1. Lagrangian Interpolation
Industrial Engineering Majors
Authors: Autar Kaw, Jai Paul
Transforming Numerical Methods Education for STEM Undergraduates

2. Lagrange 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. Lagrangian Interpolation

6. Example
A curve needs to be fit through the given points to fabricate the cam. If the cam follows a straight line profile between x = 1.28 to x = 0.66, what is the value of y at x=1.1 ? Find using the Lagrange method and linear interpolation.
Point
x (in.)
y (in.) 1
2.20
0.00 2
1.28
0.88 3
0.66
1.14 4
1.20 5
–0.60
1.04 6
–1.04
0.60 7
–1.20 1 2 3 5 6 7 X Y 4

7. Linear Interpolation

8. Linear Interpolation (contd)

9. Quadratic Interpolation

10. Example
A curve needs to be fit through the given points to fabricate the cam. If the cam follows a straight line profile between x = 1.28 to x = 0.66, what is the value of y at x=1.1 ? Find using the Lagrange method and quadratic interpolation.
Point
x (in.)
y (in.) 1
2.20
0.00 2
1.28
0.88 3
0.66
1.14 4
1.20 5
–0.60
1.04 6
–1.04
0.60 7
–1.20 1 2 3 5 6 7 X Y 4

11. Quadratic Interpolation (contd)

12. Quadratic Interpolation (contd)
The absolute relative approximate error obtained between the results from the first and second order polynomial is

13. Comparison Table

14. Example
A curve needs to be fit through the given points to fabricate the cam. If the cam follows a straight line profile between x = 1.28 to x = 0.66, what is the value of y at x=1.1 ? Find using the Lagrange method and sixth order polynomial.
Point
x (in.)
y (in.) 1
2.20
0.00 2
1.28
0.88 3
0.66
1.14 4
1.20 5
–0.60
1.04 6
–1.04
0.60 7
–1.20 1 2 3 5 6 7 X Y 4

15. Sixth Order Interpolation

16. Sixth Order Interpolation (contd)

17. Sixth Order Polynomial (contd)

18. Sixth Order Polynomial (contd)

19. Sixth Order Polynomial (contd)

20. Sixth 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

22. THE END