Revealing the Hidden Mathematical Beauties of the Cayley-Hamilton Method

The inversion of a non-singular square matrix applying a Computer Algebra System (CAS) is straightforward. The CASs make the numeric computation efficient but mock the mathematical characteristics. The algorithms conducive to the output are sealed and inaccessible. In practice, other than the CPU timing, the applied inversion method is irrelevant. This research-oriented article discusses one such process, the Cayley-Hamilton (C.H.) [1]. Pursuing the process symbolically reveals its unpublished hidden mathematical characteristics even in the original article [1]. This article expands the general vision of the original named method without altering its practical applications. We have used the famous CAS Mathematica [2]. We have briefed the theory behind the method and applied it to different-sized symbolic and numeric matrices. The results are compared to the named CAS’s sealed, packaged library commands. The codes are given, and the algorithms are unsealed.

Diagonally loaded SMI algorithm based on inverse matrix recursion

The derivation of a diagonally loaded sample-matrix inversion （LSMI） algorithm on the busis of inverse matrix recursion （i.e.LSMI-IMR algorithm） is conducted by reconstructing the recursive formulation of covariance matrix. For the new algorithm, diagonal loading is by setting initial inverse matrix without any addition of computation. In addition, a corresponding improved recursive algorithm is presented, which is low computational complexity. This eliminates the complex multiplications of the scalar coefficient and updating matrix, resulting in significant computational savings. Simulations show that the LSMI-IMR algorithm is valid.

Precise integration method without inverse matrix calculation for structural dynamic equations

The precise integration method proposed for linear time-invariant homogeneous dynamic systems can provide accurate numerical results that approach an exact solution at integration points. However, difficulties arise when the algorithm is used for non-homogeneous dynamic systems due to the inverse matrix calculation required. In this paper, the structural dynamic equalibrium equations are converted into a special form, the inverse matrix calculation is replaced by the Crout decomposition method to solve the dynamic equilibrium equations, and the precise integration method without the inverse matrix calculation is obtained. The new algorithm enhances the present precise integration method by improving both the computational accuracy and efficiency. Two numerical examples are given to demonstrate the validity and efficiency of the proposed algorithm.

New recursive algorithm for matrix inversion

To reduce the computational complexity of matrix inversion, which is the majority of processing in many practical applications, two numerically efficient recursive algorithms （called algorithms I and II, respectively） are presented. Algorithm I is used to calculate the inverse of such a matrix, whose leading principal minors are all nonzero. Algorithm II, whereby, the inverse of an arbitrary nonsingular matrix can be evaluated is derived via improving the algorithm I. The implementation, for algorithm II or I, involves matrix-vector multiplications and vector outer products. These operations are computationally fast and highly parallelizable. MATLAB simulations show that both recursive algorithms are valid.

The General and Centro(Kewsy) Symmetric Solutions to a System of Matrix Equations over an Arbitrary Skew Field

Necessary and sufficient conditions are given for the existence of the general solution, the centrosymmetric solution, and the centroskewsymmetric solution to a system of linear matrix equations over an arbitrary skew field. The representations of such the solutions of the system are also derived.

A MATRIX EQUATION FROM AN INVERSE PROBLEM OF VIBRATION THEORY

The symmetric,positive semidefinite,and positive definite real solutions of the matrix equation XA=YAD from an inverse problem of vibration theory are considered.When D=T the necessary and sufficient conditions for the existence of such solutions and their general forms are derived.

Sign Patterns That Allow the Given Matrix

Let P be a property referring to a real matrix. For a sign pattern A, if there exists a real matrix B in the qualitative class of A such that B has property P, then we say A allows P. Three cases that A allows an M matrix, an inverse M matrix and a P 0 matrix are considered. The complete characterizations are obtained.

Inverting a k-heptadiagonal matrix based on Doolitle LU factorization

The purpose of the present paper is to show a new numeric and symbolic algorithm for inverting a general nonsingular k-heptadiagonal matrix.This work is based on Doolitle LU factorization of the matrix.We obtain a series of recursive relationships then we use them for constructing a novel algorithm for inverting a k-heptadiagonal matrix.The computational cost of the algorithm is calculated.Some illustrative examples are given to demonstrate the effectiveness of the proposed method.

