Newton

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Newtons 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

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Newtons Divided Difference Method of
Interpolation http//numericalmethods.eng.us
f.edu
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What is Interpolation ?

Given (x0,y0), (x1,y1), (xn,yn), find the
value of y at a value of x that is not given.
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Interpolants

• Polynomials are the most common choice of
  interpolants because they are easy to
• Evaluate
• Differentiate, and
• Integrate.

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Newtons Divided Difference Method

• Linear interpolation Given
  pass a linear interpolant through the data
• where

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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 Newtons Divided Difference method
  for linear interpolation.

Figure 2 Location of holes on the rectangular
plate.
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Linear Interpolation
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Linear Interpolation (contd)
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Quadratic Interpolation
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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 Newtons Divided Difference method
  for quadratic interpolation.

Figure 2 Location of holes on the rectangular
plate.
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Quadratic Interpolation (contd)


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Quadratic Interpolation (contd)
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Quadratic Interpolation (contd)
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General Form
where


Rewriting
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General Form
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General form
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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 Newtons Divided Difference method
  for a fifth order polynomial.

Figure 2 Location of holes on the rectangular
plate.
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Example

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Example
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Example
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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

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