II high-speed electronic computing machines have been being constructed. It used to be transparent that the mathematical facets of com putation had to be reexamined as a way to make effective use of high-speed electronic pcs for mathematical computations. therefore, below the management of Min a Rees, John Curtiss, and others, an Institute for Numerical research was once manage on the collage of California at l. a. lower than 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 gang at UCLA for a interval of 2 years. in this interval we geared up a seminar at the learn of solu tions of simultaneous linear equations and at the selection of eigen values. G. Forsythe, W. Karush, C. Lanczos, T. Motzkin, L. J. Paige, and others attended this seminar. We stumbled on, for instance, that even Gaus sian removing was once no longer good understood from a laptop perspective and that no powerful laptop orientated removal set of rules were constructed. in this interval Lanczos constructed his three-term dating and that i had the nice fortune of suggesting the strategy of conjugate gradients. We dis lined in a while that the fundamental principles underlying the 2 systems are basically an identical. the idea that of conjugacy used to be no longer new to me. In a joint paper with G. D.

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The Hessian is positive definite when y < x 2 + e/2 and indefinite when y > x 2 + e/2. The singular curve y = x 2 + e/2 passes through the point (1, 1 + e/2), which is quite close to the minimum point (1, 1), The Newton algorithm for f is given by _ x=x+ e(l - x) J ' _ y -- x 2 2ex(1 - x) + ------'-J' With ( - 2, 1) as the initial point we obtain the following results. 999998 2 X 10- 12 5 X 10- 6 1 1 0 0 Observe that in this case the function f does not descrease monotonically. It was the experience of the author that restoring monotonicity of f by introducing a relaxation factor considerably slowed the rate of convergence.

The Newton sequence {(Xk' Yk)} converges to the minimum point (2, 0) when Xl > b and to the minimum point (-2,0) when Xl < -b. When -b < Xl < b, the sequence {(Xk' Yk)} can converge to any of the points ( - 2, 0), (0, 0), and (2, 0). It will converge to (0, 0) if and only if -c < Xl < c, where c = (~)1/2, and fails to converge when Xl = ±c. 5) with ak = 1. When Xk is sufficiently close to a non degenerate minimum point Xo we always have f(xk+ 1) < f(xk)' However, occasionally we can have f(xk+ 1) ~ f(xk) if Xk is not close to Xo.

In the 2-dimensional xy-plane the function F(x,y) = -2x + x 2 + y2 is a quadratic approximation at (0, 0) of each of the functions Jl(X, y) = F(x, y) ° + X4 + y\ J2(X, y) = F(x, y) - X4 - y\ J3(X, y) = F(x, y) + x 3 + y3. The circle F(x, y) = encloses the level curveJl(x, y) = 0, is enclosed by the level curve J2(X, y) = 0, and intersects the level curve J3(X, y) = 0, as one readily verifies. 2. A level surface ofJean have several components. To illustrate this phenomenon we consider the function EXAMPLE J(x, y) = (x 2 - 4)2 + 4y2 of two real variables x and y.