# Algorithm for multidimensional constrained root finding

• Breakgate
In summary, the speaker is looking for an algorithm in Fortran for multidimensional constrained root finding to find a steady-state solution for a dynamic model with n state variables and n coupled differential equations. They are currently using an adapted version of Newton-Raphson from Numerical Recipes, but are having trouble with convergence on impossible values due to lack of support for bounds. They have found an implementation of BFGS that supports constraints, but are unsure if it is suitable for root finding. The speaker is seeking advice on whether to use the BFGS algorithm or if there is a better algorithm available.
Breakgate
Hi all,

I'm looking for an algorithm for multidimensional constrained root finding, implemented in Fortran. It's intended for finding a steady-state solution for a dynamic model. I have n state variables and n coupled differential equations (n~=60), and I need to find the value for the state variables at which the rate of change is zero.

Currently I'm working with an adapted version of Newton-Raphson taken from Numerical Recipes, but this algorithm doesn't support bounds and the solver has a tendency to converge on impossible values. Actually, it seems that not many multidimensional algorithms support bounds. I found an implementation of BFGS (http://hod.greeley.org/papers/Unsorted/lbfgsb.pdf) that supports constraints, but I'm not sure if this suitable for my purpose. From what I understand, not every minimization algorithm is suitable for root finding.
Can anyone tell me if the BGFS algorithm is suitable for root finding, or suggest a better algorithm?

It is probably best to implement it and test it. Should be faster than searching for other algorithms without knowing the outcome.

## 1. What is an algorithm for multidimensional constrained root finding?

An algorithm for multidimensional constrained root finding is a computational method used to determine the roots or solutions of a system of equations with multiple variables, while taking into account certain constraints or limitations on the variables.

## 2. How does the algorithm work?

The algorithm works by iteratively updating the values of the variables in the system of equations, based on a set of rules and constraints, until a solution that satisfies all the constraints is found. This process is repeated until the desired level of accuracy is achieved.

## 3. What are some common constraints used in multidimensional constrained root finding?

Some common constraints used in multidimensional constrained root finding include bounds on the values of the variables, fixed values for certain variables, and relationships between the variables, such as linear or nonlinear constraints.

## 4. Is the algorithm efficient?

The efficiency of the algorithm depends on the specific implementation and the complexity of the system of equations. However, in general, multidimensional constrained root finding algorithms are considered to be efficient and can often find solutions in a reasonable amount of time.

## 5. What are some applications of this algorithm?

This algorithm has various applications in fields such as physics, engineering, economics, and optimization. It can be used to solve complex systems of equations in order to model and analyze real-world problems, and to find optimal solutions that satisfy certain constraints.

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