Is it possible to solve for y using non-linear methods?

[pjoseph98]

Hi there,

I am no expert in linear algebra (and I don't think this problem is linear anyway).

I am trying to solve the following for y: [A]y = C

A is an 8x2 matrix (fully known)
C is an 8x1 matrix (fully known)

B is an 2x1 matrix (whose terms are a function of the single unknown y).

The two terms in are: b1*e^(b2 + y) and b3*e^(b4 + y) where b1, b2, b3, and b4 are fully known.

Is it possible to solve for y? Do I use my favorite method--brute force or is there something more elegant. The problem (as I understand it) is that the matrices are not symmetric, far less, square.

And I need to solve this at each integration point in a Finite Element Analysis mesh...with up to 10,000 integration points, so ideally a brute force method would not be my preference...

Paul

 

[Simon Bridge]

y has to be 1x1 (a scalar) - but that is what you had in mind.
Note: taking out the common factor...$$B=\begin{pmatrix}b_1e^{b_2}\\ b_3e^{b_4}\end{pmatrix}e^y$$
Put $AB=Xe^y$ so that $Xye^y=C$ (X and C will be fully known) and compare terms.

 

[lurflurf]

The first thing I would suggest is pulling y out of

[A](*e^-y)y*e^y = C
where (*e^-y) does not depend on y

The next problem is we do not know if C is a multiple of [A](*e^-y)
if not we can use least squares
that is instead of solving
$$\mathbf{Ax=b} \\
\text{we instead solve}\\
\min_x \mathbf{\|b-Ax \|_2}$$ but we must make sure this solution is reasonable to use
otherwise we may need to reformulate the problem with y as a matrix

finally we need the Lambert W function

http://en.wikipedia.org/wiki/Linear_least_squares_(mathematics)
http://en.wikipedia.org/wiki/Moore–Penrose_pseudoinverse
http://en.wikipedia.org/wiki/Lambert_W_function