Twisted factorization of a banded matrix

The twisted factorization of a tridiagonal matrix T plays an important role in inverse iteration as featured in the MRRR algorithm. The twisted structure simplifies the computation of the eigenvector approximation and can also improve the accuracy. A tridiagonal twisted factorization is given by T=M _k Δ _k N _k where Δ _k is diagonal, M _k ,N _k have unit diagonals, and the k-th column of M _k and the k-th row of N _k correspond to the k-th column and row of the identity, that is . This paper gives a constructive proof for the existence of the twisted factorizations of a general banded matrix A. We show that for a given twist index k, there actually are two such factorizations. We also investigate the implications on inverse iteration and discuss the role of pivoting.      

Laguerre deconvolution with unknown matrix operator

In this paper we consider the convolutionmodel Z = X + Y withX of unknown density f, independent of Y, when both random variables are nonnegative. Our goal is to estimate the unknown density f of X from n independent identically distributed observations of Z, when the law of the additive process Y is unknown. When the density of Y is known, a solution to the problem has been proposed in [17]. To make the problem identifiable for unknown density of Y, we assume that we have access to a preliminary sample of the nuisance process Y. The question is to propose a solution to an inverse problem with unknown operator. To that aim, we build a family of projection estimators of f on the Laguerre basis, well-suited for nonnegative random variables. The dimension of the projection space is chosen thanks to a model selection procedure by penalization. At last we prove that the final estimator satisfies an oracle inequality. It can be noted that the study of the mean integrated square risk is based on Bernstein’s type concentration inequalities developed for random matrices in [23].     

实对称带状矩阵逆特征值问题

研究了一类实对称带状矩阵逆特征值问题：给定三个互异实数λ,μ和v及三个非零实向量x,y和z,分别构造实对称五对角矩阵T和实对称九对角矩阵A,使其都具有特征对(λ,x),(μ,y)和(v,z)．给出了此类问题的两种提法,研究了问题的可解性以及存在惟一解的充分必要条件,最后给出了数值算法和数值例子．     

A method of solving systems of nonlinear equations of large dimension with banded Jacobian matrix

We consider the concept of bandedness in a matrix. We study iterative methods of solving systems of linear and nonlinear equations of large dimension with a banded Jacobian matrix. We propose a method of computing the row submatrix of a certain full Jacobian matrix that is efficient in a certain sense for numerical differentiation of a vector-valued function of several arguments.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 21–24.     

On the evaluation of the Eigenvalues of a banded toeplitz block matrix

Let A = (a_ij) be an n × n Toeplitz matrix with bandwidth k + 1, K = r + s, that is, a_ij = a_j−i, i, J = 1,… ,n, a_i = 0 if i > s and if i < -r. We compute p(λ)= det(A - λI), as well as p(λ)/p′(λ), where p′(λ) is the first derivative of p(λ), by using O(k log k log n) arithmetic operations. Moreover, if a_i are m × m matrices, so that A is a banded Toeplitz block matrix, then we compute p(λ), as well as p(λ)/p′(λ), by using O(m³k(log² k + log n) + m²k log k log n) arithmetic operations. The algorithms can be extended to the computation of det(A − λB) and of its first derivative, where both A and B are banded Toeplitz matrices. The algorithms may be used as a basis for iterative solution of the eigenvalue problem for the matrix A and of the generalized eigenvalue problem for A and B.     

Impossible differential cryptanalysis using matrix method

The general strategy of impossible differential cryptanalysis is to first find impossible differentials and then exploit them for retrieving subkey material from the outer rounds of block ciphers. Thus, impossible differentials are one of the crucial factors to see how much the underlying block ciphers are resistant to impossible differential cryptanalysis. In this article, we introduce a widely applicable matrix method to find impossible differentials of block cipher structures whose round functions are bijective. Using this method, we find various impossible differentials of known block cipher structures: Nyberg’s generalized Feistel network, a generalized CAST256-like structure, a generalized MARS-like structure, a generalized RC6-like structure, Rijndael structures and generalized Skipjack-like structures. We expect that the matrix method developed in this article will be useful for evaluating the security of block ciphers against impossible differential cryptanalysis, especially when one tries to design a block cipher with a secure structure.     

