Solving a set of nonlinear quadratic equations

[Tilfani]

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

 

[Mark44]

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
$Y_1 = 1/\alpha (\vec Y \cdot \vec Y - b_1)$
$Y_2 = 1/\alpha (\vec Y \cdot \vec Y - b_2)$
.
.
.
$Y_n = 1/\alpha (\vec Y \cdot \vec Y - b_n)$

 

[Tilfani]

Thank you for your reply. I did a typing error in the formulation, the right ones is
ϒϒ'C – ϒα = B
The product Y'Y is a (n,n) matrix
Reference https://www.physicsforums.com/threads/solving-a-set-of-nonlinear-quadratic-equations.948843/

 

[StoneTemplePython]

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

So your 'equation' is:

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 let's 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
$\propto ϒ$

 

[Tilfani]

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

So your 'equation' is:

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 let's 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]

B is nonzero column, there is a way to solve that?

then $B \propto ϒ$ or this is not an equation

 

[Tilfani]

then $B \propto ϒ$ or this is not an equation

B is a constant.

 

[StoneTemplePython]

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.

 

[Tilfani]

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

 

[StoneTemplePython]

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.

 

[Tilfani]

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]

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.

 

[Tilfani]

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)