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The bisection method finds a zero of a function by repeatedly halving a root bracket. Let $f$ be a real-valued continuous function on the closed interval $[a,b]$, with $a<b$. First check for endpoint zeros of a function; otherwise require strictly opposite signs at the endpoints. Evaluate the midpoint and retain the half with strictly opposite endpoint signs, stopping immediately if the midpoint is a zero of a function. The intermediate value theorem ensures that every retained root bracket still contains a zero of a function. This is the construction in MIT's root-finding notes.^[ref]

For a worked square root calculation, use the polynomial $f(x)=x^2-2$ on $[1,2]$. Write $[a_k,b_k]$ for the root bracket after $k$ halvings, with $[a_0,b_0]=[1,2]$. The candidate evaluated during update $k$ is the old midpoint $m_{k-1}$:The initial midpoint is $m_0=1.5$. After four halvings the new midpoint is $m_4=1.40625$, and the bisection error bound is $|m_4-\sqrt{2}|\le 0.03125$. Its actual error is approximately $0.00796356237310$. The last tested candidate $m_3=1.4375$ and the reported midpoint $m_4$ are different points.

The useful feature of the bisection method is that the retained width halves on every update, independently of the shape of the function graph. The bisection error bound therefore supplies an accuracy guarantee before the zero of a function is known. No derivative is needed, and a valid initial root bracket gives convergence without a close initial guess. This makes the bisection method a dependable fallback for the faster Newton root-finding iteration.

Intersecting a parametric curve with a plane reduces to isolating zeros of a scalar function. A certified range for $F$ rejects intervals that do not contain zero. On an interval whose derivative range excludes zero, a sign-changing endpoint bracket gives exactly one root, which bisection method or a safeguarded Newton root-finding iteration refines. Other intervals are subdivided conservatively; tangencies and coplanar pieces need multiplicity or identity tests supported by the representation. An unresolved small interval is a possible contact, not a proof of an intersection. With finitely many transverse intersections and convergent enclosures, isolation terminates to any prescribed tolerance.

At a regular differentiable interior nearest point of a parametric curve, the displacement from the query point is perpendicular to the tangent vector. A branch and bound search uses distances to certified enclosing bounding volumes as lower bounds, and evaluated limit points as upper bounds. Refine the most promising restricted pieces until the upper bound differs from the global lower bound by the chosen distance tolerance. Safeguarded Newton root-finding iteration accelerates local candidates, but endpoints, nonregular points, ties and other stationary branches must not be omitted.

Suppose $f\in C^2([a,r])$, $f(r)=0$, $f^{\prime }\ge m>0$ and $-K\le f^{\prime \prime }<0$. Starting at $x_0\in [a,r)$, the Newton root-finding iteration remains below $r$ and increases strictly to it. To prove this, assume $x_n<r$. Strict increase of $f$ gives $f(x_n)<0$, so the Newton step is positive. Taylor's theorem between $x_n$ and $r$ gives, with $e_n=x_n-r$,Its coefficient is negative and at least $-K/(2m)$, proving both the stated error bound and $x_n<x_{n+1}<r$. Mathematical induction establishes the invariant. The increasing bounded iterates have a limit $L\le r$; continuity in $x_{n+1}-x_n=-f(x_n)/f^{\prime }(x_n)$ forces $f(L)=0$, hence $L=r$. The error bound supplies local quadratic convergence.

For the strictly convex polynomial $f(x)=x^2/2+x^4/4$, applying Newton root-finding iteration to $f^{\prime }(x)=x+x^3$ givesThe unique minimum is at zero. For any sufficiently small nonzero initial point, the errors satisfy $|x_{k+1}|/|x_k{|}^3\to 2$, so the order is cubic, not merely quadratic. This provides a simple counterexample to treating the usual quadratic convergence theorem as an upper bound on possible orders. In general, the quadratic coefficient for Newton optimization near a nondegenerate stationary point is $f^{(3)}(x_{\ast })/(2f^{\prime \prime }(x_{\ast }))$; when it vanishes, a higher order is possible.

