TY - JOUR
T1 - Accurate polynomial fitting and evaluation via Arnoldi
AU - Zhang, Lei Hong
AU - Su, Yangfeng
AU - Li, Ren Cang
N1 - Funding Information:
Zhang is supported in part by the National Natural Science Foundation of China NSFC- 12071332. Su is supported in part by by the National Natural Science Foundation of China NSFC-11971122. Li is supported in part by USA NSF DMS-1719620 and DMS-2009689.
Publisher Copyright:
© 2024, American Institute of Mathematical Sciences. All rights reserved.
PY - 2024/9
Y1 - 2024/9
N2 - Given sample data points {(xj, fj)}Nj=1, in [Brubeck, Nakatsukasa, and Trefethen, SIAM Review, 63 (2021), pp. 405-415], an Arnoldi-based procedure is proposed to accurately evaluate the best fitting polynomial, in the least squares sense, at new nodes {sj }Mj=1, based on the Vandermonde basis. Numerical tests indicated that this procedure can in general achieve high accuracy. The main purpose of this paper is to perform a forward rounding error analysis in finite precision. Our result establishes sensitivity factors regarding the accuracy of the algorithm, and provides a theoretical justification for why the algorithm works. For least-squares approximation on an interval, we propose a variant of this Arnoldi-based evaluation by using the Chebyshev polynomial basis. Numerical tests are reported to demonstrate our forward rounding error analysis.
AB - Given sample data points {(xj, fj)}Nj=1, in [Brubeck, Nakatsukasa, and Trefethen, SIAM Review, 63 (2021), pp. 405-415], an Arnoldi-based procedure is proposed to accurately evaluate the best fitting polynomial, in the least squares sense, at new nodes {sj }Mj=1, based on the Vandermonde basis. Numerical tests indicated that this procedure can in general achieve high accuracy. The main purpose of this paper is to perform a forward rounding error analysis in finite precision. Our result establishes sensitivity factors regarding the accuracy of the algorithm, and provides a theoretical justification for why the algorithm works. For least-squares approximation on an interval, we propose a variant of this Arnoldi-based evaluation by using the Chebyshev polynomial basis. Numerical tests are reported to demonstrate our forward rounding error analysis.
KW - Arnoldi process
KW - Lanczos process
KW - least-squares
KW - polynomial interpolation
KW - Rounding error analysis
KW - SOAR
KW - Vandermonde matrix
UR - http://www.scopus.com/inward/record.url?scp=85168649370&partnerID=8YFLogxK
U2 - 10.3934/naco.2023002
DO - 10.3934/naco.2023002
M3 - Journal article
AN - SCOPUS:85168649370
SN - 2155-3289
VL - 14
SP - 526
EP - 546
JO - Numerical Algebra, Control and Optimization
JF - Numerical Algebra, Control and Optimization
IS - 3
ER -