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How to Write Code for the Marquardt Algorithm

The Levenberg-Marquardt algorithm is a method used in computer science and mathematics to find the local minimum of a function expressed as a sum of least squares of nonlinear functions. It is frequently used to solve linear regression problems and to find the minimum of functions that can be represented as a sum of least squares, as well as an optimizer for functions that cannot be decomposed into least squares. The algorithm is iterative, meaning that it keeps repeating until a stopping condition is met, which is taken to be the optimal solution. You can write code for the Levenberg-Marquardt algorithm using functions in MatLab, or by using functions exposed in solutions written in Fortran and C++.

Things You'll Need

MatLab
C++ editor
Computer

Instructions

- 1
  Launch the "MatLab" application. "MatLab," or Matrix Laboratory, is a mathematical software application that is especially useful for solving least squares equations. It serves as a programming language for mathematical applications and contains functions that are specifically designed for the Levenberg-Marquardt algorithm.
- 2
  Write code to use the lsqnonlin function in "MatLab" to solve nonlinear least squares problems using the Levenberg-Marquardt algorithm. According to the MathWorks documentation, the syntax is:
  x = lsqnonlin(fun,x0)
  
  The function starts at the point ̶0;x0̶1; and finds a minimum of the sum of squares of the functions described in ̶0;fun.̶1; To use this function specifically for the Levenberg-Marquardt algorithm, use the ̶0;ScaleProblem̶1; option in the function definition.
- 3
  Write code to use the fsolve function in "MatLab" to solve systems of nonlinear equations using the using the Levenberg-Marquardt algorithm. According to the MathWorks documentation, the syntax is:
  
  x = fsolve(fun,x0)
  x = fsolve(fun,x0,options)
  
  The function finds a root (zero) of a system of nonlinear equations by starting at a point, x0, and iteratively trying to solve the problems in the function ̶0;fun.̶1; To use this function specifically for the Levenberg-Marquardt algorithm, use the ScaleProblem option in the function definition.
- 4
  Launch your C++ editor and use it to write code to use the Levmar functions to solve linear and nonlinear problems using the Levenberg-Marquardt algorithm. The Llevmar functions are freely available under the GNU General Public License and have been developed to be compatible with "MatLab" and several common programming languages. The Levmar functions are written in C++ and expose functions and code for constrained and unconstrained optimization. You may modify the code to suit your needs and to customize it to fit to your implementation of the Levenberg-Marquardt algorithm.

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