Minimizing error function -> sloving linear system of equations

• Theraven1982
In summary, the conversation discusses finding certain coefficients for a function by minimizing an error function. The error function is defined and a linear system of equations is derived to solve for the coefficients. The most efficient method for solving the system is mentioned to be Gaussian elimination with pivoting. The use of Matlab for solving these equations is also suggested.
Theraven1982

Homework Statement

I want to find certain coefficients $$\alpha_{uv}$$ by minimizing a error function. In the end, I want to make a function of this, so iteration is not a problem. I want to estimate these coefficients to find the best estimate for $$f(x,y)$$.

Homework Equations

The error function:
$$E(\alpha)=\sum_{x,y}w(x,y)\Big( f(x,y)-\sum_{u,v=-N}^{N}\alpha_{u,v}f(x+u, y+v)\Big)^{2}$$
with
$$w(x,y)=P(f(x,y)\in M_1|f(x,y)),$$
which in turn is equal to

$$\frac{1}{\sigma\sqrt{2\pi}} \frac{exp\Big[ -\frac{1}{2\sigma^2} \Big( f(x,y) - \sum_{u,v=-N}^{N}\alpha_{u,v}f(x+u, y+v)\Big)^2\Big]}{ \sum_{i=1}^{2}P(f(x,y)|f(x,y)\in M_{i}) }$$

The Attempt at a Solution

$$\frac{\partial E}{\partial \alpha_{s,t}}=0$$

eventually gives:

$$\sum_{u,v=-N}^{N}\alpha_{u,v}=\frac{ \sum_{x,y}w(x,y)f(x+s, y+t)f(x,y) }{ \sum_{x,y}w(x,y)f(x+s, y+t)f(x+u, y+v) }$$

So, now we have a linear system of equations. I don't know how to solve this in a efficient way.

Last edited:
That's a very wide question! There are many different ways of solving a linear system- which is most efficient depends on the coefficients. Gaussian eliminaion with pivoting is probably the best general method.

I was afraid of that ;). I'll find my linear algebra book ;).
Is there anyone who has experience in solving these equations in Matlab? There are probably functions for this purpose ; again, any kick in the right direction is welcome ;)

1. What is the error function in the context of solving linear systems of equations?

The error function, also known as the cost or objective function, is a measure of how far off our calculated values are from the actual solutions of a linear system of equations. It is often defined as the sum of squared errors between the predicted and observed values.

2. Why is it important to minimize the error function when solving linear systems of equations?

Minimizing the error function is important because it allows us to find the most accurate solutions for our linear system of equations. By reducing the error, we can improve the precision and reliability of our results.

3. How is the error function minimized in the context of solving linear systems of equations?

The error function is typically minimized using optimization techniques such as gradient descent or least squares. These methods involve iteratively adjusting the parameters of the system to reduce the error until it reaches a minimum value.

4. What are some common sources of error in solving linear systems of equations?

There are a few common sources of error when solving linear systems of equations, including rounding errors, measurement errors, and computational errors. Additionally, using an inappropriate model or algorithm can also result in higher error values.

5. Can minimizing the error function guarantee an exact solution for a linear system of equations?

No, minimizing the error function does not guarantee an exact solution for a linear system of equations. It can only help us find the most accurate solution given the limitations of our model and data. It is important to choose an appropriate error threshold and consider the trade-off between accuracy and computational complexity when minimizing the error function.

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