Besides, this step can also prevent the ill-conditioned problem when applying CCA directly to the raw data.
In practice, it is possible to have rank ([X.sup.T]X)< p so that the invertibility cannot be satisfied, and directly applying eigenvalue decomposition in the raw data space may lead to the ill-conditioned problem. Therefore, some appropriate preprocessing strategies are needed in practice before applying CCA.
With using CCA, we can extract highly correlated LVs from EEG and EMG signals, but it cannot ensure that such LVs are nontrivial and we may face the ill-conditioned problem.
Accordingly, regularization techniques, which can treat the ill-conditioned problem, have been utilized in many fields [10-13].
To identify [S.sub.id], we need to get [Y.sup.c] and G; moreover, it was found that (15) is a typical ill-conditioned problem. In some cases, small varieties in the right-hand side e of (15) may lead to dramatic varieties in the solution [25-28].
Memory requirements for each solver show that the memory requirements for the PCG and SPAR solver are nearly identical while the AMG solver requires about twice as much memory for this example and increases additionally when the more ill-conditioned problem uses a more expensive preconditioner.
A second parallel iterative solver has been added in ANSYS 5.7 that gives improved convergence for ill-conditioned problems and better parallel scaling.
For ill-conditioned problems, these methods are often unstable  [Chap.
Moreover, for ill-conditioned problems, sparsity regularization is often unstable.
Simon Bolivar, Venezuela) survey modern techniques for the numerical solution of linear and nonlinear least squares, and introduce the treatment of large and ill-conditioned problems
. The material is geared toward scientists and engineers who analyze and solve least-squares problems, but is also suitable for a graduate or advanced undergraduate course for students with a working knowledge of linear algebra and basic statistics.
It is the aim of this paper to extend this theory to the case of discrete ill-conditioned problems. Such problems typically (but not exclusively) arise by discretizing ill-posed problems and are very important.
By studying some typical examples of ill-conditioned problems in Section 5.1, we can consider the question if and in what sense a noise condition holds, and how the parameter choice rules are used in practice.