For the quadratic polynomial $f(x)=(x-a{)}^2-b^2$ with $b>0$, consider the Newton root-finding iteration towards the polynomial root $x_{\ast }=a+b$. If $x_k\ne a$, its signed error $e_k=x_k-x_{\ast }$ satisfiesIndeed the iteration for $u_k=x_k-a$ is $u_{k+1}=(u_k+b^2/u_k)/2$, and subtracting $b$ yields $(u_k-b{)}^2/(2u_k)$. This exact identity explains quadratic convergence near the simple polynomial root. If an initial error is asymptotic to $C{\epsilon }^m$ and $x_k-a\to b\ne 0$ as $\epsilon \to 0$, one step changes its leading power to $2m$ and its coefficient to $C^2/(2b)$.

An approximation to a Jacobian matrix obtained using finite differences instead of analytic partial derivatives. For a vector-valued function $F$, the forward-difference entry is $J_{ij}\approx (F_i(x+h{e}_j)-F_i(x))/h$, where $e_j$ is a coordinate basis vector. The step $h$ balances truncation and rounding errors; the resulting matrix can be used in a Newton root-finding iteration.

Let $F$ be twice continuously differentiable on an interval around a zero of a function $t_{\ast }$, with $|F^{\prime }(t)|\ge m>0$ and $|F^{\prime \prime }(t)|\le M$ there. Choose a closed interval $[t_{\ast }-\rho ,t_{\ast }+\rho ]$ inside that interval with $M\rho /(2m)<1$. Starting inside it makes the Newton root-finding iteration well defined, keeps it inside, and gives the displayed quadratic convergence bound for $e_r=t_r-t_{\ast }$. The nonzero derivative and sufficiently close start are essential hypotheses. The scalar estimate follows from the Taylor remainder argument in MIT's root-finding notes.^[ref]

For a short proof outline, apply Taylor's theorem between $t_r$ and $t_{\ast }$; subtract the defining tangent line equation. The linear error cancels, leaving $e_{r+1}=F^{\prime \prime }({\xi }_r)e_r^2/(2F^{\prime }(t_r))$. The displayed bound shrinks the error and preserves the chosen interval. If $F^{\prime \prime }(t_{\ast })\ne 0$, the error ratio tends to $|F^{\prime \prime }(t_{\ast })/(2F^{\prime }(t_{\ast }))|$; if it vanishes, convergence can be faster.

For fixed vectors $a_1,\dots ,a_m$ in a finite-dimensional Euclidean space, defineThe gradient and Hessian matrix of $\Phi =\log F$ areTo derive the first identity, differentiate each exponential function and divide the sum of its gradients by $F$. For the second, differentiating a normalized weight gives $\nabla w_i=2w_i(\bar{a}-a_i)$. Thus $J_{\bar{a}}=-2{\mathrm{Cov}}_w(a_i)$, where the weighted covariance matrix is the sum in the displayed Hessian matrix. Differentiating $2(x-\bar{a})$ completes the calculation.

The weighted covariance matrix is positive semidefinite, so ${\nabla }^2\Phi ⪰2I$. Also $\Phi (x)\ge \Vert x-a_1{\Vert }^2\to \infty$ as $\Vert x\Vert \to \infty$. Therefore $F$ and $\Phi$ share a unique global minimizer, characterized by $x=\bar{a}(x)$. The weights place that minimizer in the convex hull of the centres.

For stable numerical evaluation, subtract the largest exponent before computing the normalized weights. This is the log-sum-exp principle and avoids floating-point overflow without changing the gradient. A naive iteration $x_{n+1}=\bar{a}(x_n)$ need not converge: in one dimension, centres $-1,1$ give $\bar{a}(x)=-\tanh (2x)$ with derivative $-2$ at its unique fixed point. A Newton root-finding iteration or a suitable quasi-Newton method instead uses the positive curvature of $\Phi$.

