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Formula verified · Deterministic calculation

Root Finder

Find a real root of a polynomial up to degree three inside an interval whose endpoints have opposite signs.

Root1.41
Residual
-0.00000000000061084471
Iterations
41
Final bracket
[1.4142135623724243, 1.4142135623733338]
Methodology

The certified numerical engine applies bounded bisection only to a validated sign-changing interval.

Sources
  • CalcuMint deterministic numerical solver
Solution steps
  1. Start bracket [1, 2]
  2. Bisection iterations: 41
  3. Root ≈ 1.414213562372879
  4. Residual f(root) ≈ -6.108447081487611e-13

How this calculator works

Bisection repeatedly halves the interval and retains the half that continues to bracket a sign-changing root.

Formula

m=(L+R)/2; retain the sign-changing half-interval

Finds a real root of a polynomial inside a user-supplied sign-changing bracket.

Worked example

x - 1 on [0,2] is a verified test case used by the calculation engine.

Assumptions and limitations

  • The polynomial must be non-constant.
  • Unless an endpoint is itself a root, the supplied interval must have opposite signs at its endpoints.
  • Even-multiplicity roots that do not change sign require a different method.

Methodology & sources

This calculator uses deterministic, versioned calculation logic. The formula and verified examples above are part of the calculation definition used by CalcuMint. Technical calculation version: 1.

  • CalcuMint deterministic numerical solver — Bisection method with bounded iterations and relative interval tolerance.

Frequently asked questions

Why must the interval bracket a root?

The certified bisection method relies on a sign change to guarantee the retained interval contains a root.

Which polynomial form is used?

f(x)=ax³+bx²+cx+d.

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