3. A variant of the secant method defines two sequences uk and vk such that f(u)
ID: 2254983 • Letter: 3
Question
3. A variant of the secant method defines two sequences uk and vk such that f(u) has one sign and f(vk) has the opposite sign. From these sequences and the secant method one can derive the expression ukruk)-wf(uk), k = 1,2,3, wk /(Uk)-/(uk) We define uk+1 = uk and uk +1 = Uk 1f/(uk)/(uk) > 0 and uk+1 = uk and Uk+1 = uk otherwise. Suppose that f" is continuous on the interval [uo, vo] and that for some K, f" has a constant sign in [uK,UK]. Explain why either uk = uK for all k > K or th UK for all k 2 K. Deduce that the methods converges Ii nearly.Explanation / 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)
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.