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When sets meet tangentially, a residual may still locate the solution—but only through a fractional-power error bound that predicts slower convergence.
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Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Define Hölder geometry
Route 2 · Work the rate ledgers
Route 3 · Transfer to projections
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · DEFINE HÖLDER GEOMETRY
1. When a linear bridge is too strong
A metric-subregularity bound links distance to a solution set linearly with a residual. Tangential geometry can break that linear relationship: the residual shrinks much faster than the actual distance. Convergence may continue, but a linear-rate certificate becomes dishonest.
A Hölder error bound weakens the bridge. It allows distance to be bounded by a constant times a fractional power of the residual. The exponent records how flat the geometry is near the solution and predicts a slower rate.
CHAPTER 2 OF 12 · DEFINE HÖLDER GEOMETRY
2. The Hölder error-bound form
For a fixed-point operator T, write r(x)=||x−T(x)||. A Hölder error bound on a bounded region has the form dist(x,Fix T) at most mu times r(x) raised to gamma, with gamma between zero and one.
Gamma equal to one recovers the linear error bound. A smaller exponent permits the residual to vanish more rapidly than distance. Constants and exponents are local or bounded-set claims unless a global result is proved.
CHAPTER 3 OF 12 · WORK THE RATE LEDGERS
3. A fractional-power ledger
Take T(x)=x−x³ on the nonnegative real line near zero. The solution is zero, the distance is x and the residual is x³. Therefore distance equals residual raised to one third.
| Point x | Residual x cubed | Residual to the one-third power | Distance to solution |
|---|---|---|---|
| 0.5 | 0.125 | 0.5 | 0.5 |
| 0.2 | 0.008 | 0.2 | 0.2 |
| 0.1 | 0.001 | 0.1 | 0.1 |
| 0.05 | 0.000125 | 0.05 | 0.05 |
This example continues the residual counterexample in metric subregularity and linear certificates by identifying the correct weaker exponent.
CHAPTER 4 OF 12 · WORK THE RATE LEDGERS
4. From iteration to sublinear rate
The iteration x next equals x minus x cubed decreases slowly near zero. Its continuous approximation satisfies dx/dk approximately minus x cubed, giving x around one divided by the square root of 2k. That is sublinear rather than geometric.
| Iteration count k | Approximate x from 1 over square root of 2k | Geometric benchmark 0.5 to the k | Rate reading |
|---|---|---|---|
| 10 | 0.224 | 0.00098 | Sublinear tail remains visible |
| 100 | 0.071 | about 0 | Fractional-power decay |
| 1000 | 0.022 | about 0 | Progress continues without a fixed contraction |
A straight log-linear plot is therefore the wrong diagnostic. Log-log slopes and residual-distance exponents are more informative.
CHAPTER 5 OF 12 · WORK THE RATE LEDGERS
5. Tangential intersections
Consider two sets that touch with a common tangent. Moving a small horizontal distance from the contact point can create a much smaller vertical separation, often of quadratic order. A projection residual based on that separation may therefore be proportional to distance squared or another power.
Transversal intersections behave more like a fixed angle and commonly support linear regularity. As the angle collapses into tangency, the linear constant can blow up or cease to exist. Hölder regularity records what remains.
CHAPTER 6 OF 12 · TRANSFER TO PROJECTIONS
6. Averaged operators and descent
For an averaged operator, a Fejér-type inequality can decrease squared distance by a multiple of the squared residual. Combine that descent with a Hölder error bound to derive a recurrence for distance.
When gamma is one, the recurrence yields a geometric contraction under suitable conditions. When gamma is below one, it yields a polynomial-style bound. The exact exponent depends on the theorem; cite it rather than extrapolating from one example.
CHAPTER 7 OF 12 · TRANSFER TO PROJECTIONS
7. Changed-context transfer: tangent constraint curves
Transfer the fixed-point example to alternating projections between the x-axis and a curve that touches it at the origin. Starting near the contact, successive projections may make very small normal corrections while remaining noticeably far along the tangent direction.
The transfer check is to perturb the curve so it crosses at a nonzero angle. The observed rate should improve toward linear. If it does not, another issue—nonunique projections, nonconvexity or implementation error—may be controlling behaviour.
CHAPTER 8 OF 12 · TRANSFER TO PROJECTIONS
8. Estimating an exponent responsibly
On synthetic problems, compute distance to the known solution set and fit the relationship between log distance and log residual over shrinking neighbourhoods. The slope estimates gamma only when the bound is close to tight and numerical precision is adequate.
On applied problems, exact distance may be unavailable. Use primal-dual gaps, certified lower bounds or high-accuracy references where justified. Treat an empirical exponent as descriptive evidence, not a universal theorem.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and common misconceptions
A Hölder exponent is not a topological invariant of the space. It belongs to a particular mapping, residual, set system and neighbourhood. Reparameterising a residual can change the exponent without changing the zero set.
Sublinear does not mean unusably slow, and a poor worst-case bound need not match typical performance. Conversely, a fast early phase does not prove a linear asymptotic rate. Tangency, conditioning and starting region all matter.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
State the operator, norm, fixed-point set, bounded region, constant and exponent. Record the descent inequality and the theorem translating regularity into a rate. Distinguish an upper-bound exponent from an empirically fitted slope.
Test several starting points, perturb the intersection angle, and plot both residual and independent distance evidence. Use sufficient precision near the solution. Report when numerical stopping truncates the apparent tail.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Does every tangential intersection have exponent one half?
No. The exponent depends on the order of contact and the chosen residual. Quadratic contact often suggests one half, but higher-order contact gives other powers.
Can convergence be linear even when one Hölder bound has gamma below one?
Possibly, because the bound may be non-tight or another stronger local property may hold on the actual orbit. The certificate and observed rate must be distinguished.
Is gamma estimated from residuals alone?
Not reliably. Estimating the bridge requires some independent information about distance to the solution set.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Borwein, Li and Tam analyse Hölder regularity and convergence rates for averaged fixed-point iterations. Borwein, Li and Yao derive rates for cyclic projections on semialgebraic convex sets. Lewis, Luke and Malick connect intersection geometry with local projection convergence. The scalar ledgers above are original teaching examples.
Continue through the Topology root, metric subregularity and linear certificates, Fejér monotonicity and convergence gaps and alternating projections. The Gold Standard of Education and How X Works provide the wider route.
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