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Newton’s Divided Difference Polynomial Method of Interpolation

Newton’s Divided Difference Polynomial Method of Interpolation. Chemical Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates.

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Newton’s Divided Difference Polynomial Method of Interpolation

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  1. Newton’s Divided Difference Polynomial Method of Interpolation Chemical 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. Newton’s Divided Difference 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. Newton’s Divided Difference Method Linear interpolation: Given pass a linear interpolant through the data where http://numericalmethods.eng.usf.edu

  6. 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 Newton’s divided difference method with a first order and then a second order polynomial to determine the value of the specific heat at T = 61°C. Table 1 Specific heat of water as a function of temperature. Figure 2 Specific heat of water vs. temperature. 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. Quadratic Interpolation (contd) http://numericalmethods.eng.usf.edu

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

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

  13. General Form where Rewriting http://numericalmethods.eng.usf.edu

  14. General Form http://numericalmethods.eng.usf.edu

  15. General form http://numericalmethods.eng.usf.edu

  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 Newton’s divided difference method with a third order polynomial to determine the value of the specific heat at T = 61°C. Table 1 Specific heat of water as a function of temperature. Figure 2 Specific heat of water vs. temperature. http://numericalmethods.eng.usf.edu

  17. Example http://numericalmethods.eng.usf.edu

  18. Example http://numericalmethods.eng.usf.edu

  19. Example http://numericalmethods.eng.usf.edu

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

  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 http://numericalmethods.eng.usf.edu/topics/newton_divided_difference_method.html

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

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