# Convergence and stability in multivariate fixed point iteration

• dhatfield
In summary, the speaker is seeking help with solving a simulation of a mineral processing froth flotation plant, which involves a computationally intensive solution of systems of equations. They have tried using different algorithms such as Picard iteration and Krasnoselskij iteration, but have concerns about convergence and stability. They are also looking for a recommended algorithm for accelerating convergence and increasing stability without the use of derivatives. They have found a potentially helpful link on the topic.
dhatfield
Hi, I'm new to posting questions on forums, so I apologise if the problem is poorly described.

My problem is solving a simulation of the state of a mineral processing froth flotation plant. In the form x@i+1 = f(x@i), f represents the flotation plant. f is a computationally intensive solution of systems of simultaneous linear equations (material balancing) and evaluation of the state of each unit in the process. x contains ~20 elements, depending on user configuration of the process.

For some (user defined) configurations of the function f, this system iterates rapidly to convergence by back-substitution ie. Picard iteration. For the last couple of years I have been using Krasnoselskij iteration (EMA filter) and the system converges in most, but not all situations. Also, convergence is slow (200+ iterations) for some configurations. Since f is computationally expensive, on the order of 10ms, calculation of a good x@i+1 is crucial. I have also investigated Mann iterations. I have implemeted Direct Inversion in the Iterative Subspace (DIIS) but still have concerns about the uniqueness of solutions from DIIS which I am investigating.

A single calculation of the Jacobian would take longer than most simulations take to converge by back-substitution.

Is there a recommended algorithm for accelerating convergence and increasing the stability of a multidimensional fixed point iteration problem without derivatives?

## 1. How do you define convergence in multivariate fixed point iteration?

Convergence in multivariate fixed point iteration refers to the process of approaching a fixed point or a solution to a system of equations as the number of iterations increases. It means that the values in the iteration sequence are getting closer and closer to the fixed point, and the difference between consecutive iterations is becoming smaller.

## 2. What is the significance of stability in multivariate fixed point iteration?

Stability in multivariate fixed point iteration refers to the property that the iteration process will not diverge or produce unstable results. It is crucial for ensuring that the final solution is accurate and reliable. If the iteration process is unstable, small errors in the initial values can lead to significantly different and incorrect results.

## 3. How can you determine the convergence rate in multivariate fixed point iteration?

The convergence rate in multivariate fixed point iteration can be determined by calculating the spectral radius of the iteration matrix. The spectral radius is the largest eigenvalue of the matrix, and it indicates how quickly the iteration process is approaching the fixed point. A smaller spectral radius indicates a faster convergence rate.

## 4. What are some common methods for improving convergence and stability in multivariate fixed point iteration?

Some common methods for improving convergence and stability in multivariate fixed point iteration include using efficient initial values, using an appropriate iteration scheme (such as Gauss-Seidel or Jacobi), and applying preconditioning techniques. Additionally, choosing a suitable stopping criteria and using adaptive step sizes can also help improve convergence and stability.

## 5. How does the choice of the fixed point affect the convergence and stability in multivariate fixed point iteration?

The choice of the fixed point can greatly impact the convergence and stability in multivariate fixed point iteration. In general, a good fixed point is one that is close to the solution and has a small spectral radius. If the fixed point is far from the solution or has a large spectral radius, the iteration process may be slow or unstable. It is important to carefully choose the fixed point to ensure efficient and stable convergence.

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