Numerical Methods20785 marksNumerical: Divided difference interpolationNewton's divided difference method
Given the following set of data points. Obtain the table of divided difference and use that table to estimate the value of f(1.5).
Given the following set of data points. Obtain the table of divided difference and use that table to estimate the value of $f(1.5)$.
$$\begin{array}{|c|c|c|c|c|c|}\hline x & 1 & 2 & 3 & 4 & 5 \ \hline f(x)=x^3-1 & 0 & 7 & 26 & 63 & 124 \ \hline \end{array}$$
[5]
$x$ 1 2 3 4 5 ------------------ $f(x)=x^3-1$ 0 7 26 63 124 Estimate: $f(1.5)$ --- First divided differences: $$f[1,2]=\frac{7-0}{2-1}=7,\quad f[2,3]=\frac{26-7}{3-2}=19$$ $$f[3,4]=\frac{63-26}{4-3}=37,\quad f[4,5]=\frac{124-63}{5-4}=61$$ Second divided differences: $$f[1,2,3]=\frac{19-7}{3-1}=6,...