Fixed-point iteration and numerical roots — 9709 P3
Use a stated rearrangement and check its apparent convergence against a root bracket.
Notes and key methods
Rearrange \(\displaystyle f(x)=0\) into \(\displaystyle x=F(x)\), choose the stated starting value and compute \(\displaystyle x_{n}\)₊\(\displaystyle _{1}=F(x_{n})\), retaining guard digits. A different rearrangement of the same equation may fail to converge.
Agreement between successive rounded iterates is evidence to examine, not a universal error guarantee. Check the final rounded value against the original equation and, where possible, a sign-change bracket at the rounding boundaries.
Check the conditions
Use the real cube root in this iteration.
Keep guard digits until the final stated rounding.
Three worked examples
Example 1 · Fixed-point iteration and accuracy
Solve \(x^{3} + x = 32\) using \(x_{n+1}=\sqrt[3]{32-x_n}\), with \(\displaystyle x_{0}=3. A\) calculator is appropriate; keep guard digits.
Derive the stated iteration from the cubic.
Calculate the first three iterates.
Find the root to six decimal places and justify the rounding.
Find the limiting root to at least six decimal places.
Show the reasoning
(a)
Isolate \(\displaystyle x^{3}\) and take its real cube root.
Answer: \(x=\sqrt[3]{32-x}\)
(b)
Use the recurrence in order, with guard digits retained.