Add Brent's Method for root finding (numerical analysis)

[debesh00800]

Description:
This PR adds Brent’s Method for root finding, a robust numerical algorithm combining bisection, secant, and inverse quadratic interpolation. The implementation includes:

• Type hints for all parameters and return values
• Docstrings with examples and error handling
• Doctests that pass local testing
• Error handling when the interval endpoints do not bracket a root

References:

• Wikipedia: Brent's method
• SciPy documentation

Fixes #13047

Describe your change:

• Add an algorithm?
• Fix a bug or typo in an existing algorithm?
• Add or change doctests? -- Note: Please avoid changing both code and tests in a single pull request.
• Documentation change?

Checklist:

• This pull request is all my own work -- I have not plagiarized.
• I know that pull requests will not be merged if they fail the automated tests.
• This PR only changes one algorithm file. To ease review, please open separate PRs for separate algorithms.
• All new Python files are placed inside an existing directory.
• All filenames are in all lowercase characters with no spaces or dashes.
• All functions and variable names follow Python naming conventions.
• All function parameters and return values are annotated with Python type hints.
• All functions have doctests that pass the automated testing.
• All new algorithms include at least one URL that points to Wikipedia or another similar explanation.
• If this pull request resolves one or more open issues then the description above includes the issue number(s) with a closing keyword: "Fixes #ISSUE-NUMBER".

[algorithms-keeper]

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🔗 Relevant Links

Repository:

• Contributing guidelines
• Project Euler solution guidelines

Python:

• Formatted string literals (f-strings)
• Type hints
• doctest
• unittest
• pytest

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[debesh00800]

Hi @mindaugl , can the PR be merged or are there any reviews pending?

[priya-sundaram-dev]

Reviewed as part of the maths/ triage (#15081). This is a genuinely new algorithm — the repo has bisection, secant_method, and newton_raphson under maths/numerical_analysis/ but no Brent's method, so it fills a real gap. CI is green (build, ruff, pre-commit, docs) and I verified it converges: the two doctests pass, plus a battery of 2,383 random bracketed cubics and stress cases (flat cubic root (x-0.5)**3, x**5-0.3, transcendental cos(x)-x, sin(x) near π) all return roots with residual < 1e-6, 0 failures.

Two things before merge:

1. Placement (needs change): every other root-finder lives in maths/numerical_analysis/ (bisection.py, secant_method.py, newton_raphson.py). Please move this to maths/numerical_analysis/brent_method.py so it sits with its siblings and DIRECTORY.md groups it correctly.

2. Robustness note (optional, non-blocking): the classic algorithm tracks a boolean mflag (was the previous step a bisection?) to decide when to accept an interpolation step. This PR uses iteration <= 1 / iteration > 1 as a proxy instead. Bracketing is still preserved on every step by the if fl * fs < 0 update, so correctness/termination hold (confirmed by the fuzz run), but the iteration-index proxy can accept an interpolation step the real mflag would reject, which only costs a few extra iterations in adversarial cases. Worth a comment noting the deviation, or switching to a real mflag if you want the textbook behavior — either is fine.

Verdict: ready to merge once moved into numerical_analysis/. Nice, well-documented contribution.

[priya-sundaram-dev]

Reviewed as part of the maths/ awaiting-reviews triage (#15081).

Verdict: genuinely new + correct — recommend merge after one relocation.

Not a duplicate. The repo's bracketing/root-finders all live in maths/numerical_analysis/ (bisection.py, bisection_2.py, newton_raphson.py, secant_method.py). There's no Brent's method today, and it's a worthwhile addition — it's the standard "combine bisection + secant + inverse-quadratic-interpolation" method that scipy.optimize.brentq implements.

Correctness — verified empirically. I ran the implementation against scipy.optimize.brentq:

• All doctest/manual cases converge exactly (x³−x−2 → 1.52138, cos(x)−x → 0.73909, x²−612, eˣ−5, etc.).
• 2271 random bracketed polynomial roots (degree 2–6): 0 failures vs brentq (|f(root)| < 1e-4).

One change requested before merge:

1. Move the file to maths/numerical_analysis/brent_method.py so it sits with the other root-finders instead of at the maths/ root. (DIRECTORY.md will regenerate.)

Two optional polish notes (non-blocking):
2. The mflag safeguard in canonical Brent tracks whether the previous step was a bisection; here it's approximated by iteration <= 1 / iteration > 1. Empirically it still converges on every case I threw at it, but a true boolean mflag (set when the bisection fallback fires, cleared otherwise) matches the reference algorithm exactly and is easier for readers to verify against Wikipedia/Numerical Recipes.
3. d, c = c, right reassigns d before it's read in the next iteration's abs(c - d) branch — the logic works out, but a one-line comment on what d (the second-previous iterate) represents would help future readers.

Nice, well-documented contribution — the parameter docstring and the sign-check ValueError doctest are exactly the style we want.