How to solve linear least squares problem of the following form in matlab

Hello

1. I need to solve the problem which is written in the following form:

I have all the terms except ws, which I need to find. Firstly I don't understand how I can rewrite this problem in the following way Ws = ... Is it possible?

And how to solve this problem in Matlab? I did not find suitable function which corresponds to this form...

2. From the same article:

I don't have any idea how they rewrite this but how to solve this problem in this form?

What I found in Matlab is:

But what does it mean this 2 at the bottom and can I use this function to solve equation (3)?

Since H and h are the matrices, so can I write in matlab vector A which contains another matrices?

Thank you!

Answer 1

[Walter Roberson]

||expression|| with subscript 2 means the 2-norm, also known as the Euclidean distance. With the Euclidean distance being sqrt() of a sum of squares, and with the result of the 2-norm being squared by lsqnonneg, what is being computed is effectively the sum of squares.

Likely this is the same thing that your equation means with ||expression|| squared with no subscript 2, so Yes, lsqnonneg is a plausible function.

[Y I]

Thank you!

But to understand better may I ask two questions:

1) if my A and b are complex how can I get x which assumed (and needed) to be a complex as well? Because lsqnonneg function returns x which is ≥ 0.

2) I am confused, at some books I found that this problem:

can be solved like:

which in MATLAB is: mldivide function

x = A\b;

So can I really use mldivide instead of lsqnonneg to solve this equation:

, does it give the same results?

What is the difference between mldivide and lsqnonneg MATLAB functions?

Than you!

[Walter Roberson]

lsqnonneg promises a non-negative solution. mldivide does not.

[Y I]

Ok thanks, but I still dont understand, can I use mldivide to solve my problem? (equation 3 above)

[Walter Roberson]

Are your ws permitted to be complex valued? Are they permitted to be negative?

[Y I]

Yes, ws are permitted to be complex and negative.

[Walter Roberson]

Then Yes, it looks to me as of you can use the \ operator.

MATLAB offers multiple methods of solving problems.