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Lagrangian Interpolation

Lagrangian Interpolation. Civil Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Lagrange Method of Interpolation http://numericalmethods.eng.usf.edu. What is Interpolation ?.

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Lagrangian Interpolation

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

  2. Lagrange Method of Interpolationhttp://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. http://numericalmethods.eng.usf.edu

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

  5. Lagrangian Interpolation http://numericalmethods.eng.usf.edu

  6. Example To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using the Lagragian method for linear interpolation. Temperature vs. depth of a lake http://numericalmethods.eng.usf.edu

  7. Linear Interpolation http://numericalmethods.eng.usf.edu

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

  9. Quadratic Interpolation http://numericalmethods.eng.usf.edu

  10. Example To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using the Lagragian method for quadratic interpolation. Temperature vs. depth of a lake http://numericalmethods.eng.usf.edu

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

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

  13. Cubic Interpolation http://numericalmethods.eng.usf.edu

  14. Example To maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using the Lagragian method for cubic interpolation. Temperature vs. depth of a lake http://numericalmethods.eng.usf.edu

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

  16. Cubic Interpolation (contd) http://numericalmethods.eng.usf.edu

  17. Comparison Table http://numericalmethods.eng.usf.edu

  18. Thermocline What is the value of depth at which the thermocline exists? The position where the thermocline exists is given where Simply setting this expression equal to zero, we get http://numericalmethods.eng.usf.edu

  19. 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/lagrange_method.html

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

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