Solution for a system of linear equation with some numeric peculiarities

[gianluca]

Hello,

An overdetermined system of linear equation

y = A x + z

with y vector of known real numbers of dimension m; x vector of unknown real numbers of dimension n; z vector of Gaussian noise of dimension m and A the known coefficient matrix.

it is characterized by 3 aspects:
1)
The unknown x exhibits elements with order of magnitude difference among them.
example: x is 4 elements and I know in advance that two of them will be around 10^4 and 2 around 10^0

2)
The vector z is a noise and each of its element is a Gaussian number with zero mean and known variance.
Basically those are measurements coming from sensors of different "quality", i.e., different variance

3)
Eventually z is composed by elements with a predominant variance.
Example, 80% of the elements of z comes from the same sensor with the same variance and 20% from others

Question:
can someone please link me to a textbook where such numerical aspects are elaborated?
I'm not an expert but I guess that a simple pseudoinverse is not the "best" solution

Thanks in advance,
g.

 

 

[studiot]

Hello,

You have an over determined system that is linear.

There are no unique solutions to this situation however

The branch of mathematics dealing with this is called Linear Programming  ( There is also non linear programming for non linear equations).

 

The method identifies a 'convex hull' in the variable space, bounded in the linear case by lines or flat surfaces / hypersurfaces depending upon the number of variables.

The over determined system is turned into a determined system by introducing additional constraints to identify the optimum maximum or minimum solution.

 

Probabilities are dealt with by adding weighting coefficients to the equation matrix.

 

https://en.wikipedia.org/wiki/Linear_programming