Halley’s method for solving equations
E229501
Halley’s method for solving equations is an iterative numerical algorithm, related to and faster-converging than Newton’s method, used to find approximate roots of equations.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Halley’s method for solving equations canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2040182 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Halley’s method for solving equations Context triple: [Edmund Halley, knownFor, Halley’s method for solving equations]
-
A.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
-
B.
Picard iteration
Picard iteration is a successive approximation method used to construct solutions to ordinary differential equations and establish their existence and uniqueness.
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C.
Gauss–Seidel method
The Gauss–Seidel method is an iterative numerical technique used to solve systems of linear equations, particularly in large, sparse problems arising in scientific and engineering computations.
-
D.
Successive Over-Relaxation
Successive Over-Relaxation is an iterative numerical method that accelerates the convergence of the Gauss–Seidel algorithm for solving large systems of linear equations by introducing a relaxation factor.
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E.
Milstein method
The Milstein method is a numerical scheme for solving stochastic differential equations that improves on the Euler–Maruyama method by including derivative terms of the diffusion coefficient for higher accuracy.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Halley’s method for solving equations Target entity description: Halley’s method for solving equations is an iterative numerical algorithm, related to and faster-converging than Newton’s method, used to find approximate roots of equations.
-
A.
Euler’s method for numerical integration
Euler’s method for numerical integration is a simple first-order numerical procedure used to approximate solutions to ordinary differential equations by stepping forward in small increments.
-
B.
Picard iteration
Picard iteration is a successive approximation method used to construct solutions to ordinary differential equations and establish their existence and uniqueness.
-
C.
Gauss–Seidel method
The Gauss–Seidel method is an iterative numerical technique used to solve systems of linear equations, particularly in large, sparse problems arising in scientific and engineering computations.
-
D.
Successive Over-Relaxation
Successive Over-Relaxation is an iterative numerical method that accelerates the convergence of the Gauss–Seidel algorithm for solving large systems of linear equations by introducing a relaxation factor.
-
E.
Milstein method
The Milstein method is a numerical scheme for solving stochastic differential equations that improves on the Euler–Maruyama method by including derivative terms of the diffusion coefficient for higher accuracy.
- F. None of above. chosen
Statements (41)
| Predicate | Object |
|---|---|
| instanceOf |
higher-order Newton-like method
ⓘ
iterative numerical method ⓘ root-finding algorithm ⓘ |
| advantage |
cubic convergence for simple roots
ⓘ
potentially fewer iterations than Newton’s method ⓘ |
| appliesTo |
complex-valued functions
ⓘ
real-valued functions ⓘ |
| assumes |
denominator 2 (f'(x))^2 - f(x) f''(x) is nonzero at iterates
ⓘ
function is sufficiently smooth near the root ⓘ nonzero first derivative at the root ⓘ |
| basedOn | third-order Taylor expansion of the function ⓘ |
| canFailWhen |
derivatives are poorly conditioned or expensive to compute
ⓘ
initial guess is far from any root ⓘ |
| category | open methods for root finding ⓘ |
| comparedToNewton |
can require fewer iterations for similar accuracy
ⓘ
has higher per-iteration cost ⓘ uses second derivative information ⓘ |
| convergenceOrder | cubic ⓘ |
| convergenceSpeedComparedToNewton | faster local convergence under suitable conditions ⓘ |
| disadvantage |
more complex implementation than Newton’s method
ⓘ
requires evaluation of second derivatives ⓘ |
| field |
computational mathematics
ⓘ
numerical analysis ⓘ |
| generalizationOf | Newton’s method ⓘ |
| input | initial guess for the root ⓘ |
| iterationType | fixed-point iteration ⓘ |
| localConvergence | cubic when started sufficiently close to a simple root ⓘ |
| mathematicalDomain | analysis ⓘ |
| namedAfter |
Edmond Halley
ⓘ
linked to:
Edmund Halley
|
| namedEntityType | mathematical algorithm ⓘ |
| output | sequence of approximations to a root ⓘ |
| relatedTo | Newton’s method ⓘ |
| requires |
first derivative of the function
ⓘ
second derivative of the function ⓘ |
| rootType | simple roots ⓘ |
| stability | locally stable near simple roots under standard conditions ⓘ |
| updateFormula | x_{n+1} = x_n - \frac{2 f(x_n) f'(x_n)}{2 (f'(x_n))^2 - f(x_n) f''(x_n)} ⓘ |
| usedFor |
finding approximate roots of nonlinear equations
ⓘ
solving f(x) = 0 numerically ⓘ |
| usedIn |
high-precision computation of special functions
ⓘ
iterative algorithms in scientific computing ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Halley’s method for solving equations Description of subject: Halley’s method for solving equations is an iterative numerical algorithm, related to and faster-converging than Newton’s method, used to find approximate roots of equations.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.