ExerciseInterpolationintermediate

Exercise: Newton with population data

Full divided-difference table for the 1971–2011 census and estimate of the 2005 population with the degree-4 Newton polynomial.

Data and table

Year19711981199120012011
Población [M]33.95637.74339.43440.84746.816
Example

Estimate the 2005 population with degree-4 Newton.

  1. Divided differences (upper diagonal):

    b0=33.956, b1=0.3787, b2=−0.01048, b3=0.000303, b4=0.0000127b_0=33.956,\ b_1=0.3787,\ b_2=-0.01048,\ b_3=0.000303,\ b_4=0.0000127

  2. Newton polynomial:

    p4(x)= 33.956+0.3787(x−1971)−0.01048(x−1971)(x−1981)+0.000303(x−1971)(x−1981)(x−1991)+0.0000127(x−1971)(x−1981)(x−1991)(x−2001)\begin{aligned} p_4(x)=\ &33.956+0.3787(x-1971)-0.01048(x-1971)(x-1981)\\ &+0.000303(x-1971)(x-1981)(x-1991)\\ &+0.0000127(x-1971)(x-1981)(x-1991)(x-2001) \end{aligned}

Estimate at 2005:

p4(2005)≈42.315 millonesp_4(2005)\approx 42.315\ \text{millones}