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Iterative solution to non-linear equations
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laniik
science forum beginner


Joined: 19 Jun 2006
Posts: 4

PostPosted: Fri Jul 14, 2006 6:38 pm    Post subject: Iterative solution to non-linear equations Reply with quote

Hi, I have three equations with three unknowns. All of the equations
are non-linear. 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

PostPosted: Fri Jul 14, 2006 6:53 pm    Post subject: Re: Iterative solution to non-linear equations Reply with quote

On 14 Jul 2006 11:38:39 -0700, "laniik" <laniik@yahoo.com> wrote:

Quote:
Hi, I have three equations with three unknowns. All of the equations
are non-linear. I would like to try to solve for the three unknowns
via iterative solution. What are some methods for this?

Thanks.

http://www.google.com/search?q=iterative+methods+nonlinear+equations&start=0&ie=utf-8&oe=utf-8&client=firefox-a&rls=org.mozilla:en-US:official

--Lynn
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laniik
science forum beginner


Joined: 19 Jun 2006
Posts: 4

PostPosted: Fri Jul 14, 2006 7:01 pm    Post subject: Re: Iterative solution to non-linear equations Reply with quote

[Mr.] Lynn Kurtz wrote:
Quote:
On 14 Jul 2006 11:38:39 -0700, "laniik" <laniik@yahoo.com> wrote:

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

Thanks.

http://www.google.com/search?q=iterative+methods+nonlinear+equations&start=0&ie=utf-8&oe=utf-8&client=firefox-a&rls=org.mozilla:en-US:official

--Lynn

I've looked at those. They seem to be targeted towards solving one
non-linear 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

PostPosted: Fri Jul 14, 2006 7:30 pm    Post subject: Re: Iterative solution to non-linear equations Reply with quote

laniik wrote:
Quote:
[Mr.] Lynn Kurtz wrote:
On 14 Jul 2006 11:38:39 -0700, "laniik" <laniik@yahoo.com> wrote:

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

Thanks.

http://www.google.com/search?q=iterative+methods+nonlinear+equations&start=0&ie=utf-8&oe=utf-8&client=firefox-a&rls=org.mozilla:en-US:official

--Lynn

I've looked at those. They seem to be targeted towards solving one
non-linear equation via somthing like Newtons method, which I
understand. I just dont see how to apply that to a system of equations.

The Newton-Raphson 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 N-R 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
Newton-Raphson method.

R.G. Vickson
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Jim Rockford
science forum beginner


Joined: 30 Jun 2006
Posts: 3

PostPosted: Fri Jul 14, 2006 10:08 pm    Post subject: Re: Iterative solution to non-linear equations Reply with quote

C6L1V@shaw.ca wrote:
Quote:
The Newton-Raphson 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 N-R 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

PostPosted: Sat Jul 15, 2006 1:05 am    Post subject: Re: Iterative solution to non-linear equations Reply with quote

Jim Rockford wrote:
Quote:
C6L1V@shaw.ca wrote:
The Newton-Raphson 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 N-R 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


Quote:

J
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