Help with applying the least squares method for solving simultaneous equations

In summary, the conversation discusses a system of equations with 3 unknowns and the need to solve for these unknowns. However, due to experimental data and lack of additional equations, the solution will only be approximate. The speaker also mentions a standard technique for solving equations with more variables.
  • #1
yasith
14
0
Hi everyone given the system of equations

A1Cx + B1Cy + C1Cz = D1
A2Cx + B2Cy + C2Cz = D2
A3Cx + B3Cy + C3Cz = D3

I need to solve for Cx, Cy, Cz
All other variables are known and constants.
However all other variables (A,b,c,d) come from experimentally measured data and thus I cannot use RREF to derive a unique solution.

This system will only have an approximate solution. Please help me with the strategy for solving these equations.

Yasith
 
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  • #2
You have 3 equations and 3 unknowns. In the absence of any further information, and assuming there's no redundancy in the set of equations, there will be a unique, exact solution. I see no basis for allowing for experimental error.
If you had more equations then there's a standard technique, and it sounds like you're aware of that.
 

FAQ: Help with applying the least squares method for solving simultaneous equations

1. What is the least squares method?

The least squares method is a mathematical technique used to find the best fit line or curve for a set of data points. It minimizes the sum of the squared differences between the actual data points and the predicted values from the line or curve.

2. How is the least squares method used to solve simultaneous equations?

The least squares method can be applied to solve systems of linear equations by setting up a matrix equation and finding the least squares solution using matrix operations. This allows for finding an approximate solution when there is no exact solution to the system.

3. What are the advantages of using the least squares method for solving simultaneous equations?

The least squares method allows for finding an approximate solution when there is no exact solution, making it useful for real-world applications. It also takes into account all data points, not just a select few, and minimizes the error between the predicted values and the actual data, resulting in a more accurate solution.

4. Are there any limitations to using the least squares method for solving simultaneous equations?

One limitation of the least squares method is that it assumes a linear relationship between variables. If the relationship is non-linear, the least squares solution may not be the best fit. It also relies on the accuracy of the data points, so outliers or errors in the data can impact the results.

5. Can the least squares method be used for non-linear equations?

While the least squares method is typically used for linear equations, it can also be adapted for solving non-linear equations by transforming the equations into a linear form. However, this may not always result in the best fit and other methods may be more suitable for non-linear equations.

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