Newton’s Divided Difference Polynomial Method of Interpolation

Newton’s Divided Difference Method Linear interpolation: Given pass a linear interpolant through the data where http://numericalmethods.eng.usf.edu

Figure 2: Velocity vs. time data Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for linear interpolation. t v(t) s m/s 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure 2: Velocity vs. time data for the rocket example Table 1: Velocity as a function of time

Linear Interpolation http://numericalmethods.eng.usf.edu

Linear Interpolation (contd) http://numericalmethods.eng.usf.edu

Quadratic Interpolation http://numericalmethods.eng.usf.edu

Figure 2: Velocity vs. time data Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for quadratic interpolation. t v(t) s m/s 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure 2: Velocity vs. time data for the rocket example Table 1: Velocity as a function of time

Quadratic Interpolation (contd)

Quadratic Interpolation (contd) http://numericalmethods.eng.usf.edu

Quadratic Interpolation (contd)

General Form where Rewriting http://numericalmethods.eng.usf.edu

General Form

General form http://numericalmethods.eng.usf.edu

Figure 2: Velocity vs. time data Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Newton Divided Difference method for cubic interpolation. t v(t) s m/s 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure 2: Velocity vs. time data for the rocket example Table 1: Velocity as a function of time

Example The velocity profile is chosen as we need to choose four data points that are closest to http://numericalmethods.eng.usf.edu

Example

Example http://numericalmethods.eng.usf.edu

Comparison Table http://numericalmethods.eng.usf.edu

Latihan Misalkan Nilai-nilai fungsi seperti pada tabel berikut, tentukanlah nilai aproksimasi dari f(0,1), f(0,35), dan f(1,16) berturut-turut menggunakan interpolasi beda terbagi Newton linear, kuadratik, kubik. x f(x) 0,0 0,000 0,2 0,406 0,4 0,846 0,6 1,368 0,8 2,060 1,0 3,114 1,2 5,114

Distance from Velocity Profile Find the distance covered by the rocket from t=11s to t=16s ?

Acceleration from Velocity Profile Find the acceleration of the rocket at t=16s given that