Let $f\in C^2([a,b])$ have $f^{\prime },f^{\prime \prime }>0$ and $f(a)<0<f(b)$. Its unique root $r$ lies in $(a,b)$. Starting Newton root-finding iteration at $b$, strict convexity puts the tangent root strictly between $r$ and the current iterate, so the iterates decrease to $r$. Taylor expansion about the current iterate gives $e_{n+1}=f^{\prime \prime }({\xi }_n)e_n^2/[2f^{\prime }(x_n)]$, where $e_n=x_n-r$ and ${\xi }_n\in (r,x_n)$. Continuity proves the displayed positive asymptotic constant and quadratic convergence.

The finite-dimensional Newton root-finding iteration for a vector equation $G(x)=0$. At a point with invertible Jacobian matrix, solve the displayed linear system. For $G=\nabla F$, its Jacobian matrix is the Hessian matrix, giving Newton's method in optimization. The update solves the affine first-order approximation $G(x_k)+DG(x_k)s_k=0$; it can converge to any nondegenerate critical point, including a saddle point.

If a sufficiently smooth function has the form $f(x)=(x-\alpha {)}^{m}q(x)$ with $m>1$ and $q(\alpha )\ne 0$, then the Newton root-finding iteration satisfiesThus the ratio of successive nonzero errors tends to $(m-1)/m$: the usual iteration has only linear convergence. Cancelling the common factor in $f/f^{\prime }$ produces a continuous extension at the multiple root, although the original quotient is undefined there. An implementation should stop upon finding an exact zero of a function.

The Newton root-finding iteration seeks a zero of a function $F$ by repeatedly solving its tangent line approximation. At a current guess $t_r$ with derivative $F^{\prime }(t_r)\ne 0$, setGeometrically, follow the tangent line to the horizontal axis, then return to the function graph for the next tangent line. Close to a suitable zero of a function, the error becomes dramatically smaller.^[ref]

For a concrete square root calculation, take the polynomial $F(t)=t^2-2$ and start at $t_0=1$. The Newton root-finding iteration becomes $t_{r+1}=(t_r+2/t_r)/2$, givingThe errors use the unrounded iterates. After the first step, each iterate overestimates $\sqrt{2}$, and the exact identityexhibits quadratic convergence: once close, the number of correct decimal digits roughly doubles per step. This speed makes the Newton root-finding iteration useful when accurate derivatives are available.

Global convergence requires additional care. For the polynomial $F(x)=x^3-2x+2$, starting at $x_0=0$ gives $F(0)=2$, $F^{\prime }(0)=-2$, hence $x_1=1$. Then $F(1)=1$, $F^{\prime }(1)=1$ gives $x_2=0$. The sequence repeats $0,1,0,1,\dots$ forever, although a real polynomial root lies in $(-2,-1)$ by the intermediate value theorem. Neither point in the cycle is a zero of a function, and neither has zero derivative. A maintained sign-changing interval and fallback bisection method can protect a root solver from unsuitable Newton steps.

The intermediate value theorem gives a root $r\in (a,b)$, unique because $f^{\prime }>0$ makes $f$ strictly increasing. We show that Newton root-finding iteration, started at $b$, always stays to the right of $r$. If $x>r$, the mean value theorem gives $f(x)=f^{\prime }(\xi )(x-r)$ for some $\xi \in (r,x)$. Since $f^{\prime \prime }>0$, $f^{\prime }$ is strictly increasing, so $0<f^{\prime }(\xi )<f^{\prime }(x)$. It follows thatInduction gives $r<x_{n+1}<x_n\le b$, so the bounded monotone sequence theorem supplies a limit $\alpha \ge r$. The continuous positive $f^{\prime }$ is bounded away from zero on $[a,b]$, making $F$ continuous. Passing to the limit in $x_{n+1}=F(x_n)$ gives $\alpha =F(\alpha )$ and hence $f(\alpha )=0$. Uniqueness implies $\boxed{\alpha =r}$.

