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Derivation of 2nd divided difference

  1. Feb 18, 2012 #1
    I'm trying to understand how the second divided difference is formulated. I understand that the first divided difference is just the equation of a slope.

    [itex]f(x_{i},x_{i+1})=\frac{f(x_{i})-f(x_{i+1})}{x_{i}-x_{i+1}}[/itex]

    Every source that I have read always jumps to the second divided difference by saying "and by induction"

    [itex]f(x_{i},x_{i+1},x_{i+2})=\frac{f(x_{i},x_{i+1})-f(x_{i+1},x_{i+2})}{x_{i+2}-x_{i+1}}[/itex]

    How is induction used to get this equation?
     
  2. jcsd
  3. Feb 21, 2012 #2
    http://nptel.iitm.ac.in/courses/Webcourse-contents/IIT-KANPUR/mathematics-2/node112.html [Broken]

    you can check the above site if you think it would be useful.
     
    Last edited by a moderator: May 5, 2017
  4. Feb 21, 2012 #3

    HallsofIvy

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    Having got two consecutive first differences,
    [tex]\frac{f(x_i)- f(x_{i+1}}{x_i- x_{i+1}}[/tex]
    [tex]\frac{f(x_{i+1}- f(x_{i+2}}{x_{i+1}- x_{i+2}}[/tex]

    Now, for the first difference of those:
    [tex]\frac{\frac{f(x_i)- f(x_{i+1}}{x_i- x_{i+1}}- \frac{f(x_{i+1}- f(x_{i+2}}{x_{i+1}- x_{i+2}}}{x_i- x_{i+2}}[/tex]
     
  5. Feb 21, 2012 #4
    Thank you. I new it was some type of substitution but I failed to think of using the first difference with the first differences.
     
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