To solve f(x) = x² − 4 = 0, two rearrangements are tried near the positive root x* = 2: (A) g(x) = 4/x and (B) g(x) = (x + 4/x)/2. For g(x) = 4/x, g'(2) = −1; for g(x) = (x + 4/x)/2, g'(2) = 0. Which iteration converges?
ABoth converge, since both are valid rearrangements of the same equation
BNeither — fixed point iteration cannot find roots of polynomial equations
COnly (A), because it is a simpler expression
DOnly (B), because |g'(2)| = 0 < 1 for (B) while |g'(2)| = 1 for (A)
Convergence requires |g'(x*)| < 1. For (A), |g'(2)| = 1 — exactly at the boundary — so iteration oscillates and does not converge. For (B), |g'(2)| = 0, guaranteeing convergence (and in fact quadratic convergence, since this is Newton's method in disguise). The key insight is that the same equation produces entirely different convergence behavior depending on how it is rearranged — the equation determines x*, but the choice of g determines whether iteration finds it.
Question 2 Multiple Choice
A fixed point iteration scheme has |g'(x*)| = 0.05. A second scheme for the same problem has |g'(x*)| = 0.9. How do their convergence rates compare?
ABoth converge at the same rate since they solve the same underlying problem
BThe second converges faster since 0.9 is closer to 1 and therefore 'stronger'
CThe first converges much faster — errors shrink by roughly 95% each iteration versus only 10% for the second
DNeither converges — both values must equal zero for fixed point iteration to work
The convergence rate of fixed point iteration is determined by |g'(x*)|: at each step, the error scales by approximately this factor. With |g'| = 0.05, errors shrink by ~95% per iteration. With |g'| = 0.9, errors shrink by only ~10% per iteration — roughly 18 times slower. This is why achieving |g'(x*)| = 0, as Newton's method does by construction, gives quadratic convergence: the error doesn't merely scale, it squares at each step.
Question 3 True / False
The convergence of fixed point iteration depends on which rearrangement x = g(x) is used, not just on the equation f(x) = 0 being solved.
TTrue
FFalse
Answer: True
The explainer demonstrates this with f(x) = x² − 2: one rearrangement gives |g'(√2)| ≈ 1.83 (diverges), another gives |g'| = 1 (boundary, oscillates), and Newton's method gives |g'| = 0 (quadratic convergence). All three start from the same equation. The equation f(x) = 0 determines what x* is; the choice of g determines whether iteration converges to it and how fast.
Question 4 True / False
If fixed point iteration converges, it usually converges at a quadratic rate.
TTrue
FFalse
Answer: False
Basic fixed point iteration achieves linear convergence: errors decrease by a constant factor |g'(x*)| at each step. Quadratic convergence — where the error roughly squares at each step — is special, occurring only when |g'(x*)| = 0. Newton's method achieves this by careful construction, but it is the engineered refinement, not the norm. The explainer states: 'Fixed point iteration in its basic form... converges [linearly]. Newton's method achieves |g'(x*)| = 0... which is why it converges quadratically.'
Question 5 Short Answer
Why can the same equation f(x) = 0 produce both convergent and divergent fixed point iterations depending on how it is rearranged?
Think about your answer, then reveal below.
Model answer: Because convergence depends on |g'(x*)| < 1 for the specific function g used in the rearrangement x = g(x). Different algebraic rearrangements of the same equation produce different functions g with different derivatives at the fixed point. When |g'(x*)| < 1, g is a contraction near x* — nearby iterates are pulled toward the solution. When |g'(x*)| > 1, g repels nearby iterates and iteration diverges. The equation determines what x* is; the rearrangement determines whether iteration converges to it.
This is why 'just rearrange and iterate' is not a reliable strategy. The contraction condition |g'(x*)| < 1 must be verified for each specific rearrangement, and ideally |g'(x*)| should be as small as possible for fast convergence. The art of fixed point methods lies in finding a g that satisfies this condition — Newton's method is essentially an algorithm for constructing a g where g'(x*) = 0 exactly.