Numerical Analysis/Newton form example

We'll find the interpolating polynomial passing through the points $(1,-6)=(x_{0},y_{0})$, $(2,2)=(x_{1},y_{1})$, $(4,12)=(x_{2},y_{2})$, using the Newton form of the interpolation polynomial.

The Newton form is given by the formula $p(x)=\sum _{j=0}^{k}a_{j}n_{j}(x)$, where $a_{j}=[y_{0},\ldots ,y_{j}]$ and $n_{j}(x)=\prod _{i=0}^{j-1}(x-x_{i})$, with $n_{0}(x)=1$. We start by finding each $n_{j}(x)$.

$n_{0}(x)=1$

$n_{1}(x)=x-1$

$n_{2}(x)=(x-1)(x-2)=x^{2}-3x+2$

Next, we find the necessary divided differences. First, $[y_{0}]=-6$, $[y_{1}]=2$, and $[y_{2}]=12$. For the next level, we have:

$[y_{0},y_{1}]={\frac {2+6}{2-1}}=8$

$[y_{1},y_{2}]={\frac {12-2}{4-2}}=5$

Finally, we can find:

$[y_{0},y_{1},y_{2}]={\frac {5-8}{4-1}}=-1$.

Now, we can find the coefficients $a_{j}$.

$a_{0}=[y_{0}]=-6$

$a_{1}=[y_{0},y_{1}]=8$

$a_{2}=[y_{0},y_{1},y_{2}]=-1$

Substituting and simplifying, we get our interpolating polynomial:

$p(x)=-6+8(x-1)-(x^{2}-3x+2)=-x^{2}+11x-16$.

Adding a point

Now let's add the point $(3,-10)=(x_{3},y_{3})$ to our data set and find the new polynomial using the same method. Due to the formula for the Newton form, we only have to add the term $a_{3}n_{3}(x)$ to our previous interpolating polynomial.

First, we have

$n_{3}(x)=(x-1)(x-2)(x-4)=x^{3}-7x^{2}+14x-8$.

Now to find $a_{3}$ we calculate some more divided differences.

$[y_{3}]=-10$

$[y_{2},y_{3}]={\frac {-10-12}{3-4}}=22$

$[y_{1},y_{2},y_{3}]={\frac {22-5}{3-2}}=17$

$a_{3}=[y_{0},\ldots ,y_{3}]={\frac {17+1}{3-1}}=9$

So, our new interpolating polynomial is:

$p(x)=-x^{2}+11x-16+9(x^{3}-7x^{2}+14x-8)=9x^{3}-64x^{2}+137x-88$.