9.4.2 Lagrange interpolation  9.4 Interpolation and extrapolation  9.4 Interpolation and extrapolation

9.4.1 Finite-difference interpolation formulas

Interpolation formulas based on the Taylor series use the finite-difference formulas of the preceding section. One might consider starting with

$$f(x-x_i)=f_i+(x-x_i)\left[{{f_{i+1}-f_i}\over{\delta}}\right]......}}\left[{{f_{i+2}-2f_{i+1}+f_i}\over{\delta^2}}\right] \cdots$$ (9.17)

 

However, a better expression results from keeping terms that are of consistent order in the expected error. Equation (9.12) can be rewritten as

$$f'_i=f'_{F}-{{\delta}\over{2!}}f''_i-{{\delta^3}\over{3!}}f'''_i+\cdots$$ (9.18)

 

where f'_F is the first-order finite difference formula (9.6) evaluated at x_i. Similarly,

$$f''_i=f''_{F}-\delta f'''_i + \cdots$$ (9.19)

 

If the third derivative is included in the interpolation formula, the third-derivative corrections to the finite difference formulas for the first and second derivatives should also be included. The result of regrouping the series to consistent order is the Gregory-Newton interpolation formula:

$$f(x-x_i) = f_i + (x-x_i)f'_{F}+{{(x-x_i)(x-x_{i+1})}\over{2!}}f''_{F}$$ (9.20)

  

$$~~~~~~~~+ {{(x-x_i)(x-x_{i+1})(x-x_{i+2})}\over{3!}}f''_{F}\cdots$$ (9.21)

 

where the finite-difference formulas with subscripts F are the forward-difference formulas with error limits of order O($\delta$) from Table 9.1, evaluated at x_i. The backward-interpolation formula is similar except backward difference formulas are used and terms involving (x-x_i+1)become (x-x_i-1), etc. Both of these interpolation formulas can be used for extrapolation beyond the limits of the available data, but usually should be used only for distances of about the data spacing.

Stirling's formula is an analogous formula evaluated with central-difference derivatives. It is usually preferable for interpolation within the range of a table. It is

$$f(x-x_i) = f_i + (x-x_i)f'_{C}+{{(x-x_i)^2}\over{2}}f''_C$$ (9.22)

  

$$~~~~~+{{(x-x_i)((x-x_i)^2-1)}\over{3!}}f'''_C + {{(x-x_i)^2((x-x_i)^2-1)}\over{4!}}f''''_C + \cdots$$ (9.23)

 

where the derivatives with subscript C are the centered-difference formulas with accuracy of order O($\delta^2$) from Table 9.1, evaluated at x_i.

   
 9.4.2 Lagrange interpolation  9.4 Interpolation and extrapolation  9.4 Interpolation and extrapolation