1. http://numericalmethods.eng.usf.edu 1 Direct Method of Interpolation Computer Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

2. Direct Method of Interpolation http://numericalmethods.eng.usf.edu http://numericalmethods.eng.usf.edu

3. 3 What is Interpolation ? Given (x 0,y 0 ), (x 1,y 1 ), …… (x n,y n ), find the value of ‘y’ at a value of ‘x’ that is not given. Figure 1 Interpolation of discrete.

4. http://numericalmethods.eng.usf.edu4 Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate

5. http://numericalmethods.eng.usf.edu5 Direct Method Given ‘n+1’ data points (x 0,y 0 ), (x 1,y 1 ),………….. (x n,y n ), pass a polynomial of order ‘n’ through the data as given below: where a 0, a 1,………………. a n 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.

6. http://numericalmethods.eng.usf.edu6 Example A robot arm with a rapid laser scanner is doing a quick quality check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below. If the laser is traversing from x = 2 to x = 4.25 in a linear path, find the value of y at x = 4 using the direct method for linear interpolation. Figure 2 Location of holes on the rectangular plate.

7. http://numericalmethods.eng.usf.edu7 Linear Interpolation Solving the above two equations gives, Hence

8. http://numericalmethods.eng.usf.edu8 Example A robot arm with a rapid laser scanner is doing a quick quality check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below. If the laser is traversing from x = 2 to x = 4.25 in a linear path, find the value of y at x = 4 using the direct method for quadratic interpolation. Figure 2 Location of holes on the rectangular plate.

9. http://numericalmethods.eng.usf.edu9 Quadratic Interpolation Solving the above three equations gives Quadratic Interpolation

10. http://numericalmethods.eng.usf.edu10 Quadratic Interpolation (contd)

11. http://numericalmethods.eng.usf.edu11 Comparison Table

12. http://numericalmethods.eng.usf.edu12 Example A robot arm with a rapid laser scanner is doing a quick quality check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below. If the laser is traversing from x = 2 to x = 4.25 in a linear path, find the value of y at x = 4 using the direct method using a fifth order polynomial. Figure 2 Location of holes on the rectangular plate.

13. http://numericalmethods.eng.usf.edu13 Fifth Order Interpolation

14. http://numericalmethods.eng.usf.edu14 Fifth Order Interpolation (contd)

15. http://numericalmethods.eng.usf.edu15 Fifth Order Interpolation (contd)

16. Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/direct_met hod.html

17. THE END http://numericalmethods.eng.usf.edu