For the rate, apply the Taylor theorem with Lagrange remainder to $f$ between $r$ and $x_n$:With $e_n=x_n-r>0$, rearrangement givesThus the monotone Newton convergence for a strictly convex function has quadratic convergence. More concretely, if $K={\max }_{[a,b]}{f}^{\prime \prime }/[2{\min }_{[a,b]}{f}^{\prime }]$, then $e_{n+1}\le K{e}_n^2$. Once $K{e}_N<1$, induction gives $e_{N+j}\le K^{-1}(K{e}_N{)}^{2^j}$, exhibiting the repeated squaring of small errors.

An efficient parametric curve interrogation interface should provide the parameter domain and open/closed status, point evaluation, one-sided endpoint enquiries, first and second derivatives wherever defined, and information about knots or singular parameters. It should also restrict a curve to an interval and subdivide it without changing its limit, return a certified bounding volume for that restriction, and supply derivative or chord error bounds when supported. These enquiries allow cheap geometric rejection before expensive exact evaluation. For a subdivision curve, limit evaluation is distinct from returning a vertex of a finite refined control polygon; the latter is usually an approximation.

For a query point $Q$, minimize $d(t)=\frac{1}{2}\Vert P(t)-Q{\Vert }^2$. At a differentiable interior minimizer,At a regular curve point this says that the query displacement is perpendicular to the tangent vector. If a second derivative exists, the local second-order test isa strictly positive value is sufficient for a strict local minimum. The stationarity equation alone also includes local maxima and is automatic at $P^{\prime }=0$, so it cannot be used as the only global search criterion.

A reliable efficient closest-point search on a subdivision curve uses branch and bound. Cover the compact curve by restricted pieces with certified enclosures $B_I$. For each piece set $L_I=\mathrm{dist}(Q,B_I)$, a lower bound on its true distance. Distances to actually evaluated limit points give upper bounds; keep the best point $P(t_{\ast })$ and distance $U$. For an axis-aligned bounding box ${\prod }_k[l_k,h_k]$, the squared lower bound is inexpensive:For positive partition-of-unity refinement, a convex hull of the active control points is an enclosure. A negative-weight scheme requires a separate certified enclosure; its control hull need not contain the limit.

The algorithm is:

Because ${\min }_I{L}_I\le d_{\min }\le U$, the reported point is at most $ϵ$ farther from $Q$ than a true nearest point. Include endpoints, nondifferentiable joins, and singular parameters in the candidate bookkeeping, or split them into separate pieces. With enclosures converging in diameter under subdivision on a compact continuous curve, the bounds converge to arbitrary prescribed positive distance tolerance. Interval bounds or outward-rounded arithmetic are needed if the guarantee is intended to be certified in floating-point computation. A single Newton root-finding iteration from a single seed supplies no such global guarantee.

An efficient subdivision surface interrogation interface needs the mesh topology and patch adjacency, boundary and trimming information, patch-local parameter charts and their transition maps, and local restriction/refinement with the complete influencing control net. Numerical enquiries should supply limit positions $S$, first derivatives $S_u,S_v$, and second derivatives where defined; these determine normal vectors and curvatures at regular points. It must also return certified patch bounding volumes and convergent geometric/derivative error bounds. Near an extraordinary subdivision vertex, use the local subdivision matrix and limit/tangent evaluation appropriate to that scheme; second derivatives need not exist merely because a tangent plane does. Finite refined mesh vertices are not exact limit positions.

Represent the cutting plane as $n\cdot x=d$, with $n$ normalized. On each chart setThe desired plane section of a subdivision surface is the image of the zero set $g=0$. For a bounding volume $B$, compute a certified interval $[g_{\min },g_{\max }]$ for the plane functional over $B$. If that interval is strictly positive or strictly negative, reject the patch. With positive partition-of-unity weights, the minimum and maximum of $n\cdot p_i-d$ over all influencing control points already bound its limit patch. For masks without this convex hull property, the interface must provide a different valid enclosure.

An adaptive algorithm that isolates components before tracing is:

This is exhaustive to a chosen tolerance when certified subdivision isolates finitely many regular transverse branches and enclosures converge. If $\Vert \nabla g\Vert \ge \gamma >0$ locally and $\Vert DS\Vert \le M$, a residual tolerance $|g|\le \delta$ corresponds locally to a geometric correction bounded by $M\delta /\gamma$, using the gradient flow toward the level set. Bounds on second derivatives similarly control curve-segment chord error bound. As $\gamma$ approaches zero, a fixed residual tolerance no longer gives a fixed geometric error, so the step and stopping rules must reflect conditioning.

