Compression of Lookup Table for Piecewise Polynomial Function Evaluation

2014 ◽  
Author(s):  
Shen-Fu Hsiao ◽  
Chia-Sheng Wen ◽  
Po-Han Wu
Keyword(s):  
Polynomial Function ◽  
Lookup Table ◽  
Function Evaluation ◽  
Piecewise Polynomial ◽  
Piecewise Polynomial Function

Dual-Channel Multiplier for Piecewise-Polynomial Function Evaluation for Low-Power 3-D Graphics

2019 ◽  
Vol 27 (4) ◽  
pp. 790-798 ◽  
Author(s):  
Dina M. Ellaithy ◽  
Magdy A. El-Moursy ◽  
Amal Zaki ◽  
Abdelhalim Zekry
Keyword(s):  
Low Power ◽  
Polynomial Function ◽  
Function Evaluation ◽  
Piecewise Polynomial ◽  
Piecewise Polynomial Function ◽  
Dual Channel

Multivariate piecewise polynomials

Acta Numerica ◽  
1993 ◽  
Vol 2 ◽  
pp. 65-109 ◽  
Author(s):  
C. de Boor
Keyword(s):  
Approximation Theory ◽  
Variational Approach ◽  
Polynomial Function ◽  
Piecewise Polynomial ◽  
Multivariate Splines ◽  
Piecewise Polynomial Function ◽  
Multivariate Spline ◽  
Piecewise Polynomials ◽  
The Subject ◽  
Function Of Several Arguments

This article was supposed to be on ‘multivariate splines». An informal survey, taken recently by asking various people in Approximation Theory what they consider to be a ‘multivariate spline’, resulted in the answer that a multivariate spline is a possibly smooth piecewise polynomial function of several arguments. In particular the potentially very useful thin-plate spline was thought to belong more to the subject of radial basis funtions than in the present article. This is all the more surprising to me since I am convinced that the variational approach to splines will play a much greater role in multivariate spline theory than it did or should have in the univariate theory. Still, as there is more than enough material for a survey of multivariate piecewise polynomials, this article is restricted to this topic, as is indicated by the (changed) title.

Large simulation of hysteresis systems using a piecewise polynomial function

2002 ◽  
Vol 9 (7) ◽  
pp. 207-210 ◽  
Author(s):  
Jyh-Da Wei ◽  
Chuen-Tsai Sun
Keyword(s):  
Polynomial Function ◽  
Piecewise Polynomial ◽  
Piecewise Polynomial Function

Correction to "Simulation of hysteresis systems using a piecewise polynomial function"

2002 ◽  
Vol 9 (9) ◽  
pp. 295-295 ◽  
Author(s):  
J.-D. Wei ◽  
C.-T. Sun
Keyword(s):  
Polynomial Function ◽  
Piecewise Polynomial ◽  
Piecewise Polynomial Function

scholarly journals A diagrammatic approach to Kronecker squares

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Ernesto Vallejo
Keyword(s):  
Polynomial Function ◽  
Young Tableaux ◽  
Piecewise Polynomial ◽  
Piecewise Polynomial Function ◽  
Isomorphism Classes ◽  
Nous Montrons ◽  
Il Y A ◽  
International Audience ◽  
Kronecker Coefficients ◽  
Standard Young Tableaux

