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How to solve for unknown.

  1. Apr 27, 2014 #1
    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. jcsd
  3. Apr 27, 2014 #2

    Simon Bridge

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    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.
     
  4. Apr 27, 2014 #3

    lurflurf

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    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
     
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