Given stepsizes h 1 and h 2 , develop a numerical scheme to approximate f ′ ′ ( x 0 ) with function values f ( x 0 ) , f ( x 0 + h 1 ) and f ( x 0 + h 2 ) . Under what conditions will your method not work?

Question

Given stepsizes       h    1   and       h    2   , develop a numerical scheme to approximate       f          ′      ′        (      x    0    ) with function values   f  (      x    0    ) ,   f  (      x    0    +      h    1    ) and   f  (      x    0    +      h    2    ) . Under what conditions will your method not work?

Lisbonaid · Accepted Answer

Step 1If       h    1    =  −      h    2   , you can use second derivative midpoint formula      f                          ″              (      x    0    )  =      1          h      2        [  f  (      x    0    −  h  )  −  2  f  (      x    0    )  +  f  (      x    0    +  h  )  ]  +  o  (      h    3    )If       h    1    ≠  −      h    2   , you can do like this:  f  (      x    0    +      h    1    )  =  f  (      x    0    )  +      h    1        f    ′    (      x    0    )  +            h      1      2        2        f                          ″              (      x    0    )  +  o  (      h    1    3    )  f  (      x    0    +      h    2    )  =  f  (      x    0    )  +      h    2        f    ′    (      x    0    )  +            h      2      2        2        f                          ″              (      x    0    )  +  o  (      h    2    3    )Minus the two equations  f  (      x    0    +      h    2    )  −  f  (      x    0    +      h    1    )  =  (      h    2    −      h    1    )      f    ′    (      x    0    )  +                    h        2        2            −              h        1        2              2        f                          ″              (      x    0    )  +  o  (      h    2    3    −      h    1    3    )and      f    ′    (      x    0    )  =            f      (              x        0            +              h        2            )      −      f      (              x        0            +              h        1            )                      h        2            −              h        1              +  o  (      h    2    2    −      h    1    2    )But, considering the accuracy, in general, we pick       h    1    =  −      h    2   , which cancels out the       f    ′    (      x    0    ) that has a low accuracy in a certain sense.An alternative way to find       f                          ″              (      x    0    ) using only   f  (      x    0    ) ,   f  (      x    0    +      h    1    ) and   f  (      x    0    +      h    2    ) and not involving       f    ′    (      x    0    ) is that you should express       f    ′    (      x    0    ) with   f  (      x    0    ) ,   f  (      x    0    +      h    1    ) and   f  (      x    0    +      h    2    ) , so a potential formula is  f  (      x    0    +      h    1    )  =  f  (      x    0    )  +            h      1                      h        1            +              h        2              [  f  (      x    0    +      h    1    )  +  f  (      x    0    +      h    2    )  −  2  f  (      x    0    )  −  (                    h        2        2            −              h        1        2              2        f                          ″              (      x    0    )  )  ]  +            h      1      2        2        f                          ″              (      x    0    )

Kiana Dodson · Accepted Answer

Step 1      f    1    ≈      f    0    +      h    1        f    0    ′    +            h      1      2        2        f    0    ″    ,      f    2    ≈      f    0    +      h    2        f    0    ′    +            h      2      2        2        f    0    ″    .Then by elimination of       f    0    ′   , you get      h    2    (      f    1    −      f    0    )  −      h    1    (      f    2    −      f    0    )  =                    h        1                    h        2            (              h        1            −              h        2            )        2        f    0    ″    .

Given stepsizes h 1 </msub> and h 2 </msub> , develop a nume

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