Numerical Rank Deficiency in Matrix Products

2018 ◽  
Vol 39 (1) ◽  
pp. 365-399
Author(s):  
Michael Stewart
Keyword(s):  
Rank Deficiency ◽  
Numerical Rank ◽  
Matrix Products

scholarly journals A low-area high-efficiency video coding inverse transform core using resource and time sharing architecture

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Yuan-Ho Chen ◽  
Chieh-Yang Liu
Keyword(s):  
Video Coding ◽  
Large Scale ◽  
High Efficiency ◽  
Vlsi Design ◽  
Cmos Technology ◽  
Oxide Semiconductor ◽  
Time Sharing ◽  
High Efficiency Video Coding ◽  
Low Area ◽  
Matrix Products

AbstractIn this paper, a very-large-scale integration (VLSI) design that can support high-efficiency video coding inverse discrete cosine transform (IDCT) for multiple transform sizes is proposed. The proposed two-dimensional (2-D) IDCT is implemented at a low area by using a single one-dimensional (1-D) IDCT core with a transpose memory. The proposed 1-D IDCT core decomposes a 32-point transform into 16-, 8-, and 4-point matrix products according to the symmetric property of the transform coefficient. Moreover, we use the shift-and-add unit to share hardware resources between multiple transform dimension matrix products. The 1-D IDCT core can simultaneously calculate the first- and second-dimensional data. The results indicate that the proposed 2-D IDCT core has a throughput rate of 250 MP/s, with only 110 K gate counts when implemented into the Taiwan semiconductor manufacturing (TSMC) 90-nm complementary metal-oxide-semiconductor (CMOS) technology. The results show the proposed circuit has the smallest area supporting the multiple transform sizes.

scholarly journals Infinite Matrix Products and the Representation of the Matrix Gamma Function

2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
J.-C. Cortés ◽  
L. Jódar ◽  
Francisco J. Solís ◽  
Roberto Ku-Carrillo
Keyword(s):  
Gamma Function ◽  
Infinite Product ◽  
Infinite Matrix ◽  
Convergence Results ◽  
Matrix Products ◽  
The Matrix ◽  
Definition Of

We introduce infinite matrix products including some of their main properties and convergence results. We apply them in order to extend to the matrix scenario the definition of the scalar gamma function given by an infinite product due to Weierstrass. A limit representation of the matrix gamma function is also provided.

scholarly journals Factorizations for q-Pascal matrices of two variables

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Thomas Ernst
Keyword(s):  
Matrix Products ◽  
Natural Way

AbstractIn this second article on q-Pascal matrices, we show how the previous factorizations by the summation matrices and the so-called q-unit matrices extend in a natural way to produce q-analogues of Pascal matrices of two variables by Z. Zhang and M. Liu as followsWe also find two different matrix products for

scholarly journals $$\ell _2$$-penalized approximate likelihood inference in logit mixed models for regional prevalence estimation under covariate rank-deficiency

Metrika ◽  
2021 ◽  
Author(s):  
Joscha Krause ◽  
Jan Pablo Burgard ◽  
Domingo Morales
Keyword(s):  
Mixed Model ◽  
Mean Squared Error ◽  
Routine Data ◽  
Parametric Bootstrap ◽  
Monte Carlo Simulation Study ◽  
Rank Deficiency ◽  
Prevalence Estimation ◽  
Fitting Algorithm ◽  
Individual Contributions ◽  
The Individual

AbstractRegional prevalence estimation requires the use of suitable statistical methods on epidemiologic data with substantial local detail. Small area estimation with medical treatment records as covariates marks a promising combination for this purpose. However, medical routine data often has strong internal correlation due to diagnosis-related grouping in the records. Depending on the strength of the correlation, the space spanned by the covariates can become rank-deficient. In this case, prevalence estimates suffer from unacceptable uncertainty as the individual contributions of the covariates to the model cannot be identified properly. We propose an area-level logit mixed model for regional prevalence estimation with a new fitting algorithm to solve this problem. We extend the Laplace approximation to the log-likelihood by an $$\ell _2$$ ℓ 2 -penalty in order to stabilize the estimation process in the presence of covariate rank-deficiency. Empirical best predictors under the model and a parametric bootstrap for mean squared error estimation are presented. A Monte Carlo simulation study is conducted to evaluate the properties of our methodology in a controlled environment. We further provide an empirical application where the district-level prevalence of multiple sclerosis in Germany is estimated using health insurance records.

Domains of Interference Between Working Bodies in Mechanisms and Manipulators

1995 ◽  
Author(s):  
Edward J. Haug ◽  
Frederick A. Adkins ◽  
Chi-Mei Luh ◽  
Jia-Yi Wang
Keyword(s):  
Numerical Methods ◽  
Jacobian Matrix ◽  
Underlying Mechanism ◽  
Rank Deficiency ◽  
Ground Vehicle ◽  
Complementary Problems ◽  
Stewart Platforms ◽  
Working Bodies ◽  
Set Of Points

Abstract Criteria for the set of all points in a pair of working bodies in a mechanism or manipulator that can coincide for any kinematically admissible configuration of the underlying mechanism, called the domain of interference between the bodies, are formulated. Kinematic equations for the mechanism and parameterizations of the domains of the working bodies are used to derive analytical criteria for domains of interference. Three complementary problems are formulated and analyzed to characterize (1) the set of points in one of the interfering bodies that are occupied by any point in the second body, (2) the set of points in one of the interfering bodies that are occupied by any point on the boundary of the second body, and (3) the set of all points in space that are simultaneously occupied by points in the interfering bodies; each condition occurring for any kinematically admissible configuration of the mechanism. Analytical criteria for the boundaries of domains of interference for each of the three problems arc derived, based on row-rank deficiency of a sub-Jacobian matrix associated with the kinematic equations for each of the problems. Numerical methods for mapping boundaries of domains of interference are presented and illustrated for planar Stewart platforms with domes attached that are characteristic of flight or ground vehicle simulators.

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