Matakuliah Tahun : S0262-Analisis Numerik : 2010 Interpolation Pertemuan 11 Material Outline • Interpolation – Linear interpolation – Quadratic interpolation – Lagrange interpolation INTERPOLATION • As in the case of Curve Fitting, Interpolation is a tool to estimate intermediate values between precise data points. One of the methods that frequently used is Polynomial Interpolation. nth order Polynomial f(x)= a0 + a1x + a2x2 + …anxn For n+1 data points, there is only one nth order Polynomial that fit the data points. In this slides, only the linear and quadratic interpolations will be discussed. • • • • 4 LINEAR INTERPOLATION • • This interpolation is the simplest method to estimate the values between 2 data points. The estimation was done by connecting the two points with a straight line: Assume the coordinate of the two data points are : (x1,y1), (x2,y2) then the intermediate values (f(x)) for x1<x<x2 can be written as: y2 y1 f ( x) y1 ( x x1 ) x2 x1 y2 f(x) y1 x1 x x2 5 LINEAR INTERPOLATION • Example: Assume the population of a country in Africa is given in the following table: Year 1998 2000 2002 Population 72,4 million 75,1 million 77 million Use a linear interpolation to estimate the population of the country in the year of 1999 and 2001. Solution: 6 LINEAR INTERPOLATION Year 1998 2000 2002 Population 72,4 million 75,1 million 77 million y2 y1 ( x x1 ) x2 x1 75,1 72,4 f (1999) 72,4 (1999 1998) 73,75million 2000 1998 Solution: f ( x) y1 77 75,1 f (2001) 75,1 (2001 2000) 76,05million 2002 2000 7 QUADRATIC INTERPOLATION • • It is obvious that the weakness of linear interpolation is that the method force the linear trend of Interpolate the intermediate values. To improve this obvious weakness, the quadratic interpolation will be discussed: y3 y2 f(x) y1 x1 x x2 x3 8 • Referring to the following figure, the formula to approximate the intermediate values f(x) for x1<x<x3 is given below: f ( x) y1 b0 ( x x1 ) b1 ( x x1 )( x x2 ) in which : y2 y1 b0 x2 x1 y3 y2 y2 y1 x3 x2 x2 x1 b1 x3 x1 y3 y2 f(x) y1 x1 x x2 x3 9 QUADRATIC INTERPOLATION • Example: Assume the population of a country in Africa is given in the following table: Year 1998 2000 2002 Population 72,4 million 75,1 million 77 million Use a quaratic interpolation to estimate the population of the country in the year of 1999 and 2001. Solution: 10 QUADRATIC INTERPOLATION Year 1998 2000 2002 Population 72,4 million 75,1 million 77 million Solution: : x1=1998, x2= 2000, x3= 2002 y1=72.4 y2= 75.1 75.1 72.4 b0 1.35; b1 2 y3= 77 7775.1 2 75.172.4 2 0.1 4 f (1999) 72.4 1.35 0.1(1)( 1) 73.85 million f (2001) 11 EXERCISE 1: Use a linear interpolation to fill in the missing data from the following table: x 2 3 4,5 5 6 7,5 9 y 7 …… 9,5 …. 8,5 …. 10,5 EXERCISE 2: Use a quadratic interpolation to fill in the missing data from the following table: x 2 3 4,5 5 6 7,5 9 y 7 …… 9,5 …. 8,5 …. 10,5 12 LAGRANGE INTERPOLATION • This interpolation is the reformulation of the Newton Polynomials and can be written as: n f n ( x) Li ( x) f ( xi ) i 0 Example : x x0 x x1 for n 1 f1 ( x) f ( x0 ) f ( x1 ) x0 x1 x1 x0 x x1 x x2 f ( x ) x0 x1 x0 x2 0 x x0 x x2 f ( x ) x x0 x x1 f ( x ) x1 x0 x1 x2 1 x2 x0 x2 x1 2 for n 2 f 2 ( x) Li ( x) n x xj j 0, j i xi x j LAGRANGE INTERPOLATION Example: Use Lagrange interpolation of 1st and 2nd order to evaluate f(2) from the following data: i xi F(x i) 0 1 0 1 4 1.3863 2 6 1.79176