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Newton-Raphson and Bisection Methods

  1. Newton-Raphson Method
     
    We have find the solution of f (xy) = 0 and g (xy) = 0
     
    First find an approximate solution of equation (x0y0) by graphical or trial and error method.
     
    From this we find more approximate values (x1y1) such that
     
    x1 = x0 + h and y1 = y0 + k
     
    ∴ f (x0 + hy0 + k) = 0 and g (x0 + hy0 + k) = 0
     
    From Taylor’s series we get,
     
    Description: 398.png = 0 and Description: 403.png = 0
     
    Solving these equations we get h and k.
     
    Thus we find x1 and y1
     
    From this value of x1 and y1 we find x2 and y2 in similar way
     
  2. Bisection Method
     
    For a non-linear equation f (x) and for any two number a and b if f (a), f (b) > 0, then there is at least one root for the equation f (x) in between a and b.
     
    Now let x1 = a and x2 = b
     
    We find the mid-point of x1 and x2 i.e. x0
     
    x0 Description: 408.png
     
    If f (x0) = 0, x0 is the root
     
    If f (x0f (x1) < 0, root is in between x0 and x1
     
    If f (x0f (x2) < 0, root is in between x0 and x2
     
    and this process in continued with either (x0x1) or with (x0x2).




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