Lab 8 · Lecture 8 · Fixed-Point Iteration
Fixed-Point Lab
Draw cobwebs, test rearrangements, and see linear convergence in numbers.
Type g(x) or pick a rearrangement. Step the iteration and watch the cobweb. The note below the plot checks |g′| at the current point.
Fixed-point iteration
Try this
Check g′ before iterating
Try this
Q1 Find a root of by the iteration method, correct to 4 decimal places.
Root in (5, 6). , . From : 5.477226, 5.648797, 5.709219, 5.730347, 5.737716, 5.740284, 5.741179, 5.741491, 5.741599, 5.741637. Root ≈ 5.7416.
Q2 State the condition for convergence of the iteration method and prove it briefly.
near the root. Proof: , so .
Q3 Solve by iteration, 4 decimal places.
, . From : 0.608445, 0.606846, 0.607150, 0.607092, 0.607103, 0.607101 → 0.6071.
Q4 Why does fail for the root 5.7417 of ?
, so : each step multiplies the error by about 2.87, so the iterates move away.
Q5 Fixed-point iteration has errors 0.08, 0.024, 0.0072. Estimate and the number of further steps to reach .
Ratio 0.3, so . Need : , so 8 more steps.