The approximation of a totally positive band matrix by a strictly banded totally positive one

Every nonsingular totally positive m-banded matrix is shown to be the product of m totally positive one-banded matrices and, therefore, the limit of strictly m-banded totally positive matrices. This result is then extended to (bi)infinite m-banded totally positive matrices with linearly independent rows and columns. In the process, such matrices are shown to possess at least one diagonal whose principal sections are all nonzero. As a consequence, such matrices are seen to be approximable by strictly m-banded totally positive ones.     

Coalescing points for eigenvalues of banded matrices depending on parameters with application to banded random matrix functions

Numerical Algorithms - In this work, we develop and implement new numerical methods to locate generic degeneracies (i.e., isolated parameters’ values where the eigenvalues coalesce) of banded...     

Parallel factorization of banded linear matrices using a systolic array processor

A parallel algorithm for calculating theQR factorization of a banded system of linear equations using a systolic array processor is presented and an application to spline fitting is given. The major advantage of the method is that the size of the processor array is fixed by the size of the bandwidth of the system to be solved. This allows the factorization of large systems using small systolic arrays. The cost of the method, in terms of storage and time, is optimal.     

Prediction of a glucose appearance function from foods using deconvolution

The glycaemic response of an insulin-treated diabetic patientgoes through many transitory phases, leading to a steady stateglycaemic profile following a change in either insulin regimenor diet. Most models attempting to model the glucose and insulinrelationship try to model the effect of oral or injected glucoserather than that from the digestion of food. However, it isclear that a better understanding of the glycaemic responsewould arise from consideration of intestinal absorption fromthe gut. It is assumed that this type of absorption can be modelledby a so-called glucose appearance function (systemic appearanceof glucose via glucose absorption from the gut) predicting theglucose load from the food. Much research has been carried outin the areas of hepatic balance, insulin absorption and insulinindependent/dependent utilization. However, little is knownabout intestinal absorption patterns or their correspondingglucose appearance profiles. The strategy under investigation herein is to use deconvolutionor backward engineering. By starting with specific results i.e.blood glucose and insulin therapy, it is possible to work backwardsto predict the glucose forcing functions responsible for theoutcome. Assuming compartmental consistency, this will allowa clearer insight into the true gut absorption process, If successful,the same strategy can be applied to more recent glucose andinsulin models to further our understanding of the food to bloodglucose problem. This paper investigates the Lehmann-Deutsch modified model ofglucose and insulin interaction, created from the model proposedby Berger-Rodbard. The model attempts to simulate the steadystate glycaemic and plasma insulin responses, independent ofthe initial values from which the simulation is started. Glucoseenters the model via both intestinal absorption and hepaticglucose production. We considered a 70kg male insulindependentdiabetic patient with corresponding hepatic and insulin sensitivityparameters of 0.6 and 0.3 respectively. Net hepatic glucosebalance was modelled piecewise by linear and symmetric functions.A first-order Euler method with step size of 15 minutes wasemployed. For the simulation, only Actrapid and NPH injectionswere considered. The injection of insulin and the glucose fluxto the gut were started simultaneously to avoid any delay associatedwith gastric emptying. The systemic appearance of glucose was compared from two viewpoints, not only to assess the strategic principle, but alsoto assess the suitability of the modifications made by Lehmannand Deutsch. The first is a forward prediction using the compartmentalstructure. This analysis involves the rate of gastric emptyingwithout time delay. The second is a backward prediction fromexperimentally observed blood glucose profiles. Investigationsinvolved porridge, white rice and banana containing the samecarbohydrate content (25 g). Results obtained from the firstanalysis were dependent on the rate of gastric emptying, especiallyits ascending and descending branches. Results from the secondanalysis were dependent on the dose and type of insulin administered.Both predicted profiles showed consistency with physiologicalreasoning, although it became apparent that such solutions couldbe unstable. Furthermore, both types of prediction were similarin structure and appearance, especially in simulations for porridgeand banana. This emphasized the consistency and suitabilityof both analyses when investigating the compartmental accuracyand limitations within a model. The new strategic approach was deemed a success within the model,and the modifications made by Lehmann and Deutsch appropriate.We suggest that a gastric emptying curve with a possible gastricdelay is the way forward in regulating the appearance of glucosevia gut absorption. The Lehmann-Deutsch gastric curve is describedby either a trapezoidal or triangular function dependent onthe carbohydrate content of the meal. However, it was clearfrom the results obtained that carbohydrate content is onlyone factor in carbohydrate absorption, and further progressmust inevitably involve other food characteristics and propertiesif we are to improve the glucose flux.     

