Title: Lagrangian%20Interpolation
1Lagrangian Interpolation
- Industrial Engineering Majors
- Authors Autar Kaw, Jai Paul
- http//numericalmethods.eng.usf.edu
- Transforming Numerical Methods Education for STEM
Undergraduates
2Lagrange Method of Interpolation
http//numericalmethods.eng.usf.edu
3What is Interpolation ?
Given (x0,y0), (x1,y1), (xn,yn), find the
value of y at a value of x that is not given.
4Interpolants
- Polynomials are the most common choice of
interpolants because they are easy to - Evaluate
- Differentiate, and
- Integrate.
5Lagrangian Interpolation
6Example
- 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 x1.1 ? 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 0.00 1.20
5 0.60 1.04
6 1.04 0.60
7 1.20 0.00
7Linear Interpolation
8Linear Interpolation (contd)
9Quadratic Interpolation
10Example
- 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 x1.1 ? 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 0.00 1.20
5 0.60 1.04
6 1.04 0.60
7 1.20 0.00
11Quadratic Interpolation (contd)
12Quadratic Interpolation (contd)
The absolute relative approximate error obtained
between the results from the first and second
order polynomial is
13Comparison Table
14Example
- 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 x1.1 ? 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 0.00 1.20
5 0.60 1.04
6 1.04 0.60
7 1.20 0.00
15Sixth Order Interpolation
16Sixth Order Interpolation (contd)
17Sixth Order Polynomial (contd)
18Sixth Order Polynomial (contd)
19Sixth Order Polynomial (contd)
20Sixth Order Polynomial (contd)
21Additional 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/lagran
ge_method.html
22- THE END
- http//numericalmethods.eng.usf.edu