By J. B. Tatum
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Extra info for Celestial Mechanics Vol.1
A simple application of these considerations arises if you have to solve a polynomial equation f(z) = 0, where there are no real roots, and all solutions for z are complex. You then merely write z = x + iy and substitute this in the polynomial equation. 22 and solve them for x and y. For example, find the roots of the equation z4 − 5z + 6 = 0. 25 It will have been observed that, in order to obtain the last equation, we have divided through by y, which is permissible, since we know z to be complex.
5 x8 y8 Let us suppose that we want to find y for a value of x that is a fraction θ of the way from x4 to x5. 3 Here the Bn are the Besselian interpolation coefficients, and the successive terms in parentheses in the expansion are the sums of the numbers in the boxes in the table. 4 if n is odd. 5 ( ) means the coefficient of xm in the binomial expansion of (1 + x)n. m n B1 = θ − 12 B2 = 1 2 θ(θ − 1) / 2 ! 6 = B3 = (θ − 12 )θ(θ − 1) / 3! 8 34 B4 = 1 2 (θ + 1)θ(θ − 1)(θ − 2) / 4 ! = θ(2 + θ(−1 + θ(−2 + θ))) / 48 B5 = (θ − 12 )(θ + 1)θ(θ − 1)(θ − 2) / 5 !
538 469 310 106 and 0. 14. Let’s try it. 000 000 000 04, and might presumably have come even closer to 1 had we given xl , i and cl , i to more significant figures. You should now write a computer program for Gaussian quadrature – you will have to store the xl , i and cl , i , of course. You have presumably already written a program for Simpson’s rule. 35 2 0 π/4 0 0 1 3 π/ 4 0 e 3 x sin 2 x dx 2x dx x −4 x dx 2 x −4 2 cos 2 x dx 52 All of these can be integrated analytically, so I am going to invite the reader to evaluate them first analytically, and then numerically by Simpson’s rule and again by Gaussian quadrature, and to see at how many points the integrand has to be evaluated by each method to achieve nine or ten figure precision.
Celestial Mechanics Vol.1 by J. B. Tatum