Different methods of solving equations compared. From the Excel tables of each method, we know that method 1 (change of sign method) takes 28 steps to find the root, while method 2 (Newton-Raphson method) and method 3 (rearrangement) take 4 and 17 steps r

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Numerical Methods

Method 1: Change of sign method

Equation 1:



  Use the Autograph and it gives us the overall view of the graph of the function.

  If we zoom in on the x and y axes, we can see that the graph shows that the equation has a root between x=0 and x=1. Bisect the interval between 0 and 1, then bisect the new interval where the sign changes and we can see that the two boundary lines get closer and closer to each other and will meet at a point finally (in theory).

  However, in practice, we are not able to find out the exact values of the roots but we can still use this method to find the approximate values. Excel can help us with that.

 As shown in the table below by inputting initial values of a=0, b=1, we can calculate that f(a)>0, f(b)<0. Then divide the interval in half and get the midpoint c, where c==0.5, and also the value of f(c). Since f(c)>0, we then can get a subinterval between c and b. The previous steps are then repeated. This method is applied recursively to the subinterval where the sign change occurs.

  The table below shows the data with accuracy of 8 decimal places.

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Error bound: ±0.000000005 (9dp)

The table below shows part of the formulas used in the Excel.

Failure in this method

  This method cannot always be applied to every equation successfully. For example in the following case:

  Equation A:


  The graph above shows us that the roots of the equation are very close. Zoom in on the x and y axes, we can see that they all lie between x=0 and x=1. Use Autograph that follows the same steps ...

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