Newton

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Newtons 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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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
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
61C. The specific heat of water is given as a
function of time in Table 1. Use Newtons divided
difference method with a first order and then a
second order polynomial to determine the value of
the specific heat at T 61C.
Table 1 Specific heat of water as a function of
temperature.
Temperature, Specific heat,
22 42 52 82 100 4181 4179 4186 4199 4217
Figure 2 Specific heat of water vs.
temperature.
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Linear Interpolation
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Linear Interpolation (contd)
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Quadratic Interpolation
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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
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
61C. The specific heat of water is given as a
function of time in Table 1. Use Newtons divided
difference method with a third order polynomial
to determine the value of the specific heat at T
61C.
Table 1 Specific heat of water as a function of
temperature.
Temperature, Specific heat,
22 42 52 82 100 4181 4179 4186 4199 4217
Figure 2 Specific heat of water vs.
temperature.
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Example

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Example
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Example
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Comparison Table
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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