**6. Concluding remarks**

In this paper, we presented v*ε*-PRINCIPALS that accelerates the convergence of PRINCIPALS by using the v*<sup>ε</sup>* algorithm. The algorithm generates the v*<sup>ε</sup>* accelerated sequence {**X**˙ <sup>∗</sup>(*t*)} using {**X**∗(*t*)}*t*≥<sup>0</sup> but it does not modify the estimation equations in PRINCIPALS. Therefore the algorithm enables an acceleration of the convergence of PRINCIPALS, while still preserving the stable convergence property of PRINCIPALS. The v*ε* algorithm in itself is a fairly simple computational procedure and, at each iteration, it requires only *O*(*np*) arithmetic operations. For each iteration, the computational complexity of the v*ε* algorithm may be less expensive than that for computing a matrix inversion and for solving the eigenvalue problem in PRINCIPALS.

The most appealing points of the v*ε* algorithm are that, if an original sequence converges to a limit point then the accelerated sequence converges to the same limit point as the original sequence and its speed of convergence is faster than the original sequence. In all the numerical experiments, the v*<sup>ε</sup>* accelerated sequence {**X**˙ <sup>∗</sup>(*t*) }*t*≥<sup>0</sup> converges to the final value of {**X**∗(*t*) }*t*≥0 after the significantly fewer number of iterations than that of PRINCIPALS.

The numerical experiments employing simulated data in Section 4 demonstrated that v*ε* acceleration for PRINCIPALS significantly speeds up the convergence of {**X**∗(*t*)}*t*≥<sup>0</sup> in terms of the number of iterations and the computation time. In particular, the v*ε* acceleration effectively works to speed up the convergence for the larger number of iterations of PRINCIPALS. Furthermore, we evaluate the performance of the v*ε* acceleration for PRINCIPALS by applying to variable selection in M.PCA of qualitative data. Numerical experiments using simulated and real data showed that v*ε*-PRINCIPALS improves the speed of convergence of ordinary PRINCIPALS and enables greatly the reduction of computation times in the variable selection for finding a suitable variable set using Backward elimination and Forward selection procedures. The results indicate that the v*ε* acceleration well works in saving the computational time in variable selection problems.

The computations of variable selection in M.PCA of qualitative data are partially performed by the statistical package VASpca(VAriable Selection in principal component analysis) that was developed by Mori, Iizuka, Tarumi and Tanaka in 1999 and can be obtained from Mori's website in Appendix C. We will provide VASpca using v*ε*-PRINCIPALS as the iterative algorithm for PCA and M.PCA of qualitative data.
