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

2. Lagrange 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.

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 Lagrangian Interpolation

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 Lagrange method for linear interpolation. Figure 2 Location of holes on the rectangular plate.

7. http://numericalmethods.eng.usf.edu7 Linear Interpolation

8. http://numericalmethods.eng.usf.edu8 Linear Interpolation (contd)

9. http://numericalmethods.eng.usf.edu9 Quadratic Interpolation

10. http://numericalmethods.eng.usf.edu10 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 Lagrange method for quadratic interpolation. Figure 2 Location of holes on the rectangular plate.

11. http://numericalmethods.eng.usf.edu11 Quadratic Interpolation

12. http://numericalmethods.eng.usf.edu12 Quadratic Interpolation (contd)

13. http://numericalmethods.eng.usf.edu13 Comparison Table

14. http://numericalmethods.eng.usf.edu14 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 a fifth order Lagrange polynomial. Figure 2 Location of holes on the rectangular plate.

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

16. http://numericalmethods.eng.usf.edu16 Fifth Order Interpolation (contd)

17. http://numericalmethods.eng.usf.edu17 Fifth Order Polynomial (contd)

18. http://numericalmethods.eng.usf.edu18 Fifth Order Polynomial (contd)

19. http://numericalmethods.eng.usf.edu19 Fifth Order Polynomial (contd)

20. http://numericalmethods.eng.usf.edu20 Fifth Order Polynomial (contd)

21. 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/lagrange_ method.html

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