Seeding must not rely just on corner signs. For example $S(u,v)=(u,v,u^2+v^2-r^2)$ cut by $z=0$ has a closed circle wholly inside a patch whose corner values are all positive. Certified range tests plus subdivision retain it; a corner-only marching rule can miss it. Tangencies also change the nature of the output: $S=(u,v,u^2+v^2)$ meets $z=0$ in one isolated point, while a coplanar patch contributes an area rather than a curve. Thus a general plane section is not necessarily a union of regular curves; the algorithm should classify these cases instead of silently fabricating or deleting a branch.

For efficiency, cache subdivision stencils, patch extraction, limit-evaluation coefficients and bounds; reuse child data instead of reconstructing the entire refined mesh. Use local influencing neighborhoods, with the halo dictated by mask support, and keep adjacency/boundary topology separate from changing geometric coordinates. Plane-functional bounds can be computed by refining scalar signed-distance controls, which is cheaper than refining all three coordinates. Combine coarse inexpensive boxes with tighter local convex hull or interval arithmetic tests only where needed. Adapt steps to curvature and transversality, reuse branch seeds, and use regular spline evaluation away from extraordinary points. Stable edge ownership, outward-rounded bounds, scale-aware tolerances, and explicit degeneracy flags are correctness requirements as well as performance issues. The result is a limit-surface intersection computation; merely intersecting a finitely refined control polyhedron with the plane supplies only an uncontrolled approximation.

A parametric curve interrogation interface first supplies the parameter domain and the point $C(t)$. At regular parameters, it also supplies $C^{\prime }(t)$ and, when needed, higher derivatives. The unit tangent vector, speed and curvature are then obtained fromin three dimensions. The formulas require $C^{\prime }\ne 0$; a Frenet normal additionally requires nonzero curvature. A practical interface can restrict the curve to a parameter interval, provide endpoints, and give a certified bounding volume or a derivative-range bound. Arc length, closest-point and intersection queries are higher-level operations built from these elementary enquiries rather than consequences of point evaluation alone.

For a plane $n\cdot x=d$ with $n\ne 0$, defineThis reduces certified curve-plane intersection to scalar root isolation. A subdivision curve can expose the parametric enquiries by evaluating its limit at a parameter address: refine the corresponding piece until its certified positional enclosure reaches the requested accuracy, or use its exact local spline representation. Derivative evaluation similarly needs the derivative refinement rule and its error bound. A finite control vertex is not automatically a point of the limit curve.

One reliable algorithm is range-based isolation followed by safeguarded Newton iteration:

For a transverse root, the derivative is nonzero nearby, so sufficiently fine intervals have monotone $F$ and the sign bracket isolates it. Bisection provides a global error bound; near a simple root Newton iteration gives quadratic convergence. Good enclosures discard most of the curve before expensive limit evaluation, and the method does not require a specific subdivision mask.

The weaknesses are equally important. Near tangency $F^{\prime }$ is small and Newton steps are ill-conditioned; same-sign endpoints may conceal a tangent root or two crossings. An arc lying in the plane has infinitely many intersections and should be returned as an interval, not a point. Finite-precision evaluations require error enclosures before a sign is certified. For an arbitrary smooth point-evaluation oracle, no finite sampling scheme can certify all intersections: a smooth bump between sampled parameters can create extra crossings without changing any sampled values or derivatives. With finitely many isolated transverse intersections and convergent certified enclosures, the algorithm terminates to any prescribed tolerance. For tangencies it either uses stronger representation tests or reports unresolved possible-contact intervals, rather than silently omitting them.

