Development of numerical linear algebra algorithms in dynamic fixed-point format: a case study of Lanczos tridiagonalization

AbstractVarious applications in numerical linear algebra and computer science are related to selecting the $$r\times r$$ r × r submatrix of maximum volume contained in a given matrix $$A\in \mathbb R^{n\times n}$$ A ∈ R n × n . We propose a new greedy algorithm of cost $$\mathcal O(n)$$ O ( n ) , for the case A symmetric positive semidefinite (SPSD) and we discuss its extension to related optimization problems such as the maximum ratio of volumes. In the second part of the paper we prove that any SPSD matrix admits a cross approximation built on a principal submatrix whose approximation error is bounded by $$(r+1)$$ ( r + 1 ) times the error of the best rank r approximation in the nuclear norm. In the spirit of recent work by Cortinovis and Kressner we derive some deterministic algorithms, which are capable to retrieve a quasi optimal cross approximation with cost $$\mathcal O(n^3)$$ O ( n 3 ) .

Download Full-text

Students’ Use of Resources in a Challenge-Based Learning Context Involving Mathematics

International Journal of Research in Undergraduate Mathematics Education ◽

10.1007/s40753-021-00136-x ◽

2021 ◽

Author(s):

Birgit Pepin ◽

Zeger-jan Kock

Keyword(s):

Linear Algebra ◽

Complex Environments ◽

Learning Context ◽

University Teachers ◽

Study Approach ◽

Use Of Resources ◽

Challenge Based Learning ◽

Traditional Courses ◽

Education Of Students

AbstractIn this study we use a case study approach to investigate what kinds of resources were self-reported to have been selected by students working on their Challenge-Based bachelor end projects, and how they used and orchestrated these resources. In a previous study, we had explored students’ use of resources and their Actual Student Study Paths in Calculus and Linear Algebra courses (Pepin and Kock 2019); in this study we turned to students working on their Challenge-Based bachelor end projects. Results show that the students working on Challenge-Based projects used resources outside the realm of curriculum resources offered to them in traditional courses, and the Actual Student Study Paths were iterative. We contend that ‘linear’ study paths (found e.g. in traditionally taught Linear Algebra courses) appeared to have been beneficial for students for studying particular mathematical content, which they needed for solving open challenges. In the Challenge-Based projects, the crucial resource emerged to be the tutor. We claim that this has implications for the introduction of a more Challenge-Based curriculum, where such projects would be the norm, both in terms of the professionalization of university teachers (who coach students through such projects), and for the education of students who opt for such projects: teachers have to be supported to develop as appropriate coaches in such complex environments; students have to become self-directed learners, and need to be supported with challenged-based trained coaches and teachers, in addition to be provided with suitable curricular, technological, and social resources.

Download Full-text