(a) Newtons method for approximating a root of an equation

Chapter , Problem 6

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(a) Newtons method for approximating a root of an equation fsxd 0 (see Section 4.8) can be adapted to approximating a solution of a system of equations fsx, yd 0 and tsx, yd 0. The surfaces z fsx, yd and z tsx, yd intersect in a curve that intersects the xy-plane at the point sr, sd, which is the solution of the system. If an initial approximation sx1, y1d is close to this point, then the tangent planes to the surfaces at sx1, y1d intersect in a straight line that intersects the xy-plane in a point sx2, y2 d, which should be closer to sr, sd. (Compare with Figure 4.8.2.) Show that x2 x1 2 fty 2 fy t fx ty 2 fy tx and y2 y1 2 fx t 2 ftx fx ty 2 fy tx where f, t, and their partial derivatives are evaluated at sx1, y1d. If we continue this procedure, we obtain successive approximations sxn, yn d. (b) It was Thomas Simpson (17101761) who formulated Newtons method as we know it today and who extended it to functions of two variables as in part (a). (See the biography of Simpson on page 520.) The example that he gave to illustrate the method was to solve the system of equations xx 1 yy 1000 xy 1 yx 100 In other words, he found the points of intersection of the curves in the figure. Use the method of part (a) to find the coordinates of the points of intersection correct to six decimal places. 7et14ppx06 05/12/10 MasterID: 01651 FIGURE FOR PROBLEM 6 y 4 2 0 2 4 x