Solving a set of nonlinear quadratic equations

Main Question or Discussion Point

I would like to solve this system, which is a sets of non linear quadratic equations, the system needed to be solved can be expressed in general as follow:

ϒϒ'C – ϒα = B

Where ϒ=(ϒ1,ϒ2,....ϒn)’ is a column vector and ϒ’ its transpose

C=(c1,c2,…,cn)’ and B=(b1,b2,…bn)’ are a columns vector

And α is a reel scalar

I would like to solve for ϒ, with approximatively about 30

Can someone propose me an algorihm/method to solve this system.
also a code to do it wil be very useful.
Bests

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Mark44
Mentor
I would like to solve this system, which is a sets of non linear quadratic equations, the system needed to be solved can be expressed in general as follow:

ϒ’ϒC – ϒα = B

Where ϒ=(ϒ1,ϒ2,....ϒn)’ is a column vector and ϒ’ its transpose

C=(c1,c2,…,cn)’ and B=(b1,b2,…bn)’ are a columns vector

And α is a reel scalar

I would like to solve for ϒ, with approximatively about 30

Can someone propose me an algorihm/method to solve this system.
also a code to do it wil be very useful.
Bests
Y' Y is a scalar, ##\vec Y \cdot \vec Y = |\vec Y|^2##. If you know the value of ##\vec Y \cdot \vec Y## (but don't know the components of ##\vec Y##), you can rewrite the equation above as a system of equations:
##\alpha Y_1 = \vec Y \cdot \vec Y - b_1##
##\alpha Y_2 = \vec Y \cdot \vec Y - b_2##
.
.
.
##\alpha Y_n = \vec Y \cdot \vec Y - b_n##

Divide both sides by ##\alpha## to get
##Y_1 = 1/\alpha (\vec Y \cdot \vec Y - b_1)##
##Y_2 = 1/\alpha (\vec Y \cdot \vec Y - b_2)##
.
.
.
##Y_n = 1/\alpha (\vec Y \cdot \vec Y - b_n)##

StoneTemplePython
Gold Member
2019 Award
so we're dealing with real scalars here.
- - - -

ϒϒ'C – ϒα = B
or

ϒϒ'C = ϒα + B

so you have a real symmetric rank one matrix on the Left Hand side (LHS).

The issue is that every possible c you can choose on the LHS gets mapped to zero or is an eigenvector (i.e. ϒ) or a linear combination of the two aforementioned things. So lets hope that B is either a scaled version of ϒ or else the zero vector. If your B is the zero vector, it should be pretty easy. Otherwise you have problems.

More issues: For starters, why write ϒα + B on the Right hand side.... why not just write
##\propto ϒ##

so we're dealing with real scalars here.
- - - -

ϒϒ'C – ϒα = B
or

ϒϒ'C = ϒα + B

so you have a real symmetric rank one matrix on the Left Hand side (LHS).

The issue is that every possible c you can choose on the LHS gets mapped to zero or is an eigenvector (i.e. ϒ) or a linear combination of the two aforementioned things. So lets hope that B is either a scaled version of ϒ or else the zero vector. If your B is the zero vector, it should be pretty easy. Otherwise you have problems.

More issues: For starters, why write ϒα + B on the Right hand side.... why not just write
##\propto ϒ##
B is nonzero column, there is a way to solve that?

StoneTemplePython
Gold Member
2019 Award
B is nonzero column, there is a way to solve that?
then ##B \propto ϒ## or this is not an equation

then ##B \propto ϒ## or this is not an equation
B is a constant.

StoneTemplePython
Gold Member
2019 Award
B is a constant.
I don't know what this means. Your original post, and a quick dimensional check say B is a a column vector.

What I am trying to tell you is your original post is analogous to

## 2 = 3##

or

## 2 = 3 +x##
for real ## x \geq 0##

this is not an equation. It is just wrong.

Yes i mean B is a constant column vector. Do you think that is wrong to?

StoneTemplePython
Gold Member
2019 Award
You're not hearing me. It is one of the 3 options

option a)
## B = \mathbf 0##

option b)
##B \propto ϒ##

option c)
this is not an equation. It is just wrong.
- - - - -
I have nothing more to say on the matter. Good luck.

Ok, please look at eq (2 19) page 8 on this link, this paper, maybe some thing wrong.
Portfolio Theory: Origins, Markowitz and CAPM Based Selection - Springer
PDFhttps://www.springer.com › document

Mark44
Mentor
Ok, please look at eq (2 19) page 8 on this link, this paper, maybe some thing wrong.
Portfolio Theory: Origins, Markowitz and CAPM Based Selection - Springer
PDFhttps://www.springer.com › document
Please provide the actual link to the document. The link you show is just to the Springer site.

Please provide the actual link to the document. The link you show is just to the Springer site.
Please find enclosed the document. Go to page 7 to see the original problem, the resolution of lagrangian (which may be wrong) lead to equation posted which is (2 19)

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