Pseudospectral time-domain method for pressure-release rough surface scattering using a surface transformation and image method

When numerically analyzing acoustic scattering at a pressure-release rough surface, the conventional pseudospectral time domain (PSTD) method using Fourier transform requires rigorous stability conditions in order to solve the spatial derivative in the wave equation on the irregular boundaries between the two media due to the Gibbs phenomenon and short wavelength in air. To eliminate such disadvantages, a new algorithm is proposed based on the Fourier PSTD method utilizing a surface boundary transformation and an image method. Irregular surface boundaries are flattened by transformation and then an image method is applied to the half-space domain. The efficiency and accuracy of the proposed PSTD method are better than the conventional Fourier PSTD method. Numerical results are presented for a sloped and a sinusoidal pressure-release surface.     

Solutions to deconvolution equations using nonperiodic sampling

Explicit, compactly supported solutions, {v_i, ϕ} _i=1 ^m , to the deconvolution (or Bezout) equation

High-order total variation-based Poissonian image deconvolution with spatially adapted regularization parameter

Images captured by image acquisition systems using photon-counting devices such as astronomical imaging, positron emission tomography and confocal microscopy imaging, are often contaminated by Poisson noise. Total variation (TV) regularization, which is a classic regularization technique in image restoration, is well-known for recovering sharp edges of an image. Since the regularization parameter is important for a good recovery, Chen and Cheng (2012) proposed an effective TV-based Poissonian image deblurring model with a spatially adapted regularization parameter. However, it has drawbacks since the TV regularization produces staircase artifacts. In this paper, in order to remedy the shortcoming of TV of their model, we introduce an extra high-order total variation (HTV) regularization term. Furthermore, to balance the trade-off between edges and the smooth regions in the images, we also incorporate a weighting parameter to discriminate the TV and the HTV penalty. The proposed model is solved by an iterative algorithm under the framework of the well-known alternating direction method of multipliers. Our numerical results demonstrate the effectiveness and efficiency of the proposed method, in terms of signal-to-noise ratio (SNR) and relative error (RelRrr).     

Spectral decomposition of a matrix using the generalized sign matrix

An algorithm for spectral decomposition is presented which does not require knowledge of eigenvalues and eigenvectors. A set of eigenprojectors are defined which covers the entire spectrum of a matrix, and special attention is given to the projection on the zero eigenvalue. Some useful applications are discussed in the paper.     

TESTS OF COVARIANCE MATRIX BY USING PROJECTION PURSUIT AND BOOTSTRAP METHOD

Testing equality of covariance matrix has long been an interesting issue in statistics inference, To overcome the sparseness of data points in high-dimensional space and deal with the general cases, the author suggests several projection pursuit type statistics. Some results on the limiting distidbutions of the statistics are obtained. Some properties of bootstrap approximation are investigated. Furthermore, for computational reasons an approximation for the statistics based on number-theoretic roethod is applied. Several simulation experiments are performed.     

Optimization of image quality for decoded images using three-dimensional smoothing method

Joint Photographic Experts Group (JPEG) baseline algorithm is widely used because of its high compression capability. The algorithm has a characteristic that the degradation of image quality tends to be perceived as the compression ratio becomes high. The eyesore degradations are false contour and mosquito. A method of improving the image quality is proposed. The first step: a domain of false contours is extracted from the image and the second step: the domain is smoothed by a fitting process. It is confirmed that this method is effective in improving the image quality degraded by the JPEG high compression.