How do we know the correct root in bisection method?
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The bisection method
Numerical analysis > The bisection method
The bisection method is based on the theorem of existence of roots for continuous functions, which guarantees the existence of at least one root {\displaystyle \alpha } \alpha of the function {\displaystyle f} f in the interval {\displaystyle [a,b]} {\displaystyle [a,b]} if {\displaystyle f(a)} {\displaystyle f(a)} and {\displaystyle f(b)} {\displaystyle f(b)} have opposite sign. If in {\displaystyle [a,b]} [a,b] the function {\displaystyle f} f is also monotone, that is {\displaystyle f'(x)>0\;\forall x\in [a,b]} {\displaystyle f'(x)>0\;\forall x\in [a,b]}, then the root of the function is unique. Once established the existence of the solution, the algorithm defines a sequence {\displaystyle x_{k}} {\displaystyle x_{k}} as the sequence of the mid-points of the intervals of decreasing width which satisfy the hypothesis of the roots theorem.