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Solving Equations

A zero, x, of a real-valued continuous function, f(x), i.e., such that f(x)=0, can be found by an application of the binary search algorithm. Values ‘Lo’ and ‘Hi’ are chosen such that (i) Lo<Hi, and (ii) f(Lo)<0 and f(Hi)>0 or v.v..

Mid=(Lo+Hi)/2 and f(Mid) is computed. If f(Mid) has the same sign as f(Lo) then Lo is moved up to Mid. If it has the same sign as f(Hi) then Hi is moved down to Mid. The algorithm terminates when Hi-Lo is "small". It is not a good idea to wait until f(Mid)=0.

The HTML FORM below can be used to solve a cubic polynomial (we'll ignore the fact that there are better ways to solve cubics). Change the coefficients of the powers of x and click on the solve button to find a zero:

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f(x)=x3+ x2+ x1+ =0
-trace
x  =  [  ]
f(x)=[  ]

Exercises

  1. Find a solution to x3+x2+x+1=0. What are the other two solutions?
  2. Multiply out f(x)=(x-1)(x-2)(x-3), by hand, and use the FORM to find one of the solutions to f(x)=0.
  3. What happens if the coefficient of x3 is set to zero, and the other coefficients are set to 1? Why does this happen? Could the problem be fixed?
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