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    "# NonlinearConjugateGradientOptimizer Class Documentation\n",
    "\n",
    "*Disclaimer: This documentation was generated by AI and may require human revision for accuracy and completeness.*\n",
    "\n",
    "## Overview\n",
    "\n",
    "The `NonlinearConjugateGradientOptimizer` class in GTSAM is an implementation of the nonlinear conjugate gradient method for optimizing nonlinear functions. This optimizer is particularly useful for solving large-scale optimization problems where the Hessian matrix is not easily computed or stored. The conjugate gradient method is an iterative algorithm that seeks to find the minimum of a function by following a series of conjugate directions.\n",
    "\n",
    "## Key Features\n",
    "\n",
    "- **Optimization Method**: Implements the nonlinear conjugate gradient method, which is an extension of the linear conjugate gradient method to nonlinear optimization problems.\n",
    "- **Efficiency**: Suitable for large-scale problems due to its iterative nature and reduced memory requirements compared to methods that require the Hessian matrix.\n",
    "- **Flexibility**: Can be used with various line search strategies and conjugate gradient update formulas.\n",
    "\n",
    "## Main Methods\n",
    "\n",
    "### Constructor\n",
    "\n",
    "- **NonlinearConjugateGradientOptimizer**: Initializes the optimizer with a given nonlinear factor graph and initial values. The user can specify optimization parameters, including the choice of line search method and conjugate gradient update formula.\n",
    "\n",
    "### Optimization\n",
    "\n",
    "- **optimize**: Executes the optimization process. This method iteratively updates the solution by computing search directions and performing line searches to minimize the objective function along these directions.\n",
    "\n",
    "### Accessors\n",
    "\n",
    "- **error**: Returns the current error value of the objective function. This is useful for monitoring the convergence of the optimization process.\n",
    "- **values**: Retrieves the current estimate of the optimized variables. This allows users to access the solution at any point during the optimization.\n",
    "\n",
    "## Mathematical Background\n",
    "\n",
    "The nonlinear conjugate gradient method seeks to minimize a nonlinear function $f(x)$ by iteratively updating the solution $x_k$ according to:\n",
    "\n",
    "$$ x_{k+1} = x_k + \\alpha_k p_k $$\n",
    "\n",
    "where $p_k$ is the search direction and $\\alpha_k$ is the step size determined by a line search. The search direction $p_k$ is computed using the gradient of the function and a conjugate gradient update formula, such as the Fletcher-Reeves or Polak-Ribiere formulas:\n",
    "\n",
    "- **Fletcher-Reeves**: \n",
    "  $$ \\beta_k^{FR} = \\frac{\\nabla f(x_{k+1})^T \\nabla f(x_{k+1})}{\\nabla f(x_k)^T \\nabla f(x_k)} $$\n",
    "  \n",
    "- **Polak-Ribiere**: \n",
    "  $$ \\beta_k^{PR} = \\frac{\\nabla f(x_{k+1})^T (\\nabla f(x_{k+1}) - \\nabla f(x_k))}{\\nabla f(x_k)^T \\nabla f(x_k)} $$\n",
    "\n",
    "The choice of $\\beta_k$ affects the convergence properties of the algorithm.\n",
    "\n",
    "## Usage Notes\n",
    "\n",
    "- The `NonlinearConjugateGradientOptimizer` is most effective when the problem size is large and the computation of the Hessian is impractical.\n",
    "- Users should choose an appropriate line search method and conjugate gradient update formula based on the specific characteristics of their optimization problem.\n",
    "- Monitoring the error and values during optimization can provide insights into the convergence behavior and help diagnose potential issues.\n",
    "\n",
    "This class provides a robust framework for solving complex nonlinear optimization problems efficiently, leveraging the power of the conjugate gradient method."
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