Lagrangian Interpolation Major All Engineering Majors Authors Autar

  • Slides: 21
Download presentation
Lagrangian Interpolation Major: All Engineering Majors Authors: Autar Kaw, Jai Paul http: //numericalmethods. eng.

Lagrangian Interpolation Major: All Engineering Majors Authors: Autar Kaw, Jai Paul http: //numericalmethods. eng. usf. edu Transforming Numerical Methods Education for STEM Undergraduates http: //numericalmethods. eng. usf. edu 1

Lagrange Method of Interpolation http: //numericalmethods. eng. usf. edu

Lagrange Method of Interpolation http: //numericalmethods. eng. usf. edu

What is Interpolation ? Given (x 0, y 0), (x 1, y 1), ……

What is Interpolation ? Given (x 0, y 0), (x 1, y 1), …… (xn, yn), find the value of ‘y’ at a value of ‘x’ that is not given. 3 lmethods. eng. usf. edu http: //numerica

Interpolants Polynomials are the most common choice of interpolants because they are easy to:

Interpolants Polynomials are the most common choice of interpolants because they are easy to: Evaluate Differentiate, and Integrate. 4 lmethods. eng. usf. edu http: //numerica

Lagrangian Interpolation 5 lmethods. eng. usf. edu http: //numerica

Lagrangian Interpolation 5 lmethods. eng. usf. edu http: //numerica

Example The upward velocity of a rocket is given as a function of time

Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Lagrangian method for linear interpolation. Table Velocity as a function of time (s) 0 10 15 20 22. 5 30 6 (m/s) 0 227. 04 362. 78 517. 35 602. 97 901. 67 Figure. Velocity vs. time data for the rocket examplelmethods. eng. usf. edu http: //numerica

Linear Interpolation 7 lmethods. eng. usf. edu http: //numerica

Linear Interpolation 7 lmethods. eng. usf. edu http: //numerica

Linear Interpolation (contd) 8 lmethods. eng. usf. edu http: //numerica

Linear Interpolation (contd) 8 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation 9 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation 9 lmethods. eng. usf. edu http: //numerica

Example The upward velocity of a rocket is given as a function of time

Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Lagrangian method for quadratic interpolation. Table Velocity as a function of time (s) 0 10 15 20 22. 5 30 10 (m/s) 0 227. 04 362. 78 517. 35 602. 97 901. 67 Figure. Velocity vs. time data for the rocket examplelmethods. eng. usf. edu http: //numerica

Quadratic Interpolation (contd) 11 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation (contd) 11 lmethods. eng. usf. edu http: //numerica

Quadratic Interpolation (contd) The absolute relative approximate error obtained between the results from the

Quadratic Interpolation (contd) The absolute relative approximate error obtained between the results from the first and second order polynomial is 12 lmethods. eng. usf. edu http: //numerica

Cubic Interpolation 13 lmethods. eng. usf. edu http: //numerica

Cubic Interpolation 13 lmethods. eng. usf. edu http: //numerica

Example The upward velocity of a rocket is given as a function of time

Example The upward velocity of a rocket is given as a function of time in Table 1. Find the velocity at t=16 seconds using the Lagrangian method for cubic interpolation. Table Velocity as a function of time (s) 0 10 15 20 22. 5 30 14 (m/s) 0 227. 04 362. 78 517. 35 602. 97 901. 67 Figure. Velocity vs. time data for the rocket examplelmethods. eng. usf. edu http: //numerica

Cubic Interpolation (contd) 15 lmethods. eng. usf. edu http: //numerica

Cubic Interpolation (contd) 15 lmethods. eng. usf. edu http: //numerica

Cubic Interpolation (contd) The absolute relative approximate error obtained between the results from the

Cubic Interpolation (contd) The absolute relative approximate error obtained between the results from the first and second order polynomial is 16 lmethods. eng. usf. edu http: //numerica

Comparison Table 17 Order of Polynomial 1 2 3 v(t=16) m/s 393. 69 392.

Comparison Table 17 Order of Polynomial 1 2 3 v(t=16) m/s 393. 69 392. 19 392. 06 Absolute Relative Approximate Error ---- 0. 38410% 0. 033269% lmethods. eng. usf. edu http: //numerica

Distance from Velocity Profile Find the distance covered by the rocket from t=11 s

Distance from Velocity Profile Find the distance covered by the rocket from t=11 s to t=16 s ? 18 lmethods. eng. usf. edu http: //numerica

Acceleration from Velocity Profile Find the acceleration of the rocket at t=16 s given

Acceleration from Velocity Profile Find the acceleration of the rocket at t=16 s given that , 19 lmethods. eng. usf. edu http: //numerica

Additional Resources For all resources on this topic such as digital audiovisual lectures, primers,

Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, Math. Cad and MAPLE, blogs, related physical problems, please visit http: //numericalmethods. eng. usf. edu/topics/lagrange_ method. html

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

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