Roundoff Error Analysis of an Algorithm Based on Householder Bidiagonalization for Total Least Squares Problems

For large-scale problems, how to establish an algorithm with high accuracy and stability is particularly important. In this paper, the Householder bidiagonalization total least squares (HBITLS) algorithm and nonlinear iterative partial least squares for total least squares (NIPALS-TLS) algorithm were established, by which the same approximate TLS solutions was obtained. In addition, the propagation of the roundoff error for the process of the HBITLS algorithm was analyzed, and the mixed forward-backward stability of these two algorithms was proved. Furthermore, an upper bound of roundoff error was derived, which presents a more detailed and clearer approximation of the computed solution.

Effect of Externally Cued Training on Dynamic Stability Control During the Sit-to-Stand Task in People With Parkinson Disease

Background Previous studies have shown that people with Parkinson disease (PD) have difficulty performing the sit-to-stand task because of mobility and stability-related impairments. Despite its importance, literature on the quantification of dynamic stability control in people with PD during this task is limited. Objective The study objective was to examine differences in dynamic stability control between people with PD and people who were healthy and the extent to which externally cued training could improve such control during the sit-to-stand task in people with PD. Design This was a quasi-experimental controlled trial. Methods The performance of 21 people with PD was compared with that of 12 older adults who dwelled in the community. People with PD were randomly assigned to 2 groups: a group that did not receive training and a group that received audiovisually cued training (3 times per week for 4 weeks) for speeding up performance on the sit-to-stand task. Outcome measures recorded at baseline and after 4 weeks included center-of-mass position, center-of-mass velocity, and stability against either backward or forward balance loss (backward or forward stability) at seat-off and movement termination. Results Compared with people who were healthy, people with PD had greater backward stability resulting from a more anterior center-of-mass position at seat-off. This feature, combined with decreased forward stability at movement termination, increased their risk of forward balance loss at movement termination. After training, people with PD achieved greater backward stability through increased forward center-of-mass velocity at seat-off and reduced the likelihood of forward balance loss at movement termination through a posterior shift in the center-of-mass position. Limitations The study applied stability limits derived from adults who were healthy to people with PD, and the suggested impact on the risk of balance loss and falling is based on these theoretical stability limits. Conclusions For people with PD, postural stability against backward balance loss at task initiation was increased at the expense of possible forward balance loss at task termination. Task-specific training with preparatory audiovisual cues resulted in improved overall dynamic stability against both forward and backward balance loss.