Numerical Methods20795 marksNumerical: Divided difference tableDerivative estimation using interpolation formulas
Divided Difference Table and Derivatives Construct the divided difference table for the following data and find first and second order derivatives at x=2.
Divided Difference Table and Derivatives
Construct the divided difference table for the following data and find first and second order derivatives at $x=2$.
$$\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 2 & 4 & 8 & 10 \ \hline y & 0 & 1 & 5 & 21 & 27 \ \hline \end{array}$$
[5]
$x$ 1 2 4 8 10 ------------------ $y$ 0 1 5 21 27 $$f[x0,x1]=\frac{1-0}{2-1}=1$$ $$f[x1,x2]=\frac{5-1}{4-2}=2$$ $$f[x2,x3]=\frac{21-5}{8-4}=4$$ $$f[x3,x4]=\frac{27-21}{10-8}=3$$ $$f[x0,x1,x2]=\frac{2-1}{4-1}=\frac{1}{3}$$ $$f[x1,x2,x3]=\frac{4-2}{8-2}=\frac{2}{6}=\frac{1}{3}$$ $$f[x2,x3,x4]=\frac...