Discussion
To find a real root of the equation \(x^{3} + 4x^{2} - 10 = 0\) in the interval \(\left(1, \frac{3}{2}\right)\) by using the fixed-point iteration scheme, consider the following two statements:
S1: The iteration scheme \(x_{k+1} = \sqrt{\dfrac{10}{4 + x_{k}}}\), \(k = 0, 1, 2, \ldots\), converges for any initial guess \(x_{0} \in \left(1, \frac{3}{2}\right)\).
S2: The iteration scheme \(x_{k+1} = \frac{1}{2}\sqrt{10 - x_{k}^{3}}\), \(k = 0, 1, 2, \ldots\), diverges for some initial guess \(x_{0} \in \left(1, \frac{3}{2}\right)\).
Then, which one of the following is correct?
Answer: A
S1 is TRUE and S2 is FALSE.
A fixed-point scheme \(x_{k+1} = g(x_{k})\) converges from every start in an interval if g maps the interval into itself and \(|g'| \lt 1\) there.
S1: \(g(x) = \sqrt{10/(4 + x)}\) is decreasing, with \(g(1) = \sqrt{2} \approx 1.414\) and \(g(1.5) \approx 1.348\), so it maps the interval into itself; and \(|g'(x)| = \sqrt{10}/(2(4 + x)^{3/2}) \le 0.15\). So it converges: S1 is true.
S2: \(g(x) = \tfrac{1}{2}\sqrt{10 - x^{3}}\) is also decreasing, with \(g(1) = 1.5\) and \(g(1.5) \approx 1.287\), so it maps (1, 3/2) into itself, and \(|g'(x)| = 3x^{2}/(4\sqrt{10 - x^{3}})\) is at most about 0.66 there. It converges for every start in the interval, so it never diverges and S2 is false.
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