quietrain - don't confuse the
Newton-Raphson (N-R) method and
Taylor Series as methods for solving F(x)=0. The
Newton-Raphson method IS an iterative method for solving F(x)=0. One can construct the N-R method by employing a linear Taylor Series approximation to F(x).
Having said that, the Newton-Raphson method is of the form:
<br />
x_{\nu+1} = x_{\nu} -\frac{F(x_{\nu})}{F'(x_{\nu})}<br />
Here, F(x) is the expression you are trying to solve for zero. It's important to understand that the expression F(x^*) = 0 at the desired root x = x^*. Hence, for example, if your are trying to solve the following equation for x, you need to re-write the expression in terms of F(x) = 0.
Example: Solve for x in the following:
<br />
\tan(x) - x = A<br />
Re-write that as: Solve for x such that F(x) = 0.
<br />
F(x) = \tan(x) - x - A = 0<br />
Your iteration scheme would then be as follows:
<br />
x_{\nu+1} = x_{\nu} - \frac{(\tan(x_{\nu}) - x_{\nu} - A)}{\tan^2(x_{\nu})}<br />
To answer your questions directly:
so the gradient of the line in the form Y = MX + C ==> f(x0) - f(x1) = (x0 - x1) f '(x0)
so if i set f(x1) = 0, i will end up with the Newton-raphsod equation, which is the 1st order taylor series polynomial.
Close - more accuratly, we use a first order Taylor Series to construct the N-R method. For example, let the first order Taylor Series approximation for F(x), expanded about x_0, be
<br />
F(x) \approx F(x_0) + F'(x_0) \cdot (x - x_0)<br />
Using this approximation, what would "x" be if we assumed F(x) = 0. Re-writting the above:
<br />
\frac{F(x) - F(x_0)}{F'(x_0)} = x - x_0<br />
and since we assumed F(x) = 0, then
<br />
x = x_0 -\frac{F(x_0)}{F'(x_0)} <br />
We use this value of "x" and repeat the process (iterate) until convergence.
so is the term f(x1) equals 0 because i define my function tanx -x = 0? which means i am looking at the roots where tan x = x ?
Yes! You got it. And yes, your function is F(x) = tan(x) - x[/itex]<br />
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so does it mean if my equation is tanx - x = 5, i must set my f(x1) term as 5? which means i am looking at the roots where tan x = 5+x?
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That's one way of looking at it, with F(x) = tan(x)-x. However, its better to consider the search for the zero of a function F(x) so as not to "confuse the masses". See the example above F(x) = tan(x) - x - A. This does not necessarily mean that your initial quess to x, ie x_0 is equal to A. But you ARE looking for the x such that \tan(x) = 5+x. So you may want to graph those two curves (f_1(x) = \tan(x) and f_2(x) = 5+x) and find their approximate point of intersection (there will be and infinite number, but maybe your just interested in the first one(?)). The value of "x" at this intersection can be used for an initial estimate (ie, x_0).<br />
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so what the taylor series expansion does is actually like the Newton-raphsod method where you throw in your guess (x0) and it points you closer and closer to the actual value of f(x1)? from the equation f(x) ~ f(x0) + f'(x0)(x-x0)) + O(x^2)
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Again, N-R is the iteration method. The first order Taylor Series is the construction tool to the iteration method. Do not use the Taylor Series approximation to solve for the root - use the N-R method to iterate to the root.
