Direct Method of Interpolation

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Direct Method of Interpolation

• Industrial Engineering Majors
• Authors Autar Kaw, Jai Paul
• http//numericalmethods.eng.usf.edu
• Transforming Numerical Methods Education for STEM
  Undergraduates

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Direct 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.
Figure 1 Interpolation of discrete.
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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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Direct Method

• Given n1 data points (x0,y0), (x1,y1),..
  (xn,yn),
• pass a polynomial of order n through the data
  as given
• below
• where a0, a1,. an are real constants.
• Set up n1 equations to find n1 constants.
• To find the value y at a given value of x,
  simply substitute the value of x in the above
  polynomial.

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Example

• A curve needs to be fit through the given
  points to fabricate the cam. If the cam follows a
  straight line profile between x 1.28 to x
  0.66, what is the value of y at x1.1 ? Find
  using the direct method and linear interpolation.

Point x (in.) y (in.)
1 2.20 0.00
2 1.28 0.88
3 0.66 1.14
4 0.00 1.20
5 0.60 1.04
6 1.04 0.60
7 1.20 0.00
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Linear Interpolation






Solving the above two equations gives,
Hence
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Example

• A curve needs to be fit through the given
  points to fabricate the cam. If the cam follows a
  straight line profile between x 1.28 to x
  0.66, what is the value of y at x1.1 ? Find
  using the direct method and quadratic
  interpolation.

Point x (in.) y (in.)
1 2.20 0.00
2 1.28 0.88
3 0.66 1.14
4 0.00 1.20
5 0.60 1.04
6 1.04 0.60
7 1.20 0.00
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Quadratic Interpolation
Quadratic Interpolation

Solving the above three equations gives


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Quadratic Interpolation (contd)
The absolute relative approximate error obtained
between the results from the first and second
order polynomial is
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Comparison Table
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Example

• A curve needs to be fit through the given
  points to fabricate the cam. If the cam follows a
  straight line profile between x 1.28 to x
  0.66, what is the value of y at x1.1 ? Find
  using the direct method and a sixth order
  polynomial.

Point x (in.) y (in.)
1 2.20 0.00
2 1.28 0.88
3 0.66 1.14
4 0.00 1.20
5 0.60 1.04
6 1.04 0.60
7 1.20 0.00
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Sixth Order Interpolation




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Sixth Order Interpolation (contd)
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Sixth Order Polynomial (contd)
Solving the above seven equations gives
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Sixth 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/direct
  _method.html

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