Abstract:
The rank determination of a matrix A Î Âmxn is a computational problem that, when the matrix is close to being singular, presents serious numerical dificulties which imply that in the presence of roundoff errors, certain "rank revealing" matricial decompositions do not fulfil that aim in a satisfactory way.
The singular value decomposition (SVD) is the most robust technique for the determination of the rank of a given matrix and is implemented in the main numerical libraries currently available. In many cases, the solution obtained with decompositions that require less computation may be good enough and, for this reason, the use of these alternatives is not to be discarded; the QR decomposition with column pivoting is a serious alternative to the SVD (in its first versions, the Matlab's function rank used this decomposition, in the most recent versions it uses the SVD). In some cases, the QR decomposition with column pivoting does not detect the "near-singularity" of the matrix, but the method can be improved; in this context, the algorithm RRQR ("Rank-revealing" QR), is an important variant.
In this thesis, we study the most relevant decompositions in the context of the problem of determining the rank of a matrix, we implement some of these decompositions and we carry out a comparison of the used methods. |