User Guide 027: Difference between revisions

From CIRPwiki
Jump to navigation Jump to search
No edit summary
No edit summary
 
(One intermediate revision by the same user not shown)
Line 3: Line 3:
Piecewise polynomials in Lagrange form are given by
Piecewise polynomials in Lagrange form are given by


{{Equation|<math>l(x) = \sum_{j=1}^{n+1} y_j l_j (x)</math>|16-1}}
{{Equation|<math>L(x) = \sum_{j=1}^{n+1} y_j l_j (x)</math>|16-1}}


where
where


y<sub>j</sub>= interpolation data values corresponding to x<sub>j</sub>
: y<sub>j</sub>= interpolation data values corresponding to x<sub>j</sub>
   
   
n = order or the interpolation polynomial
: n = order or the interpolation polynomial


l<sub>j</sub>(x) = Lagrange basis polynomials given by
: l<sub>j</sub>(x) = Lagrange basis polynomials given by


{{Equation|<math> l_j (x) = \prod_{1 < k \leq n+1 \ k\neq j} \frac{x - x_k}{x_j - x_k} </math>|16-2}}
{{Equation|<math> l_j (x) = \prod_{1 < k \leq n+1  
 
\ k\neq j} \frac{x - x_k}{x_j - x_k} </math>|16-2}}


The Lagrange basis polynomials are such that
The Lagrange basis polynomials are such that


{{Equation|<math> l_{\neq 1} (x_i) = 0, l_{j=1} = 1, l(x_j) = y_i</math>| 16-3}}
{{Equation|<math> l_{j \neq 1} (x_i) = 0, l_{j=1} = 1, l(x_j) = y_i</math>| 16-3}}


One advantage of using Lagrange polynomials is that the interpolation weights (Lagrange basis functions l<sub>j</sub> ) are not a function of y<sub>j</sub> . This prop-erty is useful when many interpolations are needed for the same x<sub>j</sub>  but different y<sub>j</sub>  such as in the case of interpolating spatial datasets in time.
One advantage of using Lagrange polynomials is that the interpolation weights (Lagrange basis functions l<sub>j</sub> ) are not a function of y<sub>j</sub> . This prop-erty is useful when many interpolations are needed for the same x<sub>j</sub>  but different y<sub>j</sub>  such as in the case of interpolating spatial datasets in time.

Latest revision as of 21:45, 8 May 2015

16 Appendix F: Piecewise Lagrangian Polynomial Interpolation

Piecewise polynomials in Lagrange form are given by

  (16-1)

where

yj= interpolation data values corresponding to xj
n = order or the interpolation polynomial
lj(x) = Lagrange basis polynomials given by
  (16-2)

The Lagrange basis polynomials are such that

  ( 16-3)

One advantage of using Lagrange polynomials is that the interpolation weights (Lagrange basis functions lj ) are not a function of yj . This prop-erty is useful when many interpolations are needed for the same xj but different yj such as in the case of interpolating spatial datasets in time.