Question

1. Bisection Method(15 pts) . The Bisection method is unique in that you can prescribe error tolerance in either z, ?, or in y, ey (a) Commonly, we use ?, as the criteria for our solution convergence. In this case, how do we conclude that our root finding algorithm is "done"

Answer 1

for part (a)

In bisection method, for each iteration, we calculate the mid point of the current bracket and verify the sign of function value at this mid points with the values at the bracket. For the opposite sign value, we update the corresponding bracket with the midpoint.

On successive mid point creation, as soon as the values of the function at midpoint is close to required value witihin the tolerance i.e. . We say the root finding algorithm is done!

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(f) One such example can be $y=x^2$ . for the interval [-2 1]. Though it has the root within the interval i.e. x=0. But as the function's sign is always evaluated to positive, it will never land on the actual root.