{ "openapi": "3.1.1", "info": { "title": "Numerical Equation Solver & Optimizer API", "version": "1.0.0", "summary": "Every root of f(x) = 0 in an interval, Newton or bisection steps, systems and minima.", "description": "Every endpoint performs the same calculation as the human page and returns inputs, result, units, assumptions, sources, warnings and computedAt. MCP endpoint: /mcp." }, "servers": [ { "url": "https://prooflineequationsolver.usefulweb.app" } ], "paths": { "/api/v1/calculate": { "post": { "operationId": "calculate", "summary": "Every root of f(x) = 0 in an interval, Newton or bisection steps, systems and minima.", "description": "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.\n\nMethodology:\n- Expressions are read by Proofline’s own parser (no eval): + − * / ^, implicit products, the usual functions; ln is the natural logarithm, log is base 10.\n- 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.\n- 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.\n- Systems: damped Newton method with a forward-difference Jacobian and LU decomposition with partial pivoting; the step is halved until the largest residual decreases.\n- 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.\n\nNumerical 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.", "tags": [ "apps" ], "x-mcp-tool": "proofline_equation_solver", "x-owner-app": "proofline", "requestBody": { "required": true, "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineEquationSolverInput" }, "examples": { "example1": { "summary": "All roots of cos(x) = x", "value": { "problem": "equation", "equations": "cos(x) = x", "from": -5, "to": 5 } }, "example2": { "summary": "Three roots of a cubic", "value": { "problem": "equation", "equations": "x^3 - 6x^2 + 11x - 6", "from": 0, "to": 4 } }, "example3": { "summary": "Newton’s method for √2 with iteration table", "value": { "problem": "equation", "equations": "x^2 = 2", "method": "newton", "start": "1" } }, "example4": { "summary": "Bisection steps for x³ − 2x − 5 = 0", "value": { "problem": "equation", "equations": "x^3 - 2x - 5 = 0", "method": "bisection", "from": 2, "to": 3, "tolerance": 0.000001 } }, "example5": { "summary": "Circle meets hyperbola: nonlinear system", "value": { "problem": "system", "equations": "x^2 + y^2 = 4\nx*y = 1", "variables": "x, y", "start": "2, 0.5" } }, "example6": { "summary": "Minimum of x⁴ − 3x² + x", "value": { "problem": "minimum", "equations": "x^4 - 3x^2 + x", "from": -3, "to": 3 } } } } } }, "responses": { "200": { "description": "Calculation result envelope.", "content": { "application/json": { "schema": { "$ref": "#/components/schemas/ProoflineEquationSolverResult" } } } }, "400": { "description": "Malformed JSON body.", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "422": { "description": "Invalid input (see errors[]).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } }, "429": { "description": "Rate limit exceeded (see Retry-After).", "content": { "application/problem+json": { "schema": { "$ref": "#/components/schemas/Problem" } } } } } } }, "/api/v1/tool": { "get": { "operationId": "getToolDefinition", "summary": "Inputs, output schema and examples for this tool", "responses": { "200": { "description": "Tool definition.", "content": { "application/json": { "schema": { "type": "object" } } } } } } } }, "components": { "schemas": { "Problem": { "type": "object", "description": "RFC 9457 problem details. 422 responses include per-field `errors`.", "required": [ "type", "title", "status", "detail" ], "properties": { "type": { "type": "string" }, "title": { "type": "string" }, "status": { "type": "integer" }, "detail": { "type": "string" }, "code": { "type": "string" }, "errors": { "type": "array", "items": { "type": "object", "required": [ "field", "message" ], "properties": { "field": { "type": "string" }, "message": { "type": "string" } } } } } }, "ProoflineEquationSolverInput": { "type": "object", "additionalProperties": false, "properties": { "problem": { "title": "Problem", "type": "string", "enum": [ "equation", "system", "minimum", "maximum" ], "default": "equation" }, "equations": { "title": "Equation, function or system", "description": "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.", "type": "string", "maxLength": 4000 }, "variables": { "title": "Unknowns", "description": "Default