Optimal H2 order-one reduction by solving eigenproblems for polynomial equations

A method is given for solving an optimal H2 approximation problem for SISO linear time-invariant stable systems. The method, based on constructive algebra, guarantees that the global optimum is found; it does not involve any gradient-based search, and hence avoids the usual problems of local minima. We examine mostly the case when the model order is reduced by one, and when the original system has distinct poles. This case exhibits special structure which allows us to provide a complete solution. The problem is converted into linear algebra by exhibiting a finite-dimensional basis for a certain space, and can then be solved by eigenvalue calculations, following the methods developed by Stetter and Moeller. The use of Buchberger's algorithm is avoided by writing the first-order optimality conditions in a special form, from which a Groebner basis is immediately available. Compared with our previous work the method presented here has much smaller time and memory requirements, and can therefore be applied to systems of significantly higher McMillan degree. In addition, some hypotheses which were required in the previous work have been removed. Some examples are included.

Statistics anxiety, state anxiety during an examination, and academic achievement.

BACKGROUND: A large proportion of students identify statistics courses as the most anxiety-inducing courses in their curriculum. Many students feel impaired by feelings of state anxiety in the examination and therefore probably show lower achievements. AIMS: The study investigates how statistics anxiety, attitudes (e.g., interest, mathematical self-concept) and trait anxiety, as a general disposition to anxiety, influence experiences of anxiety as well as achievement in an examination. SAMPLE: Participants were 284 undergraduate psychology students, 225 females and 59 males. METHODS: Two weeks prior to the examination, participants completed a demographic questionnaire and measures of the STARS, the STAI, self-concept in mathematics, and interest in statistics. At the beginning of the statistics examination, students assessed their present state anxiety by the KUSTA scale. After 25 min, all examination participants gave another assessment of their anxiety at that moment. Students' examination scores were recorded. Structural equation modelling techniques were used to test relationships between the variables in a multivariate context. RESULTS: Statistics anxiety was the only variable related to state anxiety in the examination. Via state anxiety experienced before and during the examination, statistics anxiety had a negative influence on achievement. However, statistics anxiety also had a direct positive influence on achievement. This result may be explained by students' motivational goals in the specific educational setting. CONCLUSIONS: The results provide insight into the relationship between students' attitudes, dispositions, experiences of anxiety in the examination, and academic achievement, and give recommendations to instructors on how to support students prior to and in the examination.