Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Fast structured Jacobi-Jacobi transforms


Authors: Jie Shen, Yingwei Wang and Jianlin Xia
Journal: Math. Comp.
MSC (2010): Primary 65T50, 65D05, 65N35, 65F30
DOI: https://doi.org/10.1090/mcom/3377
Published electronically: August 27, 2018
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Jacobi polynomials are frequently used in scientific and engineering applications, and often times, one needs to use the so-called Jacobi-Jacobi transforms which are transforms between two Jacobi expansions with different indices. In this paper, we develop a fast structured algorithm for Jacobi-Jacobi transforms. The algorithm is based on two main ingredients. (i) Derive explicit formulas for connection matrices of two Jacobi expansions with arbitrary indices. In particular, if the indices have integer differences, the connection matrices are relatively sparse or highly structured. The benefit of simultaneous promotion or demotion of the indices is shown. (ii) If the indices have non-integer differences, we explore analytically or numerically a low-rank property hidden in the connection matrices. Combining these two ingredients, we develop a fast structured Jacobi-Jacobi transform with nearly linear complexity, after a one-time precomputation with quadratic complexity, between coefficients of two Jacobi expansions with arbitrary indices. An important byproduct of the fast Jacobi-Jacobi transform is the fast Jacobi transform between the function values at a set of Chebyshev-Gauss-type points and coefficients of the Jacobi expansion with arbitrary indices. Ample numerical results are presented to illustrate the computational efficiency and accuracy of our algorithm.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65T50, 65D05, 65N35, 65F30

Retrieve articles in all journals with MSC (2010): 65T50, 65D05, 65N35, 65F30


Additional Information

Jie Shen
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: shen7@purdue.edu

Yingwei Wang
Affiliation: Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706
Email: wywshtj@gmail.com

Jianlin Xia
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: xiaj@purdue.edu

DOI: https://doi.org/10.1090/mcom/3377
Keywords: Jacobi-Jacobi transform, Chebyshev-Jacobi transform, recurrence formula, connection matrix, low-rank property, structured matrix
Received by editor(s): April 16, 2017
Received by editor(s) in revised form: November 14, 2017, February 8, 2018, and April 11, 2018
Published electronically: August 27, 2018
Additional Notes: The work of the first author was partially supported by NSF grants DMS-1620262, DMS-1720442 and AFOSR FA9550-16-1-0102.
The research of the third author was supported in part by NSF CAREER Award DMS-1255416.
Article copyright: © Copyright 2018 American Mathematical Society

American Mathematical Society