# Convergence issue in this Least Squares calculation

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• ChiPi
In summary, the conversation discusses computing the trajectory of a moving body using a net composed of 5 stations and observations of difference in time of arrival (DTOA). The speaker is using a Least Squares algorithm with a linear model and an iterative process, and has encountered a problem with convergence depending on the initialization point. They are seeking an explanation for this issue and suggest adding iterative rms error data to the code for further analysis.
ChiPi
TL;DR Summary
I am not able to reach convergence if I initialize the starting point with coordinates outside the stations net. The observations are Difference in Time of Arrival.
I'm computing the trajectory of a moving body and my net is composed by 5 stations.
My observations are DTOA: difference in time of Arrival (they have been linearized).
I am trying to use Least Squares with a linear model: Y = Ax + b, where Y are the observed measurements (DTOA), A the design matrix, b is the known terms vector and X is a vector of estimates. The algorithm processes data according to the epoch considered and iterate the process up to a value of 20 times, unless it reaches before a 1mm convergence.

Since it is an iterative process, the system requires an initialization at the starting point with approximated values for the unknowns, and here comes my problem: if I initialize the starting point with coordinates within the polygon formed by the five stations convergence is reached and the solution is successful, but if I initialize the point with coordinates outside the area formed by the stations the convergence is not reached and I’m not able to determine the trajectory.

Does anyone of you have an explanation for this?

I have a pretty good idea but you should figure this out. Do you have access to the iterative rms error data? Does is say anything to you?

hutchphd said:
I have a pretty good idea but you should figure this out. Do you have access to the iterative rms error data? Does is say anything to you?
I don't have access to it but I can add it at the code if it can help in figuring out the problem. What was your idea?

## What is the convergence issue in Least Squares calculation?

The convergence issue in Least Squares calculation refers to the problem of the algorithm failing to find a solution or taking a very long time to converge to a solution. This can happen when the data is poorly conditioned or when the model is not a good fit for the data.

## How does the convergence issue affect the accuracy of the results?

If the convergence issue is not addressed, it can lead to inaccurate results. The algorithm may not find the optimal solution or may take a long time to converge, resulting in a less precise solution. This can impact the overall reliability of the analysis and the conclusions drawn from it.

## What are some possible solutions to address the convergence issue?

One solution is to use a different optimization algorithm that is better suited for the specific data and model. Another solution is to preprocess the data to improve its conditioning. Additionally, adjusting the algorithm's parameters or using a different initial guess can also help improve convergence.

## How can I identify if a convergence issue is present in my Least Squares calculation?

One way to identify a convergence issue is by monitoring the error or cost function during the optimization process. If the error does not decrease or fluctuates significantly, it could indicate a convergence issue. Additionally, if the algorithm takes a very long time to converge or fails to converge, it could also be a sign of a convergence issue.

## Is there a way to prevent the convergence issue from occurring in the first place?

While it is not always possible to prevent the convergence issue, there are some steps that can be taken to minimize the likelihood of it occurring. These include choosing an appropriate model for the data, ensuring the data is well-conditioned, and using an optimization algorithm that is suitable for the problem. It is also important to carefully select the initial guess for the algorithm and monitor the convergence process closely.

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