Hi everybody.
I have a problem with solving the characteristic equation of an acoustic absorption coating. I have a 4*4 matrix that its determinant must be solved and the answer (the roots of the characteristic equation) is going to be used inside the transfer matrix of an absorbent coating.
My problem is that I can't solve the Det(A)=0 and extract the roots. It seems that the matrix A is ill-conditioned. Does anybody know any go-arounds on how I can extract the roots from the ill-conditioned A matrix (Eq. 10)?
For more information, the matrix elements are complex and include the Bessel and Newmann functions with complex arguments.
the paper is attached.
You can perform a singular value decomposition (SVD) of any matrix, whether it's symmetric or not, positive definite or not, or even ill-conditioned. The "rank" of the matrix may not equal the "order" and it might take considerable effort, but it can be done. The SVD also yields eigenvectors, which may not be symmetric. This process can also be done with complex values. There are several approaches, including: power method, Householder rotations, and Givens rotations. With these keywords you will find many articles on the web in the subject of Linear Algebra. If this doesn't help, send me the 4x4 matrix in an Excel spreadsheet or txt file and I'll send you a C source code that will decompose the matrix that you could use on other similar matrices.
You can use an efficient matrix decomposition (such as SVD) to simply the problem as Dudley suggested above. Here is another solution:
Determinant of a 4*4 matrix has an analytic solution for any general matrix with any elements (including complex). If you cannot compute, use Maple or Mathematica or Matlab Symbolic Toolbox. That will give you a 4th order polynomial with complex confections.
You can use Ferrari's method for solving the general quadratic applies with complex coefficients (could be very tedious), or again leave it to a computer algebra system (Maple, Mathematica). Matlab "roots" command also accepts polynomials with complex coefficients and does some preconditioning before solving.
Just apply Singular Values Decomposition SVD, you will get a diagonal matrix with its entries roots of the equation. Also you can use spectral decomposition or PCA for finding the roots of such matrices.
The problem you are facing is due to the fact that all or some of of columns/rows of your matrix are not mutually orthogonal....SVD is therefore a remedy....it will clean the data ... it plays the role of a filter...reducing noise.
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