Lagrangian Interpolation

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Lagrangian 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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Lagrange Method of Interpolation
http//numericalmethods.eng.usf.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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Lagrangian 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 Lagrange 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 Lagrange method for quadratic
  interpolation.

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


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Quadratic Interpolation (contd)
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Comparison Table
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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 a fifth order Lagrange polynomial.

Figure 2 Location of holes on the rectangular
plate.
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Fifth Order Interpolation




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Fifth Order Interpolation (contd)
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Fifth Order Polynomial (contd)
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Fifth Order Polynomial (contd)
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Fifth Order Polynomial (contd)
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Fifth Order Polynomial (contd)
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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/lagran
  ge_method.html

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