A subdivision curve interrogation interface provides its control/topology data, a parameter interval or address, and a refinement operation returning restricted child pieces with their parameter correspondence. It also provides a certified enclosing bounding volume, endpoints or limit-point enclosures, and an error estimate for replacing a sufficiently small piece by its chord. Positional and derivative limit queries may be implemented by refinement or by conversion to the appropriate spline. Nonnegative masks often imply a control-point convex hull bound; a generic scheme with negative weights needs a separate bound.

These enquiries can be applied to a parametric curve by using interval restriction as the subdivision operation and constructing its enclosure from its representation. For example, a Bézier curve is split by De Casteljau's algorithm, and each child's control-point convex hull encloses exactly that child arc. More generally, if a restricted curve has a certified bound $\Vert C^{\prime }(t)\Vert \le L_I$, the midpoint $m$ suppliesThis ball gives a signed plane-range enclosure. Such an upper derivative bound must be supplied or proved; taking the largest derivative seen in a few samples is not a certificate.

The recursive intersection algorithm is as follows. Start with the whole domain. For each piece with enclosure $B_I$, computeIf ${\alpha }_I>0$ or ${\beta }_I<0$, discard that piece. Otherwise split it and process the children, unless a root has already been uniquely isolated and can be polished as in Question 3. For a convex hull bound, the extrema of this affine plane function are attained among its control points, so only dot products are required. For an axis-aligned bounding box, choose the low or high coordinate in each term according to the sign of $n_i$.

At a small flat piece, the chord-plane parameter iswhen the denominator is nonzero. It supplies an initial estimate, which is corrected on the actual curve. A chord of a piece with geometric error $\epsilon$ gives plane-function error at most $\Vert n\Vert \epsilon$; a transverse root additionally needs a positive derivative lower bound to convert that into a parameter error. A chord hit is not by itself a proof of the exact crossing count. Keep unresolved tangent contacts, merge shared-endpoint hits, and return any certified coplanar parameter intervals separately.

Good speed comes from a bounding volume hierarchy, conservative cheap rejection before accurate evaluation, caching shared subdivision work, local rather than uniform refinement, and fast root polishing only after isolation. Convex-hull or box extrema cost far less than solving every potential intersection. Tight bounds shrink rapidly with subdivision and reduce the active search to neighborhoods of actual crossings. Derivative information can prove monotonicity, accelerate with the Newton root-finding iteration, or tighten range bounds using interval arithmetic. There is no universal fixed speed guarantee when the curve is nearly coplanar or has arbitrarily many intersections: those are genuinely difficult inputs. The representation must provide meaningful bounds and restriction operations; point/derivative enquiries alone do not imply such capabilities.

The polynomial factors as $p(x)=(3x-2{)}^2(3x+1)$, so $\alpha =2/3$ is a multiple root of multiplicity two. The Newton root-finding iteration isFor $x\ne 0,2/3$, cancelling a factor givesThis expression extends continuously to $g(2/3)=2/3$. Let $I=[1/3,1]$. On this closed interval,Thus $g$ is increasing, andBy the mean value theorem, $|g(x)-g(y)|\le (16/27)|x-y|$ for all $x,y\in I$. The contraction mapping theorem states that a contraction mapping from a nonempty complete metric space into itself has a unique fixed point, and every fixed-point iteration converges to it. The closed interval $I$ with ordinary Euclidean distance is complete, and $g(2/3)=2/3$, soIn fact $|x_k-2/3|\le (16/27{)}^k|x_0-2/3|$. The Newton iteration at a multiple root here has linear convergence, since $g^{\prime }(2/3)=1/2$. Thus the convergence assertion is true for the simplified fixed-point iteration, and for the usual Newton root-finding iteration that stops upon reaching an exact root of a polynomial. If one insists on applying the unsimplified quotient at every step, it is undefined for the allowed starting value $x_0=2/3$, since both $p$ and $p^{\prime }$ vanish there; that literal version of an infinite sequence requires the stated stopping convention or continuous extension.

The Newton root-finding iteration follows the current tangent line to its horizontal-axis intersection. Since $f^{\prime }(x)=2x$, it givesA positive initial guess is appropriate for the positive square root; negative starting values instead target the negative root.