How would I carry out the newton method on the non-linear system below:
x_{1}^{2} - 10x_{1} + x_{2}^{2} + 8 = 0
x_{1}x_{2}^{2} + x_{1} - 10x_{2} + 8= 0
Perform one iteration using (0,0)^{T}
What are the steps i have to do?
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", , and approximate y= f(x) with its tangent line approximation, . Setting that equal to 0 and solving for x, giving , 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 and its derivative is the matrix so that, for a given starting point the next point would be .