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Newton’s Divided Difference Polynomial Method of Interpolation

Learn about Newton's divided difference polynomial method of interpolation, a common choice for interpolants due to its ease of evaluation, differentiation, and integration. Explore linear, quadratic, and higher order interpolations with examples of practical applications in robotic arm quality checks. Transform your numerical methods education for STEM undergraduates with resources and examples to enhance understanding.

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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 Computer 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 A robot arm with a rapid laser scanner is doing a quick quality check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below. If the laser is traversing from x = 2 to x = 4.25 in a linear path, find the value of y at x = 4 using the Newton’s Divided Difference method for linear interpolation. Figure 2 Location of holes on the rectangular plate. 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 A robot arm with a rapid laser scanner is doing a quick quality check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below. If the laser is traversing from x = 2 to x = 4.25 in a linear path, find the value of y at x = 4 using the Newton’s Divided Difference method for quadratic interpolation. Figure 2 Location of holes on the rectangular plate. http://numericalmethods.eng.usf.edu

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

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

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

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

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

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

  17. Example A robot arm with a rapid laser scanner is doing a quick quality check on holes drilled in a rectangular plate. The hole centers in the plate that describe the path the arm needs to take are given below. If the laser is traversing from x = 2 to x = 4.25 in a linear path, find the value of y at x = 4 using the Newton’s Divided Difference method for a fifth order polynomial. Figure 2 Location of holes on the rectangular plate. http://numericalmethods.eng.usf.edu

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

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

  20. Example 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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