Details
| Original language | English |
|---|---|
| Pages (from-to) | 1481 - 1486 |
| Number of pages | 7 |
| Journal | IEEE Transactions on Information Theory |
| Volume | 71 |
| Issue number | 2 |
| Early online date | 4 Dec 2024 |
| Publication status | Published - Feb 2025 |
Abstract
The Schläfli function fn(x) allows to compute volume of a regular (n - 1)-dimensional spherical simplex of the dihedral angle 2α = arcsec(x) and it has many applications. For example, it defines conjectured upper bounds on the sphere packing problem and the kissing number problem, and a lower bound on the mean-squared error in the quantizing problem. The function is defined recursively via a first-order non-linear differential relation, that makes it difficult to compute, especially for large values of n. Here we present a method for an accurate numerical computation of the Schläfli function fn(x) for n ≥ 4 in the frequently used in applications interval x ∈ [n - 1, n + 1]. The computation is based on the Chebyshev approximation of the function qn(x), which is related to the Schläfli function via a simple factor of an algebraic expression and regular in the interval. We also present the computation algorithm based on the method.
Keywords
- Chebyshev approximation, kissing number problem, quantizing problem, Schläfli function, sphere packing problem
ASJC Scopus subject areas
- Computer Science(all)
- Information Systems
- Computer Science(all)
- Computer Science Applications
- Social Sciences(all)
- Library and Information Sciences
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In: IEEE Transactions on Information Theory, Vol. 71, No. 2, 02.2025, p. 1481 - 1486.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Computation of the Schläfli Function
AU - Shoom, Andrey A.
N1 - Publisher Copyright: © 1963-2012 IEEE.
PY - 2025/2
Y1 - 2025/2
N2 - The Schläfli function fn(x) allows to compute volume of a regular (n - 1)-dimensional spherical simplex of the dihedral angle 2α = arcsec(x) and it has many applications. For example, it defines conjectured upper bounds on the sphere packing problem and the kissing number problem, and a lower bound on the mean-squared error in the quantizing problem. The function is defined recursively via a first-order non-linear differential relation, that makes it difficult to compute, especially for large values of n. Here we present a method for an accurate numerical computation of the Schläfli function fn(x) for n ≥ 4 in the frequently used in applications interval x ∈ [n - 1, n + 1]. The computation is based on the Chebyshev approximation of the function qn(x), which is related to the Schläfli function via a simple factor of an algebraic expression and regular in the interval. We also present the computation algorithm based on the method.
AB - The Schläfli function fn(x) allows to compute volume of a regular (n - 1)-dimensional spherical simplex of the dihedral angle 2α = arcsec(x) and it has many applications. For example, it defines conjectured upper bounds on the sphere packing problem and the kissing number problem, and a lower bound on the mean-squared error in the quantizing problem. The function is defined recursively via a first-order non-linear differential relation, that makes it difficult to compute, especially for large values of n. Here we present a method for an accurate numerical computation of the Schläfli function fn(x) for n ≥ 4 in the frequently used in applications interval x ∈ [n - 1, n + 1]. The computation is based on the Chebyshev approximation of the function qn(x), which is related to the Schläfli function via a simple factor of an algebraic expression and regular in the interval. We also present the computation algorithm based on the method.
KW - Chebyshev approximation
KW - kissing number problem
KW - quantizing problem
KW - Schläfli function
KW - sphere packing problem
UR - http://www.scopus.com/inward/record.url?scp=85211431984&partnerID=8YFLogxK
U2 - 10.1109/TIT.2024.3511134
DO - 10.1109/TIT.2024.3511134
M3 - Article
AN - SCOPUS:85211431984
VL - 71
SP - 1481
EP - 1486
JO - IEEE Transactions on Information Theory
JF - IEEE Transactions on Information Theory
SN - 0018-9448
IS - 2
ER -