The value of \(x_3\) is then chosen as either the midpoint of \(x_0\) and \(x_2\) or as the midpoint of \(x_2\) and \(x_1\), depending on whether \(x_0\) and \(x_2\) bracket the root, or \(x_2\) and \...The value of \(x_3\) is then chosen as either the midpoint of \(x_0\) and \(x_2\) or as the midpoint of \(x_2\) and \(x_1\), depending on whether \(x_0\) and \(x_2\) bracket the root, or \(x_2\) and \(x_1\) bracket the root. The algorithm proceeds in this fashion and is typically stopped when the increment to the left side of the bracket (above, given by \((x_1 - x_0)/2)\) is smaller than some required precision.
As we have studied limits, we have gained the intuition that limits measure 'where a function is heading.' We have seen, though, that this is not necessarily a good indicator of what the function actu...As we have studied limits, we have gained the intuition that limits measure 'where a function is heading.' We have seen, though, that this is not necessarily a good indicator of what the function actually is. This can be problematic; functions can tend to one value, but attain another. This section focuses on functions that do not exhibit such behavior.
