CHE/COSC/MATH 4340-01 Numerical Analysis
Writing Assignment 04
Due Date: Thursday, 02/26/09
Problem 1. Using Taylor expansions for f(x+h)and f(x+k), derive the following approximation to
f′(x):
f′(x)≈k2f(x+h)−h2f(x+k) + (h2−k2)f(x)
(k−h)kh .
Problem 2. If the Secant method is applied to f(x) = x2−2 with x0=0 and x1=1, what is x2?
Problem 3. Show that the formula for the Secant method can be written in the form
xn+1=f(xn)xn−1−xnf(xn−1)
f(xn)−f(xn−1).
Explain why this is inferior to the standard formula of Secant method.
Problem 4. Assume that ris a zero of multiplicity 2 of the polynomial p(x).
•Prove that Newton’s method converges linearly.
Hint: Appropriately use Taylor expansions p(r) = p(xn−en)and p′(r) = p′(xn−en). As usual
en=xn−r.
•Now consider a variation of Newton’s method
xn+1=xn−2p(xn)
p′(xn).
Prove that it converges quadratically to r.
Hint: Appropriately use Taylor expansions p(xn) = p(r+en)and p′(xn) = p′(r+en).
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