. A variant of the secant method defines two sequences u and vk such that f(u) h
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Question
. A variant of the secant method defines two sequences u and vk such that f(u) has one sign and f(%) has the opposite sign. From these sequences and the secant mothod one can derive the expression ue = uk/u)-rkf(uk), k = 1,2,3, f(vx) (k) We define uk +1 vk and vk +1 = t/k if f(tok)f(uk) > 0 and uk +1 1k and vk +1 vk otherwise. Suppose that f" is continuous on the interval uo, vol and that for some K, f" has a constant sign in uK,vK]. Explain why either K for al k2 K or vk -vK for all k2 K. Deduce that the methods converges lincarlyExplanation / Answer
ANSWER:
The Newton method is based on approximating the graph of y = f(x) with a tangent line and on then using a root of this straight line as an approximation to the root of f(x).
From this perspective, other straight-line approximation to y = f(x) would also lead to methods of approximating a root of f(x). One such straight-line approximation leads to the secant method.
To derive a formula for x2, we proceed in a manner similar to that used to derive Newton’s method: Find the equation of the line and then find its root x2. The equation of the line is given by y = p(x) f(x1) + (x x1) · f(x1) f(x0) x1 x0 Solving p(x2) = 0,
we obtain x2 = x1 f(x1) · x1 x0 f(x1) f(x0) . Having found x2, we can drop x0 and use x1, x2 as a new set of approximate values for . This leads to an improved values x3; and this can be continued indefinitely. Doing so, we obtain the general formula for the secant method xn+1 = xn xn xn1 f(xn) f(xn1) , n 1.
It is called a two-point method, since two approximate values are needed to obtain an improved value. The bisection method is also a two-point method, but the secant method will almost always converge faster than bisection.
the secant method can be considered as an approximation of Newton’s method, based on using
f 0 (xn) f(xn) f(xn1)/( xn xn1 ).
xk+1=xkf(xk)f(xk)=(xk)
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