Numerical Rank Deficiency in Matrix Products

2018 ◽  
Vol 39 (1) ◽  
pp. 365-399
Author(s):  
Michael Stewart
Author(s):  
Yuan-Ho Chen ◽  
Chieh-Yang Liu

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.


2015 ◽  
Vol 2015 ◽  
pp. 1-8 ◽  
Author(s):  
J.-C. Cortés ◽  
L. Jódar ◽  
Francisco J. Solís ◽  
Roberto Ku-Carrillo

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.


2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Thomas Ernst
Keyword(s):  

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


Metrika ◽  
2021 ◽  
Author(s):  
Joscha Krause ◽  
Jan Pablo Burgard ◽  
Domingo Morales

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.


Author(s):  
Edward J. Haug ◽  
Frederick A. Adkins ◽  
Chi-Mei Luh ◽  
Jia-Yi Wang

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.


Sign in / Sign up

Export Citation Format

Share Document