By Carl E. Froberg, Carl Erik Frhoberg
E-book by way of Froberg, Carl E., Frhoberg, Carl Erik
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I a, x"-6 I . . )2 Putting x2 = y, we obtain the new equation b2Y"_2 y" + b,Y"-' + with - + ... + b" = 0 b, _ , + 2a, , b2=a,2 -2a,a3+2a4, b3 = a7 - a37 + 2aYa, - 2aa, + 2a, or (- 1)kbk = ak - 2ak-,ak+, + 2ak-2ak+2 - ... 1) The procedure can then be repeated and is finally interrupted when the double products can be neglected, compared with the quadratic terms on formation of new coefficients. Suppose that, after m squarings, we have obtained the equa- tion x" + Ax,-' + .. + A. = 0 with the roots q q5, ...
EXAMPLE x'-8x2+17x-10=0, (xx + 17x)2 _ (8x8 + 10)2, or, putting x2 = y, y'-30y=+129y-100=0. Squaring twice again, we find zs - 642z2 + 10641 z - 10" = 0 , and u' - 390882u8 + 100390881u - 108 = 0 . 999512 . The exact roots are 5, 2, and 1. A more detailed description of the method can be found in [2]; the case of complex roots has been treated particularly by Brodetsky and Smeal [3]. As has been shown by Wilkinson [9] a well-conditioned polynomial may become ill-conditioned after a root-squaring procedure; in particular this seems to affect the complex roots.
Multiplying, we obtain: x - x* (x - x0)g'(en)gV1) ... Now suppose that Ig'(e,)I S in. Then we get Ix - x*I S m* Ix - x01, and hence we have convergence if m < 1, that is, if Ig'(x)j < 1 in the interval of interest. We attain the same result by simple geometric considerations (see Fig. 5). It is also easy to give an error estimate in a more useful form. From x - x*_, = x - x* + x* - x%_1, we get, using x - x* = g(x) - g(x*_,), Ix - x*-11 S Ix - x*I + Ix* - x*-1I 5 Ix - x*-1I IgV*-01 Hence Ix - x*-1I 5 Ix* - x*-11 1-m and Ix - x*I S I m Ix* - x*-ll .