Working definition
Root finding solves f(x)=0 numerically. Bracketing methods preserve an interval containing a sign-changing root under continuity; open methods can converge faster but require stronger local conditions and may converge to an unintended root.
Notation
find x*: f(x*) = 0aₖ ≤ x* ≤ bₖAssumptions
- The residual is defined in the search region.
- Continuity or differentiability matches the algorithm.
- Repeated and tangent roots are considered.
Invariants
- A valid sign-change bracket retains at least one root for a continuous function.
- Stopping criteria are declared in input and residual units.
- Multiple crossings require separate brackets.
Reproducible procedure
- Define a continuous residual and scan for candidate brackets.
- Refine each candidate with a safeguarded solver.
- Verify residual, bracket width, direction, and duplicate handling.
Error and boundary controls
- Sampling can miss tangent or rapid roots.
- Time error depends on local slope.
- Ephemeris and timescale errors remain in the located event.
What this does not establish
A precisely located ingress, station, phase, or boundary is an event calculation; it does not prove an interpretation attached to that event.
Explicit applications
2 cross-domain bridges
Ingress and station event location
Locate when a continuous angular residual reaches a boundary or apparent speed reaches zero.
Inputs
- ephemeris function
- search interval
- target residual
Outputs
- event instant
- crossing direction
- time tolerance
Transformation: Bracket and refine each crossing with direction and duplicate controls.
Limit: The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity.
Open connected system →Limb boundary time search
Locate when a tithi, nakṣatra, yoga, or karaṇa boundary occurs.
Inputs
- continuous limb phase
- search window
- ephemeris convention
Outputs
- boundary instant
- direction
- time tolerance
Transformation: Bracket each modular crossing and refine its instant.
Limit: A computed boundary does not establish auspiciousness.
Open connected system →Authoritative references
- [1]DLMF Chapter 3: Numerical Methods · National Institute of Standards and Technology
Establishes: Reference definitions, algorithms, convergence conditions, and error terms for interpolation, quadrature, differentiation, and nonlinear equation solving.
Boundary: A numerical method is reliable only under its stated regularity, conditioning, precision, and convergence assumptions; the reference does not validate any domain interpretation.
- [2]JPL Horizons System · NASA Jet Propulsion Laboratory
Establishes: Ephemeris products and documented observer, target, timescale, coordinate, and output conventions for reproducible Solar System state and observable calculations.
Boundary: Ephemeris agreement validates positions under declared conventions; it does not validate downstream symbolic classifications or interpretations.
Direct answer
- Root finding solves f(x)=0 numerically. Bracketing methods preserve an interval containing a sign-changing root under continuity; open methods can converge faster but require stronger local conditions and may converge to an unintended root.
Mechanism and method
- Define a continuous residual and scan for candidate brackets.
- Refine each candidate with a safeguarded solver.
- Verify residual, bracket width, direction, and duplicate handling.
What is measured
- A valid sign-change bracket retains at least one root for a continuous function.
- Stopping criteria are declared in input and residual units.
- Multiple crossings require separate brackets.
Limitations
- Sampling can miss tangent or rapid roots.
- Time error depends on local slope.
- Ephemeris and timescale errors remain in the located event.
- The residual is defined in the search region.
- Continuity or differentiability matches the algorithm.
- Repeated and tangent roots are considered.
What this does not establish
- A precisely located ingress, station, phase, or boundary is an event calculation; it does not prove an interpretation attached to that event.
Bridge: Ingress and station event location
- Locate when a continuous angular residual reaches a boundary or apparent speed reaches zero.
- Input: ephemeris function
- Input: search interval
- Input: target residual
- Output: event instant
- Output: crossing direction
- Output: time tolerance
- Limit: The mathematics makes the operation reproducible; it does not by itself establish causal interpretation or predictive validity.
Bridge: Limb boundary time search
- Locate when a tithi, nakṣatra, yoga, or karaṇa boundary occurs.
- Input: continuous limb phase
- Input: search window
- Input: ephemeris convention
- Output: boundary instant
- Output: direction
- Output: time tolerance
- Limit: A computed boundary does not establish auspiciousness.