Interpolation

What is Interpolation ? Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given. Figure 1 Interpolation of discrete. http://numericalmethods.eng.usf.edu

Interpolants Evaluate Differentiate, and Integrate Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate http://numericalmethods.eng.usf.edu

1. Direct method

Direct Method Given ‘n+1’ data points (x0,y0), (x1,y1),………….. (xn,yn), pass a polynomial of order ‘n’ through the data as given below: where a0, a1,………………. an are real constants. Set up ‘n+1’ equations to find ‘n+1’ constants. To find the value ‘y’ at a given value of ‘x’, simply substitute the value of ‘x’ in the above polynomial. http://numericalmethods.eng.usf.edu

Example 1 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 direct method for linear interpolation. Table 1 Velocity as a function of time. 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 http://numericalmethods.eng.usf.edu

Linear Interpolation Solving the above two equations gives, Hence Figure 3 Linear interpolation. Hence http://numericalmethods.eng.usf.edu

Example 2 The upward velocity of a rocket is given as a function of time in Table 2. Find the velocity at t=16 seconds using the direct method for quadratic interpolation. Table 2 Velocity as a function of time. 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure 5 Velocity vs. time data for the rocket example http://numericalmethods.eng.usf.edu

Quadratic Interpolation Figure 6 Quadratic interpolation. Solving the above three equations gives http://numericalmethods.eng.usf.edu

Quadratic Interpolation (cont.) The absolute relative approximate error obtained between the results from the first and second order polynomial is http://numericalmethods.eng.usf.edu

Example 3 The upward velocity of a rocket is given as a function of time in Table 3. Find the velocity at t=16 seconds using the direct method for cubic interpolation. Table 3 Velocity as a function of time. 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure 6 Velocity vs. time data for the rocket example http://numericalmethods.eng.usf.edu

Cubic Interpolation Figure 7 Cubic interpolation. http://numericalmethods.eng.usf.edu

Cubic Interpolation (contd) The absolute percentage relative approximate error between second and third order polynomial is http://numericalmethods.eng.usf.edu

Comparison Table Table 4 Comparison of different orders of the polynomial. http://numericalmethods.eng.usf.edu

Distance from Velocity Profile Find the distance covered by the rocket from t=11s to t=16s ? http://numericalmethods.eng.usf.edu

Acceleration from Velocity Profile Find the acceleration of the rocket at t=16s given that http://numericalmethods.eng.usf.edu

2. Spline Method

Why Splines ? http://numericalmethods.eng.usf.edu

Figure : Higher order polynomial interpolation is a bad idea Why Splines ? Figure : Higher order polynomial interpolation is a bad idea http://numericalmethods.eng.usf.edu

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

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

Figure. 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 linear splines. Table Velocity as a function of time (s) (m/s) 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure. Velocity vs. time data for the rocket example http://numericalmethods.eng.usf.edu

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

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

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

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

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

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

Quadratic Spline Example The upward velocity of a rocket is given as a function of time. Using quadratic splines Find the velocity at t=16 seconds Find the acceleration at t=16 seconds Find the distance covered between t=11 and t=16 seconds Table Velocity as a function of time (s) (m/s) 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 Figure. Velocity vs. time data for the rocket example http://numericalmethods.eng.usf.edu 29

Solution Let us set up the equations http://numericalmethods.eng.usf.edu 30

Each Spline Goes Through Two Consecutive Data Points http://numericalmethods.eng.usf.edu 31

Each Spline Goes Through Two Consecutive Data Points v(t) s m/s 10 227.04 15 362.78 20 517.35 22.5 602.97 30 901.67 http://numericalmethods.eng.usf.edu 32

Derivatives are Continuous at Interior Data Points http://numericalmethods.eng.usf.edu 33

Derivatives are continuous at Interior Data Points At t=10 At t=15 At t=20 At t=22.5 http://numericalmethods.eng.usf.edu 34

Last Equation http://numericalmethods.eng.usf.edu

Final Set of Equations http://numericalmethods.eng.usf.edu 36

Coefficients of Spline ai bi ci 1 22.704 2 0.8888 4.928 88.88 3 −0.1356 35.66 −141.61 4 1.6048 −33.956 554.55 5 0.20889 28.86 −152.13 http://numericalmethods.eng.usf.edu 37

Final Solution http://numericalmethods.eng.usf.edu 38

Velocity at a Particular Point a) Velocity at t=16 http://numericalmethods.eng.usf.edu

Acceleration from Velocity Profile b) The quadratic spline valid at t=16 is given by http://numericalmethods.eng.usf.edu

Distance from Velocity Profile c) Find the distance covered by the rocket from t=11s to t=16s. http://numericalmethods.eng.usf.edu 41

3. Newton’s Divided Differences

Newton’s Divided Difference Method Linear interpolation: Given pass a linear interpolant through the data where

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

Linear Interpolation (contd)

Quadratic Interpolation

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)

Quadratic Interpolation (contd)

General Form where Rewriting

General Form

General form

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

Example

Example

Comparison Table

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