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 |
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 |
Edmund Halley
ⓘ
surface form:
Edmond 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.