1. http://numericalmethods.eng.usf.edu 1 Newton’s Divided Difference Polynomial Method of Interpolation Mechanical 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. http://numericalmethods.eng.usf.edu6 Example A trunnion is cooled 80°F to − 108°F. Given below is the table of the coefficient of thermal expansion vs. temperature. Determine the value of the coefficient of thermal expansion at T=−14°F using the direct method for linear interpolation. Temperature ( o F) Thermal Expansion Coefficient (in/in/ o F) 806.47 × 10 −6 06.00 × 10 −6 − 60 5.58 × 10 −6 − 160 4.72 × 10 −6 − 260 3.58 × 10 −6 − 340 2.45 × 10 −6

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 trunnion is cooled 80°F to − 108°F. Given below is the table of the coefficient of thermal expansion vs. temperature. Determine the value of the coefficient of thermal expansion at T=−14°F using the direct method for quadratic interpolation. Temperature ( o F) Thermal Expansion Coefficient (in/in/ o F) 806.47 × 10 −6 06.00 × 10 −6 − 60 5.58 × 10 −6 − 160 4.72 × 10 −6 − 260 3.58 × 10 −6 − 340 2.45 × 10 −6

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

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

13. http://numericalmethods.eng.usf.edu13 Quadratic Interpolation (contd)

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

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

16. http://numericalmethods.eng.usf.edu16 General form

17. http://numericalmethods.eng.usf.edu17 Example A trunnion is cooled 80°F to − 108°F. Given below is the table of the coefficient of thermal expansion vs. temperature. Determine the value of the coefficient of thermal expansion at T=−14°F using the direct method for cubic interpolation. Temperature ( o F) Thermal Expansion Coefficient (in/in/ o F) 806.47 × 10 −6 06.00 × 10 −6 − 60 5.58 × 10 −6 − 160 4.72 × 10 −6 − 260 3.58 × 10 −6 − 340 2.45 × 10 −6

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

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

20. http://numericalmethods.eng.usf.edu20 Example

21. http://numericalmethods.eng.usf.edu21 Comparison Table

22. http://numericalmethods.eng.usf.edu22 Reduction in Diameter The actual reduction in diameter is given by where T r = room temperature (°F) T f = temperature of cooling medium (°F) Since T r = 80 °F and T r = −108 °F, Find out the percentage difference in the reduction in the diameter by the above integral formula and the result using the thermal expansion coefficient from the cubic interpolation.

23. http://numericalmethods.eng.usf.edu23 Reduction in Diameter We know from interpolation that Therefore,

24. http://numericalmethods.eng.usf.edu24 Reduction in diameter Using the average value for the coefficient of thermal expansion from cubic interpolation The percentage difference would be

25. 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

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