Guillaume: 1 revision

2008-12-03T10:39:01Z

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192.168.20.177: /* p-th derivative of polynomial interpolation of real function f */

2008-06-27T10:52:07Z

‎<mathtex>p</mathtex>-th derivative of polynomial interpolation of real function <mathtex>f</mathtex>

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Given <mathtex>n</mathtex> nodes of known values of any real function <mathtex>f</mathtex>, the '''unisolvence theorem''' states that there exists a unique polynomial of degree <mathtex>n-1</mathtex> that interpolates the nodes. Such a polynomial <mathtex>P</mathtex> is easily constructed for low degrees and is easily generalized to any degree.

==Polynomial interpolation of real function <mathtex>f</mathtex>==
By increasing degree:
* <mathtex>P^{1}(x) = f(x_1) \frac{ \left( x - x_2 \right) }{ \left( x_1 - x_2 \right) } + f(x_2) \frac{ \left( x - x_1 \right) }{ \left( x_2 - x_1 \right) } </mathtex>
* <mathtex>P^{2}(x) = f(x_1) \frac{ \left( x - x_2 \right) \left( x - x_3 \right) }{ \left( x_1 - x_2 \right) \left( x_1 - x_3 \right)} + f(x_2) \frac{ \left( x - x_1 \right) \left( x - x_3 \right) }{ \left( x_2 - x_1 \right) \left( x_2 - x_3 \right) } + f(x_3) \frac{ \left( x - x_1 \right) \left( x - x_2 \right) }{ \left( x_3 - x_1 \right) \left( x_3 - x_2 \right) } </mathtex>
* ...
* <mathtex>P^{n-1}(x) = \sum_{i=1}^{n} \frac{ f(x_i) \prod_{j \neq i}^n \left( x - x_j \right) }{ \prod_{j \neq i}^n \left( x_i - x_j \right) } </mathtex>

Where <mathtex>x_1,\, x_2,\, \ldots \, x_n,</mathtex> are the nodes where <mathtex>f</mathtex> is known.

==<mathtex>p</mathtex>-th derivative of polynomial interpolation of real function <mathtex>f</mathtex>==

* <mathtex>P^{1}(x)^{(1)} = f(x_1) \frac{ 1 }{ \left( x_1 - x_2 \right) } + f(x_2) \frac{ 1 }{ \left( x_2 - x_1 \right) } </mathtex>
* <mathtex>P^{2}(x)^{(1)} = f(x_1) \frac{ \left( x - x_2 \right) + \left( x - x_3 \right) }{ \left( x_1 - x_2 \right) \left( x_1 - x_3 \right)} + f(x_2) \frac{ \left( x - x_1 \right) + \left( x - x_3 \right) }{ \left( x_2 - x_1 \right) \left( x_2 - x_3 \right) } + f(x_3) \frac{ \left( x - x_1 \right) + \left( x - x_2 \right) }{ \left( x_3 - x_1 \right) \left( x_3 - x_2 \right) } </mathtex>
* <mathtex>P^{2}(x)^{(2)} = f(x_1) \frac{ 2 }{ \left( x_1 - x_2 \right) \left( x_1 - x_3 \right)} + f(x_2) \frac{ 2 }{ \left( x_2 - x_1 \right) \left( x_2 - x_3 \right) } + f(x_3) \frac{ 2 }{ \left( x_3 - x_1 \right) \left( x_3 - x_2 \right) } </mathtex>
* ...
* <mathtex> P^{n-1}(x)^{(p)} = \sum_{\alpha_1=1}^{n} \frac{ f(x_{\alpha_1}) \sum_{\alpha_{i+1} \neq \alpha_i,\,(i)}^{n,\,(p)} \prod_{\alpha_{p+2} \neq \alpha_{p+1}}^n \left( x - x_{\alpha_{p+2}} \right) }{ \prod_{\alpha_2 \neq \alpha_1}^n \left( x_{\alpha_1} - x_{\alpha_2} \right) } </mathtex>

In particular, if <mathtex>n-1 = p</mathtex> we have,
* <mathtex> P^{p}(x)^{(p)} = \sum_{i=1}^{n} f(x_{i}) \frac{ p }{ \prod_{j \neq i}^n \left( x_{i} - x_{j} \right) } </mathtex>

===[[Compound sum]]===
* <mathtex>\sum^{(p)} = \underbrace{ \sum \circ \sum \circ \ldots \circ \sum }_{p-\text{times}}</mathtex>,
* <mathtex> \sum^{(0)} = id </mathtex>,
* <mathtex> \sum^{n,\,(p)}_{\alpha_{i+1}=1,\,(i)} = \underbrace{ \sum^{n}_{\alpha_2=1} \circ \sum^{n}_{\alpha_3=1} \circ \ldots \circ \sum^{n}_{\alpha_{p+1}=1} }_{p-\text{times}}</mathtex>,

==External references==
*[http://en.wikipedia.org/wiki/Polynomial_interpolation Polynomial interpolation article in wikipedia]

[[Category:Science]]

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192.168.20.177: /* p-th derivative of polynomial interpolation of real function f */