Combine fast proposals with a reliable bracket

Try

A derivative can be expensive or unavailable. Secant iteration replaces it by the slope through two previously evaluated points. For f(x)=x²−2, start at 1 and 2. The line through (1,−1) and (2,2) crosses zero at 4/3. With the two latest points, the next crossing is 7/5. Only one new function evaluation is required per update after the initial pair, if earlier values are retained.

Predict

Predict whether a secant intercept must remain between its two input points. It does when their function signs are opposite, but ordinary secant iteration does not necessarily preserve such a pair. Two same-sign values can propose a distant point. Nearly equal function values make the slope denominator small, magnifying error. Compare this with bisection, whose next point remains inside a known bracket independently of slope information.

Experiment

Live experiment: a fast update needs a convergence argument

Compare bisection in a fixed sign-changing bracket with unsafeguarded Newton updates. Predict whether the chosen Newton start finds the bracketed root, another root, a cycle, or an undefined update. Equal update budgets are not equal computational cost.

Equation x²−2: a positive bracket and two rootsx³−2x+2: Newton can cycleNewton starting point -2-1012Maximum updates 124812Reset experiment

Bisection: a shrinking certified interval

The polynomial is continuous and the endpoints have opposite signs. Each midpoint is within half the displayed interval width of at least one bracketed root. This bracket contains the positive root for x²−2; Newton is not constrained to choose it.

Step	Interval before update	Midpoint	f(midpoint)	Error bound
1	[-2, -1]	-1.5	1.625	≤0.5
2	[-2, -1.5]	-1.75	0.140625	≤0.25
3	[-2, -1.75]	-1.875	-0.84179688	≤0.125
4	[-1.875, -1.75]	-1.8125	-0.3293457	≤0.0625

Bisection used 6 function evaluations, including two endpoint evaluations; cached endpoints are reused.

Newton: inspect the derivative and the actual outcome

The update is x next = x − f(x)/f′(x). A small residual |f(x)| measures equation mismatch; an error bound additionally needs suitable information about the root and derivative.

Step	x	f(x)	Derivative used for next update
0	0	2	-2
1	1	1	1
2	0	2	-2
3	1	1	1
4	0	2	Not evaluated

Update budget reached; no convergence claim. Newton used 5 function evaluations and 4 derivative evaluations. These counts exclude plotting and UI work; no elapsed-time comparison is implied.

For the cubic, start at zero and trace the two-cycle. For the square equation, start at zero to expose the undefined quotient, then at −2 to find the other root. Explain how a bracketed safeguard could reject an unsuitable Newton proposal. This widget runs the two basic methods separately; it does not implement that hybrid. Settings are temporary and all calculations use ordinary browser floating point.

The live experiment compares bisection and unsafeguarded Newton, not secant iteration. Change the function and starting value, inspect a cycle or derivative failure, and compare the bracket certificate with a residual. These controls are temporary exploration.

This is a separate fixed worked example for x²−2, not a claim about any live widget settings.

Previous pair	Secant proposal	Function value at proposal
1, 2	4/3	−2/9
2, 4/3	7/5	−1/25

Compute the next proposal and inspect whether the two latest points still bracket the positive root. Then retain an explicit opposite-sign bracket separately from the latest pair used to propose a step.

Understand

The update is xₙ₊₁=xₙ−f(xₙ)(xₙ−xₙ₋₁)/(f(xₙ)−f(xₙ₋₁)). For a sufficiently smooth function near a simple root, starting sufficiently close and avoiding degenerate denominators, the generic error relation is eₙ₊₁≈[f″(r)/(2f′(r))]eₙeₙ₋₁. When the leading coefficient is nonzero, a power-law comparison gives order p satisfying p²=p+1, hence the golden ratio. This explains superlinear convergence, slower in order than Newton’s quadratic convergence but without derivative evaluations. It is not a global guarantee, and special functions can have different leading behavior. Count function and derivative evaluations separately rather than comparing iteration counts alone.

Apply

A concrete safeguard gives an elementary global guarantee. Maintain [a,b] with opposite signs for a continuous function. Accept a finite fast proposal only inside [a+(b−a)/4,b−(b−a)/4]; otherwise use the midpoint. After evaluating, retain the side with opposite signs, or finish at an exact zero. The retained width is at most 3/4 of the previous width, so nested brackets shrink to a root. The central-region rule is stricter than many production hybrids but its proof is transparent. Merely staying inside the bracket is insufficient for a uniform shrink factor: false position can repeatedly update the same endpoint and leave a wide bracket.

Continue

Finite precision still requires nonfinite checks, slope checks, stagnation detection and an iteration budget. Report the bracket certificate separately from the residual and from whether a fast step was accepted. A live comparison of Newton and bisection illustrates contrasting mechanisms; it is not a secant implementation unless explicitly labeled as one. Written hybrid-design tasks require a learner to specify invariants and test cases before coding. Next, interpolation generalizes the two-point line to higher-degree approximation, introducing its own conditioning and error issues.

Review calculus and linear algebra when a prerequisite needs practice. Continue to Fit exact data with a unique polynomial.