# Numerical Algebraic Approach

Numerical constraint solvers function by first translating the constraints into a system of algebraic equations. This system is then solved using an iterative technique such as the Newton-Raphson method. Clearly, numerical methods are an example of instance solvers. A positive feature of this approach is that it is able to handle overconstrained but consistent problems which other techniques may not be able to solve, assuming convergence. For this reason, many constraint solvers fall back on iterative techniques when the native method is not sufficient to solve a given configuration.

However, there are some serious drawbacks with the numerical approach. First is the problem that of the potentially exponential number of solutions, iterative methods can produce only a single solution. Also, the solution to which it converges depends strongly on the initial configuration. Furthermore, because of the multiple solutions and the large number of parameters, the constraint solving problem is often ill-conditioned, making convergence difficult or impossible.

## References

• L. A. Barford [2]
• A. H. Borning [5]
• J. Gosling [36]
• G. Nelson [70]
• R. Light and D. Gossard [66]
• I. Sutherland [105]
• A. Witkin, K. Fleischer, and A. Barr [118]