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consider a problem of finding the roots of the equation given
by
f(x) = x - 2 sin x
f'(x) = 1 - 2 cos x
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xn+1 = 2( sin xn - xn cos
xn)/ ( 1 - 2 cosxn)
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let e is the error calculating xn such that
e = s - xn, where s is the correct solution
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using Taylor's series (1) can be written as xn+1 =
g(xn) = g(s) + g'(s)(xn - s) +
(1/2!) g''(s)(xn - s)2 +
...
= g(s) - eg'(s) +
(1/2)e2g''(s) + ...
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For Newton-Raphson method g'(x) =
f(x)f''(x)/f'(x)2
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since f(s) = 0, then g'(s) = 0. This shows that
Newton-Raphson method is of second order
g''(s) = f''(s)/f'(s)
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en+1 = -f''(s) e2/2
f'(s)
we see that the error at a particular stage is a multiple of
the square of the error at the previous stage