Derive a third order method for solving f(x) = 0 in a way similar to the derivation of Newton’s method, using evaluations of f(xn), f′(xn), and f′′(xn). The following remarks may be helpful in constructing the algorithm: • Use the Taylor expansion with three terms plus a remainder term. • Show that in the course of derivation a quadratic equation arises, and therefore two distinct schemes can be derived. Show that the order of convergence (under the appropriate conditions) is cubic. Can you speculate what makes this method less popular than
Derive a third order method for solving f(x) = 0 in a way similar to the derivation of Newton’s method, using evaluations of f(xn), f′(xn), and f′′(xn). The following remarks may be helpful in constructing the algorithm: • Use the Taylor expansion with three terms plus a remainder term. • Show that in the course of derivation a quadratic equation arises, and therefore two distinct schemes can be derived. Show that the order of convergence (under the appropriate conditions) is cubic. Can you speculate what makes this method less popular than
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Derive a third order method for solving f(x) = 0 in a way similar to the derivation of Newton’s
method, using evaluations of f(xn), f′(xn), and f′′(xn). The following remarks may be helpful in
constructing the algorithm:
• Use the Taylor expansion with three terms plus a remainder term.
• Show that in the course of derivation a quadratic equation arises, and therefore two distinct
schemes can be derived.
Show that the order of convergence (under the appropriate conditions) is cubic. Can you speculate
what makes this method less popular than Newton’s method, despite its cubic convergence? Give two
reasons.
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