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Bracketed root-finding helpers for scalar functions.

bisect implements standard bracketed bisection. Polynomial root isolation partitions an interval at derivative roots, then refines each sign-changing monotone window with bracketed bisection.

Types

Error

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Errors returned by root-finding helpers.

pub type Error {
  InvalidTolerance(tolerance: Float)
  InvalidMaxIterations(max_iterations: Int)
  NotBracketed(
    left: Float,
    right: Float,
    left_value: Float,
    right_value: Float,
  )
  MaxIterationsReached(estimate: Float, value: Float)
}

Constructors 

  • InvalidTolerance(tolerance: Float)

    The tolerance must be greater than zero.

  • InvalidMaxIterations(max_iterations: Int)

    The maximum iteration count must be greater than zero.

  • NotBracketed(
  left: Float,
  right: Float,
  left_value: Float,
  right_value: Float,
)

    The function values at the bracket endpoints do not have opposite signs.

  • MaxIterationsReached(estimate: Float, value: Float)

    The solver did not converge within the configured iteration count.

Options

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Options for bracketed bisection.

pub type Options {
  Options(tolerance: Float, max_iterations: Int)
}

Constructors 

  • Options(tolerance: Float, max_iterations: Int)

PolynomialOptions

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Options for isolating and refining real polynomial roots.

pub type PolynomialOptions {
  PolynomialOptions(
    coefficient_tolerance: Float,
    root_tolerance: Float,
    value_tolerance: Float,
    max_iterations: Int,
  )
}

Constructors 

  • PolynomialOptions(
  coefficient_tolerance: Float,
  root_tolerance: Float,
  value_tolerance: Float,
  max_iterations: Int,
)

QuadraticOptions

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Options for solving a quadratic equation.

pub type QuadraticOptions {
  QuadraticOptions(
    coefficient_tolerance: Float,
    repeated_root_policy: RepeatedRootPolicy,
  )
}

Constructors 

  • QuadraticOptions(
  coefficient_tolerance: Float,
  repeated_root_policy: RepeatedRootPolicy,
)

RepeatedRootPolicy

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Whether a repeated quadratic root is returned once or with its algebraic multiplicity of two.

pub type RepeatedRootPolicy {
  ConsolidateRepeatedRoot
  PreserveRepeatedRoot
}

Constructors 

  • ConsolidateRepeatedRoot
  • PreserveRepeatedRoot

Values

bisect

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pub fn bisect(
  f: fn(Float) -> Float,
  from left: Float,
  to right: Float,
) -> Result(Float, Error)

Find a root of f in a bracket using default options.

f(from) and f(to) must have opposite signs, unless either endpoint is already within tolerance of zero. from may be greater than to; the bracket is normalized before solving.

bisect_with

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pub fn bisect_with(
  f: fn(Float) -> Float,
  from left: Float,
  to right: Float,
  options options: Options,
) -> Result(Float, Error)

Find a root of f in a bracket using explicit options.

The returned value is an approximation. Convergence succeeds when either abs(f(estimate)) <= tolerance or the current bracket width is no larger than tolerance.

consolidate

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pub fn consolidate(
  roots: List(Float),
  tolerance tolerance: Float,
) -> List(Float)

Sort roots and merge neighboring values within tolerance.

cubic

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pub fn cubic(
  a: Float,
  b: Float,
  c: Float,
  d: Float,
) -> Result(List(Float), Error)

Find all distinct real roots of a*x³ + b*x² + c*x + d.

cubic_with

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pub fn cubic_with(
  a: Float,
  b: Float,
  c: Float,
  d: Float,
  options options: PolynomialOptions,
) -> Result(List(Float), Error)

Find all distinct real roots of a cubic using explicit numerical options.

default_options

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pub fn default_options() -> Options

Return the default bisection options.

default_polynomial_options

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pub fn default_polynomial_options() -> PolynomialOptions

Return the default options for polynomial root isolation.

inside

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pub fn inside(
  roots: List(Float),
  from lower: Float,
  to upper: Float,
) -> List(Float)

Keep roots inside a closed interval and return them in ascending order.

linear

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pub fn linear(a: Float, b: Float) -> List(Float)

Solve a * x + b = 0 using exact zero classification.

An identically zero or inconsistent constant equation returns no isolated roots.

polynomial_derivative

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pub fn polynomial_derivative(
  coefficients: List(Float),
) -> List(Float)

Differentiate power-basis coefficients ordered from highest power to the constant term.

polynomial_roots_with

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pub fn polynomial_roots_with(
  coefficients: List(Float),
  from lower: Float,
  to upper: Float,
  options options: PolynomialOptions,
) -> Result(List(Float), Error)

Find all distinct real roots of a power-basis polynomial in a closed interval.

Derivative roots partition the interval into monotone pieces. Sign-changing roots are refined with bracketed bisection, while roots shared with the derivative preserve even-multiplicity roots that do not change sign.

polynomial_value

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pub fn polynomial_value(
  coefficients: List(Float),
  at x: Float,
) -> Float

Evaluate a power-basis polynomial whose coefficients are ordered from the highest power to the constant term.

quadratic

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pub fn quadratic(a: Float, b: Float, c: Float) -> List(Float)

Solve a * x² + b * x + c = 0 using exact degree classification.

Real roots are returned in formula order. A repeated root is returned once.

quadratic_with

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pub fn quadratic_with(
  a: Float,
  b: Float,
  c: Float,
  options options: QuadraticOptions,
) -> List(Float)

Solve a * x² + b * x + c = 0 with explicit degree and multiplicity policy.

Coefficients whose absolute value is less than coefficient_tolerance are treated as zero. A zero or negative tolerance uses exact zero comparison.

strictly_inside

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pub fn strictly_inside(
  roots: List(Float),
  from lower: Float,
  to upper: Float,
) -> List(Float)

Keep roots strictly inside an interval and return them in ascending order.

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