Numerical solutions of equations.

Decimal Search

We will look at the polynomial equations which cannot be solved by algebraic methods.  All linear and quadratic equations can be solved algebraically so we are interested in equations with powers of x3 at least.
The first method to be looked at is the Decimal Search.  This involves finding an interval of the x- axis in which a root of the equation lies. 

Let's consider the function: y=x5+4x2-2. The region to scan to find the first root is -4<x<4.

We can identify three intervals which have a change of sign:  [-2,-1]  , [-1,0] , [0,1].
If we sketch the graph of the function three roots are confirmed in these intervals:

I’ll draw a sketch to show the curve crossing the x- axis in this interval.
I am  going to concentrate my search on this interval [0,1]  and look at interval of width 0.1 unit.

The change of sign occurs in [0.6 , 0.7] which is where the root lies.

This process can now be continued using intervals of width 0.01 then by using intervals of width 0.001 etc.
 Below I will show those intervals and I will bold and highlight those where change of sign occurs in.

Join now!

I can express this interval using the Error bounds.
Change of sign method has big advantage because they provide bounds (the two ends of the interval) within root lies. Knowing that the root lies in the interval [0.680760, 0.680770] means that you can take the root as 0.680765 with a maximum error of +/- 0.000005.
I must think about problems. Are there any occasions when this decimal search method might not find root? I’ve got few examples of that.
In interval [1,2] we have no change of sign (look to the  table above). ...

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