- #1
Hassan2
- 426
- 5
Dear all,
I have a question about root-finding. In fact my problem is about solving system of nonlinear equations but I have simplified my question as follows:
Suppose I would like to find the root of the following function by iteration:
[itex]y=f(x,p(x))[/itex]
If I can calculate the total derivative with respect to x, I can use Newton-Raphson method effectively. However, for some reason, I can't calculate the total derivative. I can calculate the partial derivative though.
I am using the naive method of finding the root by "partial derivative". To this end, in each iteration I replace p(x) by its value evaluated from the previous value of x, treating it as a constant, and apply Newton-Raphson method. Convergence is achieved in most cases but I'm not sure if this is the right way.
My question is:
1. Is there a theory behind this method? Is it related to fixed-point iteration?
2. Does the convergence depend on p(x)?
Your help would be appreciated,
Hassan
I have a question about root-finding. In fact my problem is about solving system of nonlinear equations but I have simplified my question as follows:
Suppose I would like to find the root of the following function by iteration:
[itex]y=f(x,p(x))[/itex]
If I can calculate the total derivative with respect to x, I can use Newton-Raphson method effectively. However, for some reason, I can't calculate the total derivative. I can calculate the partial derivative though.
I am using the naive method of finding the root by "partial derivative". To this end, in each iteration I replace p(x) by its value evaluated from the previous value of x, treating it as a constant, and apply Newton-Raphson method. Convergence is achieved in most cases but I'm not sure if this is the right way.
My question is:
1. Is there a theory behind this method? Is it related to fixed-point iteration?
2. Does the convergence depend on p(x)?
Your help would be appreciated,
Hassan