# Solving a set of nonlinear quadratic equations

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1. Jun 3, 2018

### 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

Last edited: Jun 3, 2018
2. Jun 3, 2018

### Staff: Mentor

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

3. Jun 3, 2018

4. Jun 3, 2018

### StoneTemplePython

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
$\propto ϒ$

5. Jun 3, 2018

### Tilfani

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

6. Jun 3, 2018

### StoneTemplePython

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

7. Jun 3, 2018

### Tilfani

B is a constant.

8. Jun 3, 2018

### StoneTemplePython

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.

9. Jun 3, 2018

### Tilfani

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

10. Jun 3, 2018

### 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.

11. Jun 3, 2018

### 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

12. Jun 4, 2018

### Staff: Mentor

Please provide the actual link to the document. The link you show is just to the Springer site.

13. Jun 4, 2018

### Tilfani

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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