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laniik science forum beginner
Joined: 19 Jun 2006
Posts: 4

Posted: Fri Jul 14, 2006 6:38 pm Post subject:
Iterative solution to nonlinear equations



Hi, I have three equations with three unknowns. All of the equations
are nonlinear. I would like to try to solve for the three unknowns
via iterative solution. What are some methods for this?
Thanks. 

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Lynn Kurtz science forum Guru
Joined: 02 May 2005
Posts: 603


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laniik science forum beginner
Joined: 19 Jun 2006
Posts: 4

Posted: Fri Jul 14, 2006 7:01 pm Post subject:
Re: Iterative solution to nonlinear equations



[Mr.] Lynn Kurtz wrote:
I've looked at those. They seem to be targeted towards solving one
nonlinear equation via somthing like Newtons method, which I
understand. I just dont see how to apply that to a system of equations. 

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C6L1V@shaw.ca science forum Guru
Joined: 23 May 2005
Posts: 628

Posted: Fri Jul 14, 2006 7:30 pm Post subject:
Re: Iterative solution to nonlinear equations



laniik wrote:
The NewtonRaphson method applies to general systems of equations. If
the system is f(x) = 0, where x \in R^r and f = (f_1, ..., f_n), and if
x_0 \in R^n is a starting point, the NR method just takes a linear
approximation and solves that: f(x) =approx= f(x_0) + H(x_0) (x  x_0),
where H(x) = the Hessian matrix, H_{ij}(x) = d f_i(x)/ dx_j , and all
vectors are regarded as column vectors. If H = H(x_0) is invertible,
the next approximation is x = x_0  H^(1) f(x_0). Of course, just as
in 1 dimension, you should start with a "reasonable" approximation to
the solution.
There are many improvements possible. Do a Google search on
NewtonRaphson method.
R.G. Vickson 

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Jim Rockford science forum beginner
Joined: 30 Jun 2006
Posts: 3

Posted: Fri Jul 14, 2006 10:08 pm Post subject:
Re: Iterative solution to nonlinear equations



C6L1V@shaw.ca wrote:
Quote:  The NewtonRaphson method applies to general systems of equations. If
the system is f(x) = 0, where x \in R^r and f = (f_1, ..., f_n), and if
x_0 \in R^n is a starting point, the NR method just takes a linear
approximation and solves that: f(x) =approx= f(x_0) + H(x_0) (x  x_0),
where H(x) = the Hessian matrix, H_{ij}(x) = d f_i(x)/ dx_j , and all
vectors are regarded as column vectors. If H = H(x_0) is invertible,
the next approximation is x = x_0  H^(1) f(x_0). Of course, just as
in 1 dimension, you should start with a "reasonable" approximation to
the solution.

Actually, H is the Jacobian matrix, not the Hessian.
J 

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C6L1V@shaw.ca science forum Guru
Joined: 23 May 2005
Posts: 628

Posted: Sat Jul 15, 2006 1:05 am Post subject:
Re: Iterative solution to nonlinear equations



Jim Rockford wrote:
Quote:  C6L1V@shaw.ca wrote:
The NewtonRaphson method applies to general systems of equations. If
the system is f(x) = 0, where x \in R^r and f = (f_1, ..., f_n), and if
x_0 \in R^n is a starting point, the NR method just takes a linear
approximation and solves that: f(x) =approx= f(x_0) + H(x_0) (x  x_0),
where H(x) = the Hessian matrix, H_{ij}(x) = d f_i(x)/ dx_j , and all
vectors are regarded as column vectors. If H = H(x_0) is invertible,
the next approximation is x = x_0  H^(1) f(x_0). Of course, just as
in 1 dimension, you should start with a "reasonable" approximation to
the solution.
Actually, H is the Jacobian matrix, not the Hessian.

You're right, and I remembered that immediately after pressing the
"post" button.
RGV


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