1. http://numericalmethods.eng.usf.edu 1 Newton’s Divided Difference Polynomial Method of Interpolation Chemical Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates

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

6. To find how much heat is required to bring a kettle of water to its boiling point, you are asked to calculate the specific heat of water at 61 ° C. The specific heat of water is given as a function of time in Table 1. Use Newton’s divided difference method with a first order and then a second order polynomial to determine the value of the specific heat at T = 61 ° C. http://numericalmethods.eng.usf.edu6 Example Table 1 Specific heat of water as a function of temperature. Figure 2 Specific heat of water vs. temperature. Temperature,Specific heat, 22 42 52 82 100 4181 4179 4186 4199 4217

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 Quadratic Interpolation (contd)

11. http://numericalmethods.eng.usf.edu11 Quadratic Interpolation (contd)

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

13. http://numericalmethods.eng.usf.edu13 General Form where Rewriting

14. http://numericalmethods.eng.usf.edu14 General Form

15. http://numericalmethods.eng.usf.edu15 General form

16. To find how much heat is required to bring a kettle of water to its boiling point, you are asked to calculate the specific heat of water at 61 ° C. The specific heat of water is given as a function of time in Table 1. Use Newton’s divided difference method with a third order polynomial to determine the value of the specific heat at T = 61 ° C. http://numericalmethods.eng.usf.edu16 Example Table 1 Specific heat of water as a function of temperature. Figure 2 Specific heat of water vs. temperature. Temperature,Specific heat, 22 42 52 82 100 4181 4179 4186 4199 4217

17. http://numericalmethods.eng.usf.edu17 Example

18. http://numericalmethods.eng.usf.edu18 Example

19. http://numericalmethods.eng.usf.edu19 Example

20. http://numericalmethods.eng.usf.edu20 Comparison Table

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/newton_div ided_difference_method.html

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