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NUMERICAL DIFFERENTIATION
Methods based on Finite Difference:
By Taylor seriesexpansion 
          E f(x) = f (x + h)  =  f(x)  +  h f '(x)  +  h2/2! f '' (x)  +  .  .  .
           
            = ( 1  +  hD  +  h2D2/2!  + . . . )  f (x)

            =  ehDf(x)       where D = d/dx


    Þ  E = ehD  (or)

           hD = log E
     
     
                =    log(1 + D) = D - 1/2D2 + 1/3D3 - . . .
    - log(1 - Ñ) = Ñ + 1/2Ñ2 + 1/3Ñ3 + . . .
    2sinh-1(d / 2) = d - (12 / 22.3!) d3 + . . .

    hD f(xi) = h f '(xi)
     
               =    Dfi- 1/2D2fi + 1/3D3fi - . . .
    Ñfi+ 1/2Ñ2fi + 1/3Ñ3fi + . . .
    dfi - (12/22.3!) d3fi+ . . .

    since  m = Ö{1+d2/4}
     
    hD = 
    m
    		 2sinh-1 (d/2)
    Ö{1+d2/4}

    = m(d - 12/3! d3 + 12.22 / 5! d5 - . . .)

    Thus we get  h f '(xi)  = mdfi- 12/3! md3fi + 12.22 / 5! md5fi - . . .

    Similarly, we obtain
     
    hrDr = 
    		  
    Dr - 
     1 
    rDr+1 + 
     r(3r + 5) 
    rDr+1 - . . .
    		    
    2
    24
    		    
     
    		            
    Ñr + 
     1 
    rÑr+1 + 
     r(3r + 5) 
    rÑr+1 + . . .
    		    
    2
    24
    		    
     
    		            
    mdr - 
    r + 3
    mdr+2 +
    r(5r + 52) + 135 
    mdr+4 - . . .,
    		  
     r odd
    24
    5760
    		  
     
    		            
    dr -
     r 
    dr+2 +
     r(5r + 22)
    		  dr+4 - . . .,  
     r even
    24
     5760
    		  

     
    h f '(xi)  = 
    		  
    Dfi - 
     1
    D2fi + 
     1 
     D3fi - . . .
    2
    3
     
    		        
    Ñfi + 
     1 
    Ñ2fi + 
     1
    Ñ3fi + . . .
    2
    3
     
    		        
    mdfi - 
     1 
    md3fi +
     1
    md5fi - . . .
    6
    30

     
    h2 f ''(xi)  =
    		  
    D2fi - 
     1
    D3fi + 
     11 
     D4fi - . . .
    2
    12
     
    		        
    Ñ2fi + 
     1 
    Ñ3fi + 
     11
    Ñ4fi + . . .
    2
    12
     
    		        
    d2fi - 
     1 
    d4fi +
     1
    d6fi - . . .
    12
    90

    Keeping only the first term in the above approximations, we get
     
    f ''(xi)  = 
    		  
    fi+1 - fi
    h
     
    fi - fi-1
    h
     
    fi+1 - fi-1
    2h

    Methods based on undetermined coefficients:

    Here, the rth derivative of the function is expressed as a linear combination of the function values at an arbitrarily chosen nodal points in the form
     
     

    a
    hr f r(xi)  =
    S
    		 ckfi+k
    for uniformly distributed points and 
    k = -a

     
    a
    hr f r(xi)  =
    S
    		 ckfi+k
    for non-uniformly distributed points
    k = ±b

    Then the truncation error can be written as
     

    [
    a
    		 ]
    Er(xi) =  1 / hr  hr f r(xi)  -
    S
    		 ckfi+k
    k = -a
    or
    [
    a
    		 ]
    Er(xi) =  1 / hr  hr f r(xi)  -
    S
    		 ckfi+k
    k = ±b

    the coefficients  ck's have to be determined depending on the order of the method required.

    For example consider
     

      h2 f ''(xi) = c-2fi-2 +  c-1fi-1 +  c0fi +  c1fi+1 +  c2fi+2
       
             = (c-2 +  c-1 +  c0 +  c1 +  c2) fi +
          h (-2c-2 +  c-1 +  c1 +  2c2) f 'i +
          h2/2 (4c-2 +  c-1 +  c1 +  4c2) f ''i +
          h3/6 (-8c-2 +  c-1 +  c1 +  8c2) f '''i +
          h4/24 (16c-2 +  c-1 +  c1 +  16c2) f 'vi + . . . +
          (by Taylor series expansion)
    Þ
      c-2 +  c-1 +  c0 +  c1 +  c2 = 0
      -2c-2 +  c-1 + c1 +  2c2 = 0 
      4c-2 +  c-1 +  c1 +  4c2 = 2
      -8c-2 +  c-1 +  c1 +  8c2 = 0
      16c-2 +  c-1 +  c1 +  16c2 = 0


    Þ     c-2 =  c2 =  -1/2

      c-1 =  c1 =  16/12
      c0 =  -30/12
    Þ f ''(xi) = 
    -fi-2 + 16fi-1 - 30fi + 16fi+1 + fi+2
    12h2

    and the truncation error is
     
    ( since
    h5
    (-32c-2 -  c-1 + c1 +  32c2) = 0)
    120

     
    we get 
    1
    		 [
    h5
    (64c-2 +  c-1 + c1 +  64c2)
    		 ] f v'(x)   =  -h4  f v'(x)
    h2
    		 720
    90

    xi-2 < x < xi+2
Solution of Transcendental Equations | Solution of Linear System of Algebraic Equations | Interpolation & Curve Fitting
Numerical Differentiation & Integration | Numerical Solution of Ordinary Differential Equations
Numerical Solution of Partial Differential Equations