1. Spline Interpolation Method
Chemical Engineering Majors
Authors: Autar Kaw, Jai Paul
Transforming Numerical Methods Education for STEM Undergraduates

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

3. What is Interpolation ?
Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given.

4. Interpolants Evaluate Differentiate, and Integrate.
Polynomials are the most common choice of interpolants because they are easy to:
Evaluate
Differentiate, and
Integrate.

5. Why Splines ?

6. Figure : Higher order polynomial interpolation is a bad idea
Why Splines ?
Figure : Higher order polynomial interpolation is a bad idea

7. Linear Interpolation

8. Linear Interpolation (contd)

9. Example
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 linear spline interpolation to determine the value of the specific heat at T = 61°C.
Table 1 Specific heat of water as a function of temperature.
Temperature,
Specific heat, 22 42 52 82
100
4181
4179
4186
4199
4217
Figure 2 Specific heat of water vs. temperature.

10. Linear Interpolation

11. Quadratic Interpolation

12. Quadratic Interpolation (contd)

13. Quadratic Splines (contd)

14. Quadratic Splines (contd)

15. Quadratic Splines (contd)

16. Example
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 quadratic spline interpolation to determine the value of the specific heat at T = 61°C.
Table 1 Specific heat of water as a function of temperature.
Temperature,
Specific heat, 22 42 52 82
100
4181
4179
4186
4199
4217
Figure 2 Specific heat of water vs. temperature.

17. Solution

18. Solution (contd)

19. Solution (contd)

20. Solution (contd)

21. Solution (contd)

22. Solution (contd)

23. Better Estimate

24. Better Estimate

25. Better Estimate

26. 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

27. THE END