# Numerical Equation Solver & Optimizer

> Every root of f(x) = 0 in an interval, Newton or bisection steps, systems and minima. Finds every root of an equation such as cos(x) = x in an interval (sign changes refined with Brent’s method, touching roots detected), shows the iteration table of bisection or Newton’s method, solves systems of up to six nonlinear equations with a damped Newton method, and locates minima and maxima with Brent’s method or golden-section search. The numeric counterpart to the exact step-by-step solver; part of the Proofline math notebook.

Part of usefulweb.app (https://usefulweb.app/). Part of [Proofline](https://proofline.usefulweb.app/) (ownerApp: proofline). Results are deterministic calculations from the inputs and the listed assumptions; the API returns exactly what the web page shows.

## Use it
- Human interface: https://prooflineequationsolver.usefulweb.app/ (interactive tool at https://prooflineequationsolver.usefulweb.app/tool)
- JSON API: POST https://prooflineequationsolver.usefulweb.app/api/v1/calculate with a JSON object of inputs (schema: https://prooflineequationsolver.usefulweb.app/api/v1/tool)
- OpenAPI: https://prooflineequationsolver.usefulweb.app/openapi.json
- MCP (Streamable HTTP): https://prooflineequationsolver.usefulweb.app/mcp - tool `proofline_equation_solver`

## Inputs
- problem (select): Problem
- equations (textarea, required): Equation, function or system - One equation (cos(x) = x, or x^3 - 2x - 5 meaning = 0), one function for minimum/maximum, or one equation per line for a system. Use + − * / ^, implicit products (2x), sin, cos, tan, exp, ln (natural log), log (base 10), sqrt, abs, pi and e. Angles are in radians.
- variables (text): Unknowns - Default x; for a system a comma-separated list such as x, y (as many as equations).
- method (select): Method - Systems always use Newton’s method.
- from (number): Interval from
- to (number): Interval to
- start (text): Start value(s) - Newton’s method: one number, or one per unknown for a system (default 1 for each unknown; for one equation the interval midpoint).
- tolerance (number): Tolerance - Stop when the bracket, the step or the residual is below this value.
- maxIterations (integer): Iteration limit

## Method
- Expressions are read by Proofline’s own parser (no eval): + − * / ^, implicit products, the usual functions; ln is the natural logarithm, log is base 10.
- All roots: the interval is sampled at 401 points; every sign change is refined with Brent’s method (inverse quadratic interpolation, secant and bisection with guaranteed convergence). Sign changes across a pole (tan x) are rejected because |f| stays large. Local minima of |f| near zero without a sign change are refined by minimising f² and reported as touching roots.
- Bisection halves a bracket with a sign change; Newton’s method uses a central-difference derivative with step ∛ε·max(1, |x|); both list every iteration.
- Systems: damped Newton method with a forward-difference Jacobian and LU decomposition with partial pivoting; the step is halved until the largest residual decreases.
- Minima and maxima: sampled local extrema refined with Brent’s minimiser (parabolic interpolation + golden section) or plain golden-section search; extrema at the interval ends are flagged.

## Limits
- Numerical methods find roots and extrema to floating-point precision; roots closer together than 1/400 of the interval can be missed, and Newton’s method may converge to a different root than expected.