Newton Method

**NFS1** · July 26th 2012, 06:37 AM

How would I carry out the newton method on the non-linear system below:

x₁² - 10x₁ + x₂² + 8 = 0

x₁x₂² + x₁ - 10x₂ + 8= 0

Perform one iteration using (0,0)^T

What are the steps i have to do?

**Prove It** · July 26th 2012, 06:52 AM

Originally Posted by NFS1

How would I carry out the newton method on the non-linear system below:

x₁² - 10x₁ + x₂² + 8 = 0

x₁x₂² + x₁ - 10x₂ + 8= 0

Perform one iteration using (0,0)^T

What are the steps i have to do?

Newton's method - Wikipedia, the free encyclopedia

**HallsofIvy** · July 26th 2012, 07:37 AM

I find it surprising that you would have a problem like this if you do not know "the steps".

In any case, the "Newton-Raphson" method for solving, say, f(x)= 0, is to start with some "test value", $x_0$ , and approximate y= f(x) with its tangent line approximation, $y= f'(x_0)(x- x_0)+ f(x_0)$ . Setting that equal to 0 and solving for x, giving $x= x_0- \frac{f(x_0)}{f'(x_0)}$ , will, under reasonable conditions, give a new value of x, closer to the root. Then we can repeat to get even closer.

Here, f is "vector" function $f(x_1, x_2)= \begin{pmatrix}x_1^2- 10x_1+ x_2^2+ 8 \\ x_1x_2^2+ x_1- 10x_2+ 8\end{pmatrix}$ and its derivative is the matrix $\begin{pmatrix}2x_1- 10 & 2x_2 \\ x_2^2+ 1 & 2x_1x_2- 10\end{pmatrix}$ so that, for a given starting point $(x_1, x_2)$ the next point would be $\begin{pmatrix}x_1\\ x_2\end{pmatrix}- \begin{pmatrix}2x_1- 10 & 2x_2 \\ x_2^2+ 1 & 2x_1x_2- 10\end{pmatrix}^{-1}\begin{pmatrix}x_1^2- 10x_1+ x_2^2+ 8 \\ x_1x_2^2+ x_1- 10x_2+ 8\end{pmatrix}$ .

Thread: Newton Method - HELP!

LinkBack

Thread Tools

Display

Newton Method - HELP!

Re: Newton Method - HELP!

Re: Newton Method - HELP!

Similar Math Help Forum Discussions

Newton's method

Newton's method

newton's method

Newton's method

Search Tags