Question

Write Stirling's central difference formula.​

Answer 1

Stirling's (Central Difference) Interpolation Formula:

The Stirling's interpolation formula for 2n + 1 (odd) equispaced arguments is:

$\mathrm{f(x)\simeq S(x)=S(x_{0}+hu)}$

$\mathrm{=y_{0}+\dfrac{u}{1!}.\dfrac{\Delta y_{-1}+\Delta y_{0}}{2}+\dfrac{u^{2}}{2!}.\Delta^{2}y_{-1}}$

$\mathrm{+\dfrac{u(u^{2}-1^{2})}{3!}.\dfrac{\Delta^{3}y_{-2}+\Delta^{3}y_{-1}}{2}}$

$\mathrm{+\dfrac{u^{2}(u^{2}-1^{2})}{4!}.{\Delta}^{4}y_{-2}}$

$\mathrm{+\dfrac{u(u^{2}-1^{2})(u^{2}-2^{2})}{5!}.\dfrac{\Delta^{5}y_{-3}+\Delta^{5}y_{-2}}{2}}$

$\mathrm{\quad\quad\quad\quad+\bold{...}}$

where $\mathrm{u=\dfrac{x-x_{0}}{h}}$

Remarks:

• We use Stirling's formula when the interpolating point x is at near of the centre and when the arguments are of odd number.

• This formula is more suitable to get an accurate result in case of odd order differences for difference table, than other central difference formulas.

Numerical analysis problems:

• Evaluate $\int\limits^6_0 {\frac{\, dx }{1+x^{2} } }$ by Trapezoidal rule. - https://brainly.in/question/9294518

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