International audience In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition $\overline{ν}$ of $d$ there is a polynomial $k_{\overline{ν}}$ with rational coefficients in variables $x_C$, where $C$ runs over the set of isomorphism classes of connected skew diagrams of size at most $d$, such that for all partitions $\lambda$ of $n$, the Kronecker coefficient $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ is obtained from $k_{\overline{ν}}(x_C)$ substituting each $x_C$ by the number of $\lambda$-removable diagrams in $C$. We present two applications. First we show that for $\rho_{k} = (k, k-1,\ldots, 2, 1)$ and any partition $\overline{ν}$ of size $d$ there is a piecewise polynomial function $s_{\overline{ν}}$ such that $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ for all $k$ and that there is an interval of the form $[c, \infty)$ in which $s_{\overline{ν}}$ is polynomial of degree $d$ with leading coefficient the number of standard Young tableaux of shape $\overline{ν}$. The second application is new stability property for Kronecker coefficients. Dans ce papier nous améliorons une méthode de Robinson-Taulbee pour calculer les coefficients de Kronecker et montrons que pour toute partition $\overline{ν}$ de $d$ il y a un polynôme $k_{\overline{ν}}$ avec coefficients rationnels dans les variables $x_C$, où $C$ est dans l’ensemble de classes d’isomorphisme des diagrammes gauches connexes de taille non plus que $d$, tel que pour toute partition $\lambda$ de $n$, le coefficient de Kronecker $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ est obtenu de $k_{\overline{ν}}(x_C)$ en substituant chaque $x_C$ pour le nombre de diagrammes $\lambda$-removables en $C$. Nous présentons deux applications. Premièrement nous montrons que pour $\rho_{k} = (k, k-1,\ldots, 2, 1)$ et une partition $\overline{ν}$ de taille $d$ il y a une fonction polynôme par morceaux $s_{\overline{ν}}$ tel que pour toute $k$ on a $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ et qu'il y a une intervalle de la forme $[c, \infty)$ dans laquelle $s_{\overline{ν}}$ est polynôme de degré $d$ avec coefficient principal le nombre de tableaux de Young standard de forme $\overline{ν}$. La seconde application est une nouveau propriété de stabilité des coefficients de Kronecker.

Minimizing Coefficients Wordlength for Piecewise-Polynomial Hardware Function Evaluation With Exact or Faithful Rounding

2017 ◽  
Vol 64 (5) ◽  
pp. 1187-1200 ◽  
Author(s):  
Davide De Caro ◽  
Ettore Napoli ◽  
Darjn Esposito ◽  
Gerardo Castellano ◽  
Nicola Petra ◽  
...  
Keyword(s):  
Function Evaluation ◽  
Piecewise Polynomial ◽  
Hardware Function

scholarly journals FPGA-Based Hardware Matrix Inversion Architecture Using Hybrid Piecewise Polynomial Approximation Systolic Cells

Electronics ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 182 ◽  
Author(s):  
Javier Vázquez-Castillo ◽  
Alejandro Castillo-Atoche ◽  
Roberto Carrasco-Alvarez ◽  
Omar Longoria-Gandara ◽  
Jaime Ortegón-Aguilar
Keyword(s):  
Polynomial Approximation ◽  
Hybrid Approach ◽  
Matrix Inversion ◽  
Lookup Table ◽  
Piecewise Polynomial ◽  
Piecewise Polynomial Approximation ◽  
Matrix Operations ◽  
High Area ◽  
The Matrix ◽  
Field Programmable

The hardware of the matrix inversion architecture using QR decomposition with Givens Rotations (GR) and a back substitution (BS) block is required for many signal processing algorithms. However, the hardware of the GR algorithm requires the implementation of complex operations, such as the reciprocal square root (RSR), which is typically implemented using LookUp Table (LUT) and COordinate Rotation DIgital Computer (CORDICs), among others, conveying to either high-area consumption or low throughput. This paper introduces an Field-Programmable Gate Array (FPGA)-based full matrix inversion architecture using hybrid piecewise polynomial approximation systolic cells. In the design, a hybrid segmentation technique was incorporated for the implementation of piecewise polynomial systolic cells. This hybrid approach is composed by an external and internal segmentation, where the first is nonuniform and the second is uniform, fitting the curve shape of the complex functions achieving a better signal-quantization-to noise-ratio; furthermore, it improves the time performance and area resources. Experimental results reveal a well-balanced improvement in the design achieving high throughput and, hence, less resource utilization in comparison to state-of-the-art FPGA-based architectures. In our study, the proposed design achieves 7.51 Mega-Matrices per second for performing 4 × 4 matrix operations with a latency of 12 clock cycles; meanwhile, the hardware design requires only 1474 slice registers, 1458 LUTs in an FPGA Virtex-5 XC5VLX220T, and 1474 slice registers and 1378 LUTs when a FPGA Virtex-6 XC6VLX240T is used.

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