Elements of Numerical Analysis by James Singer

By James Singer

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B 10 = 5/66. B12 = -691/2730.... ; Ba = B6 = B7 = ... = B 2n+! = ... = 1. Bl 1/2. B2 = = Bo 1/6. Be -1/30. Be = 1/42. O. The B's are called Bernoulli numbers; they occur frequently in power series and other computational formulas. Sometimes. the odd indexed Bernoulli numbers that are equal to zero are deleted; the remaining ones alternate in sign. The signs are dropped and the numbers are reindexed. We have. in the new classification. Bo = 1. Bl = 1/2. B2 = 1/6. Ba = 1/30. Be = 1/42..... We shall use the Bernoulli numbers as originally defined.

4. /( 1 - x)nH. 2:8 for the error term becomes (xo is again 0), (n x n +1 n! 1)1 (1 - X)n+1 + x n +1 - (n + 1)(1 - X)n+1 • t We have x = i; further, X must be chosen between 0 and so that the magnitude of the error is as large as possible, hence X is also t. 000005. The first n is n = 200,000. On the other hand, we have x n +1 x n +2 < - - + - - + ... n+I n+I x n +1 n+1 = --(1 + x n +1 x + x 2 + ... 000005. This time, n = 14. This estimate of the error gives us a far better result in the shape of a much smaller n than the previous one.

Put x x f(x) =~+2. Provef(x) = f( -x) and then show that B ln +1 = 0, n = 1,2,3, .... t2. Let f(x) = Qo + QIX + ... + QnXn be a polynomial of degree n and let Xo be any constant. S. (x) q"_I(X) q"_I(X) = = +b +b l , l , (x - xo) q"_I(X) + b"_l , b". Use these equations to prove that k = 0, 1, 2, ... (x - xo)S + ... + b,,(x - x o)". Hence justify Horner's method for the determination of the real roots of a polynomial equation. tl. More generally, let f(x) be the polynomial of the previous example and let Xo, Xl , ...

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