Let be [D] a (nxn) ill-conditioned matrix and [G] its inverse. If the inverse exists then it can be efficiently implemented by the linear solving of [D]{gj}={bj} with the LU method using partial pivoting where {gj}: is a column vector of [G] and {bj}: is a column of the identity matrix [I] for j=1 to n and (bj(i)=0 for i=1 to n such that bj(j)=1). The resulting matrix can be checked with the satisfying condition [D][G]=[I].
Hi, all depends on the size of the matrix. You can for example refer to the following papers:
Dear Epenoy,
Thank you for the detailed information from your work. The papers that you have presented are a great work. So I would like to ask you some questions:
* why are you using extremely ill-conditioned matrices with large numbers ? In practice we have needed to use the inverse in the case of material properties matrix or in the development of the power of a matrices sum.
* which kind of computer (i.e processor) are you using for these computations ? A workstation or a PC.
* The CPU times are obtained from which compiler or using which library? Because CPU times in multiprocessors Workstations are not the same to those in PC's. These last ones give CPU time = Elapsed time : when using specific functions which is approximative. It is important to notice that you have presented interesting CPU times like 1.750 for DSM (n=100) and 0.016 for VR (n=20) in Hall paper.
* I am finding in this work large condition numbers ~=6.4*10^64 (Rump paper) which means that there is no need to handle these matices. Also the UBM gives cond(H)=3.78*10^150. Is your Matlab release able to perform easily these computations ?
Hi, I am not directly involved in this topic but I think preconditioners may also help:
http://www.mathcs.emory.edu/~benzi/Web_papers/survey.pdf
Dear Epenoy,
I have used preconditionners and verified that they give better results to improve the precision: it is a good idea I agréé with you. But in your case if you use n-sized large systems of equations you need to perform at least two multiplications of nxn matrices and nx1 vectors with factorization and at least a double memory storage. This leads to expensive costs. You should use published optimized algorithms like 'acm transactions on mathematical software' to save time engineer and time machine.
If the matrices with which you are working are extremely ill conditioned, then at least some entries in your inverses are apt to have few, if any, correct digits, and possibly the entire computed inverse will be a long way from the actual inverse. So, do you really need the inverse at all, or can you bypass inverse computation by solving linear systems whose solutions are implicitly. This is important because ill-conditioning may only affect the quality of certain solutions of Ax = b, and hence for many choices of b, x will be well computed by Gaussian elimination. In contrast, I would expect that ill-conditioning could severely impact much or all of A^(-1). If you have not already done so, I suggest that you look at a book on numerical linear algebra such as Gene Golub and Charlie Van Loan's classic, which is now in its fourth or fifth edition. Finally, LU decompositions are probably not the best tool for ill-conditioned matrices. Perhaps a transformation using orthogonal rotations would help, with the singular value decomposition as the destination. Again, see Golub and Van Loan.
Dear Stuart,
The LU factorisation must exit if the inverse exists when solving [L][U]{gj}={bj} with partial pivoting Gauss method which computes a j-column vector of the [G] inverse matrix because it avoids the pivoting problems. These last ones give a poor straightforward gaussian elimination. For that reason it is important to use the partial pivoting which moves forward an eventual zero pivot to the last elimination step at this stage the pivot must not be zero which means that two rows are not linearly dependent and the solution exists effectively. I agree with you with the other points that you stated.
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