> ## Documentation Index
> Fetch the complete documentation index at: https://docs.fincept.in/llms.txt
> Use this file to discover all available pages before exploring further.

# Find Root of System of Equations

> Find a solution to a system of nonlinear equations F(x) = 0 where F and x are vectors. Supports built-in systems: linear_2x2 (system of two linear equations), nonlinear_2x2 (nonlinear system like x^2 + y^2 = 4, xy = 1). Methods: newton (multidimensional Newton-Raphson with Jacobian), broyden (quasi-Newton method, approximates Jacobian). Essential for finding equilibrium points, solving nonlinear systems in economics, and multi-asset calibration. [Tier: BASIC, Credits: 1]



## OpenAPI

````yaml api-specs/numerical.json post /quantlib/numerical/roots/find-nd
openapi: 3.1.0
info:
  title: FinceptQuantLib API - Numerical
  description: >-
    Numerical module endpoints for FinceptQuantLib API. The Numerical module
    (Basic Tier, 1 credit per request) provides comprehensive numerical methods
    including finite difference differentiation, Fast Fourier Transform (FFT),
    numerical integration (quadrature and Monte Carlo), interpolation methods
    (linear, cubic, spline), linear algebra operations (matrix decomposition,
    solving systems), ODE solvers, root finding algorithms, optimization
    methods, and nonlinear least squares fitting. Essential for quantitative
    analysis, scientific computing, and numerical model implementation.
  version: 3.0.0
  contact:
    name: Fincept API Support
    url: https://fincept.in
servers:
  - url: https://api.fincept.in
    description: Fincept API Production Server
security:
  - APIKeyHeader: []
tags:
  - name: quantlib-numerical
    description: >-
      Numerical methods including differentiation, FFT, integration,
      interpolation, linear algebra, ODE solvers, root finding, and optimization
    x-displayName: Numerical
paths:
  /quantlib/numerical/roots/find-nd:
    post:
      tags:
        - quantlib-numerical
      summary: Find Root of System of Equations
      description: >-
        Find a solution to a system of nonlinear equations F(x) = 0 where F and
        x are vectors. Supports built-in systems: linear_2x2 (system of two
        linear equations), nonlinear_2x2 (nonlinear system like x^2 + y^2 = 4,
        xy = 1). Methods: newton (multidimensional Newton-Raphson with
        Jacobian), broyden (quasi-Newton method, approximates Jacobian).
        Essential for finding equilibrium points, solving nonlinear systems in
        economics, and multi-asset calibration. [Tier: BASIC, Credits: 1]
      operationId: find_root_nd
      requestBody:
        required: true
        content:
          application/json:
            schema:
              type: object
              required:
                - system
                - x0
              properties:
                system:
                  type: string
                  description: System of equations to solve
                  enum:
                    - linear_2x2
                    - nonlinear_2x2
                  example: nonlinear_2x2
                x0:
                  type: array
                  items:
                    type: number
                  description: >-
                    Initial guess for the solution (number of elements = number
                    of equations)
                  example:
                    - 1.5
                    - 1
                method:
                  type: string
                  description: Solution method
                  enum:
                    - newton
                    - broyden
                  default: newton
                  example: newton
                tol:
                  type: number
                  description: Convergence tolerance
                  default: 1.e-10
                  example: 1.e-10
                params:
                  type: array
                  items:
                    type: number
                  description: >-
                    System-specific parameters (for linear_2x2: [a,b,c,d,e,f]
                    for ax+by+c=0, dx+ey+f=0)
                  example:
                    - 1
                    - 1
                    - -3
                    - 2
                    - -1
                    - -1
            example:
              system: nonlinear_2x2
              x0:
                - 1.5
                - 1
              method: newton
              tol: 1.e-10
      responses:
        '200':
          description: Successful Response
          content:
            application/json:
              schema:
                type: object
                properties:
                  success:
                    type: boolean
                    example: true
                  data:
                    type: object
                    properties:
                      root:
                        type: array
                        items:
                          type: number
                        description: >-
                          Solution vector (for x^2+y^2=4, xy=1 → x≈1.732,
                          y≈0.577)
                        example:
                          - 1.7320508
                          - 0.5773503
                      iterations:
                        type: integer
                        description: Number of iterations required
                        example: 5
                      converged:
                        type: boolean
                        description: Whether the method converged
                        example: true
              example:
                success: true
                data:
                  root:
                    - 1.7320508
                    - 0.5773503
                  iterations: 5
                  converged: true
        '401':
          $ref: '#/components/responses/UnauthorizedError'
        '402':
          $ref: '#/components/responses/InsufficientTierError'
        '422':
          description: Validation Error
      security:
        - APIKeyHeader: []
components:
  responses:
    UnauthorizedError:
      description: Authentication information is missing or invalid
      content:
        application/json:
          schema:
            type: object
            properties:
              detail:
                type: string
                example: Invalid API key
    InsufficientTierError:
      description: API tier insufficient for this endpoint
      content:
        application/json:
          schema:
            type: object
            properties:
              detail:
                type: string
                example: Endpoint requires Basic tier or higher
  securitySchemes:
    APIKeyHeader:
      type: apiKey
      in: header
      name: X-API-Key
      description: >-
        API key for authentication. Get your key at
        https://api.fincept.in/auth/register

````