x; for a system a comma-separated list such as x, y (as many as equations).", "type": "string", "maxLength": 200 }, "method": { "title": "Method", "description": "Systems always use Newton’s method.", "type": "string", "enum": [ "brent", "bisection", "newton", "golden" ], "default": "brent" }, "from": { "title": "Interval from", "type": "number", "default": -10 }, "to": { "title": "Interval to", "type": "number", "default": 10 }, "start": { "title": "Start value(s)", "description": "Newton’s method: one number, or one per unknown for a system (default 1 for each unknown; for one equation the interval midpoint).", "type": "string", "maxLength": 200 }, "tolerance": { "title": "Tolerance", "description": "Stop when the bracket, the step or the residual is below this value.", "type": "number", "minimum": 1e-16, "maximum": 0.1, "default": 1e-12 }, "maxIterations": { "title": "Iteration limit", "type": "integer", "minimum": 1, "maximum": 10000, "default": 100 } }, "required": [ "equations" ] }, "ProoflineEquationSolverResult": { "type": "object", "additionalProperties": false, "required": [ "schemaVersion", "tool", "computedAt", "inputs", "summary", "result", "units", "assumptions", "sources", "warnings", "presentation", "links" ], "properties": { "schemaVersion": { "type": "string", "const": "1.0" }, "tool": { "type": "object", "additionalProperties": false, "required": [ "id", "name", "version" ], "properties": { "id": { "type": "string" }, "name": { "type": "string" }, "version": { "type": "string" } } }, "computedAt": { "type": "string", "format": "date-time" }, "inputs": { "type": "object", "description": "Normalised inputs actually used, including applied defaults." }, "summary": { "type": "string" }, "result": { "type": "object", "additionalProperties": false, "required": [ "problem", "method", "variables", "functions", "interval", "solutions", "converged", "message", "evaluations", "iterationTable" ], "properties": { "problem": { "type": "string", "enum": [ "equation", "system", "minimum", "maximum" ] }, "method": { "type": "string", "enum": [ "brent", "bisection", "newton", "golden" ] }, "variables": { "type": "array", "items": { "type": "string" } }, "functions": { "type": "array", "items": { "type": "string" }, "description": "The expressions set to zero (equation lhs − rhs) or the function minimised/maximised." }, "interval": { "type": [ "object", "null" ], "additionalProperties": false, "required": [ "from", "to" ], "properties": { "from": { "type": "number" }, "to": { "type": "number" } } }, "solutions": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "x", "f", "kind", "iterations" ], "properties": { "x": { "type": "array", "items": { "type": "number" } }, "f": { "type": "array", "items": { "type": "number" }, "description": "Residuals at x (equations) or the function value (extrema)." }, "kind": { "type": "string" }, "iterations": { "type": "integer" } } } }, "converged": { "type": "boolean" }, "message": { "type": "string" }, "evaluations": { "type": "integer" }, "iterationTable": { "type": "object", "additionalProperties": false, "required": [ "columns", "rows" ], "properties": { "columns": { "type": "array", "items": { "type": "string" } }, "rows": { "type": "array", "items": { "type": "array", "items": { "type": [ "number", "null" ] } } } } } } }, "units": { "type": "object", "additionalProperties": { "type": "string" } }, "assumptions": { "type": "array", "items": { "type": "string" } }, "sources": { "type": "array", "items": { "type": "object", "additionalProperties": false, "required": [ "title" ], "properties": { "title": { "type": "string" }, "url": { "type": "string" }, "publisher": { "type": "string" }, "retrievedAt": { "type": "string" }, "note": { "type": "string" } } } }, "warnings": { "type": "array", "items": { "type": "string" } }, "presentation": { "type": "array", "items": { "type": "object" }, "description": "Display blocks used by the human page (same data)." }, "links": { "type": "object", "additionalProperties": false, "required": [ "human", "api", "openapi", "methodology" ], "properties": { "human": { "type": "string", "format": "uri" }, "api": { "type": "string", "format": "uri" }, "openapi": { "type": "string", "format": "uri" }, "methodology": { "type": "string", "format": "uri" } } } } } } } }