How to solve for unknown.

1. Apr 27, 2014

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

2. Apr 27, 2014

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.

3. Apr 27, 2014

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
$$\mathbf{Ax=b} \\ \text{we instead solve}\\ \min_x \mathbf{\|b-Ax \|_2}$$ but we must make sure this solution is reasonable to use