You are here:
- Home
- Teens
- Homework/Study Tips
- Calculus
- Newton-Raphson method
Calculus/Newton-Raphson method
Advertisement
Question
Dear Scotto,
I have this problem in Newton Raphson method and would like your help on how to solve it so as to understand some things better.
The problem aks us to demonstrate that the Newton-Raphson method iteration function for a(x) = x^(-1) - A, where A > 0, is
φ(x) = x(2 - Ax).
In addition, we have to check that X = A−1 is a fixed point of φ(x).( Χ is the solution to the function)
The Newton-Raphson method is x(n+1) = xn - a(xn)/a'(xn) where xn is on term in the series
and x(n+1) is the next term.
In this problem, we are give a(x) = 1/x - A. This says a'(x) = -1/x².
Taking xn as just x, the method says that the next term will be x - a(x)/a'(x),
and that is x - (1/x - A)/(-1/x²). Inverting and multiplying gives x + x - Ax².
Combining terms gives us 2x - Ax².
Factoring out x gives us .... the answer.
If X = A-1, the a(x) = 1/(A-1) - A = (1 + A(A-1))/(A-1) = (A² - A + 1)/(A-1).
Looking at φ(x) for x=A-1, we get
φ(x) = (A-1)(2-A(A-1))
= (A-1)(2-A²+A)
= -(A-1)(A²-A-2)
V = -(A-1)(A-2)(A+1), which is different from A-1, so A-1 is not a solution.
Related Articles
Answers by Expert:
Scotto
Expertise
Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.
Experience
Experience in the area: I have tutored students in all areas of mathematics for over 25 years.
Education/Credentials: BSand MS in Mathematics from Oregon State University, where I completed sophomore course in Physics and Chemistry. I received both degrees with high honors.
Awards and Honors: I have passed Actuarial tests 100, 110, and 135.
Publications
Maybe not a publication, but I have respond to well oveer 7,500 questions on the PC.
Well over 2,000 of them have been in calculus.
Education/Credentials
I aquired well over 40 hours of upper division courses. This was well over the number that were required.
I graduated with honors in both my BS and MS degree from Oregon State University.
I was allowed to jump into a few junior level courses my sophomore year.
Awards and Honors
I have been nominated as the expert of the month several times.
All of my scores right now are at least a 9.8 average (out of 10).
Past/Present Clients
My past clients have been students at OSU, students at the college in South Seattle, referals from a company, friends and aquantenances, people from my church, and people like you from all over the world.
©2012 About.com, a part of The New York Times Company. All rights reserved.