Generalized bipositive semidefinite solutions to a system of matrix equations

In this paper, a system of complex matrix equations was studied. Necessary and sufficient conditions for the existence and the expression of generalized bipositive semidefinite solution to the system were given. In addition, a criterion for a matrix to be generalized bipositive semidefinite was determined.

UHAS-MIDA Software Package: Mass Isotopomer Distribution Analysis-Applying the Inverse Matrix for the Determination of Stable Isotopes of Chromium Distribution during Red Blood Cell Labelling

Clinical assessment of fluid volume status in children during malaria can be taxing and often inaccurate. During malaria, changes in fluid volume are rather multifarious and estimating this parameter, especially in sick children is very challenging for clinicians who frequently rely on indices such as long capillary refill times, tachycardia, central venous pressure and decreased urine volume as guides. Here, we present the UHAS-MIDA, an open-source software tool that calculates the red blood cell (RBC) concentration and blood volume during malaria in children determined using a stable isotope of chromium (53Cr as the label) by gas chromatography-mass spectrometry in selective ion monitoring (GC/MS-SIM) analysis. A key component involves the determination of the compositions of the most abundant naturally occurring isotopes of Cr (50Cr, 52Cr, 53Cr), and converting the proportions into a 3 × 3 matrix. To estimate unknown proportions of chromium isotopic mixtures from the measured abundances of three ions, an inverse matrix was calculated. The inverse together with several inputs is then used to calculate the corrected MS ion abundances. Thus, we constructed the software tool UHAS- MIDA using HTML, CSS/Bootstrap, JavaScript, and PHP scripting languages. The tool enables the user to efficiently determine RBC concentration and fluid volume. The source code, binary packages and associated materials for UHAS-MIDA are freely available at https://github.com/bentil078/Abaye-et-al_UHASmida

Efficient matrix inversion based on VLIW architecture

Matrix inversion is a critical part in communication, signal processing and electromagnetic system. A flexible and scalable very long instruction word （VLIW） processor with clustered architecture is proposed for matrix inversion. A global register file （RF） is used to connect al the clusters. Two nearby clusters share a local register file. The instruction sets are also designed for the VLIW processor. Experimental results show that the proposed VLIW architecture takes only 45 latency to invert a 4 × 4 matrix when running at 150 MHz. The proposed design is roughly five times faster than the DSP solution in processing speed.

GENERALIZED INVERSE RATIONAL EXTRAPOLATION METHODS FOR MATRIX SEQUENCES

Assume that a convergent matrix sequence{An}:An→A（n→∞）, An,A∈C3×3.We want to form a new matrix sequence {Hn}, derived from {An}, which has also A aslimit and whose convergence is faster than the of {An}. Three rational extrapolation meth-ods for accelerating the convergence of matrix sequences {An} are presented in this paper.The underlying methods are based on the generalized inverse for matrices which is

A two-stage frequency-domain blind source separation method for underdetermined instantaneous mixtures

In order to decrease the probability of missing some data points or noises being added in the inverse truncated mixing matrix （ITMM） algorithm, a two-stage frequency- domain method is proposed for blind source separation of underdetermined instantaneous mixtures. The separation process is decomposed into two steps of ITMM and matrix completion in the view that there are many soft-sparse （not very sparse） sources. First, the mixing matrix is estimated and the sources are recovered by the traditional ITMM algorithm in the frequency domain. Then, in order to retrieve the missing data and remove noises, the matrix completion technique is applied to each preliminary estimated source by the traditional ITMM algorithm in the frequency domain. Simulations show that, compared with the traditional ITMM algorithms, the proposed two-stage algorithm has better separation performances. In addition, the time consumption problem is considered. The proposed algorithm outperforms the traditional ITMM algorithm at a cost of no more than one- fourth extra time consumption.

An Improved BP Algorithm and Its Application in Classification of Surface Defects of Steel Plate

Artificial neural network is a new approach to pattern recognition and classification. The model of multilayer perceptron （MLP） and back-propagation （BP） is used to train the algorithm in the artificial neural network. An improved fast algorithm of the BP network was presented, which adopts a singular value decomposition （SVD） and a generalized inverse matrix. It not only increases the speed of network learning but also achieves a satisfying precision. The simulation and experiment results show the effect of improvement of BP algorithm on the classification of the surface defects of steel plate.

Approximate Iteration Detection and Precoding in Massive MIMO

Massive multiple-input multiple-output provides improved energy efficiency and spectral efficiency in 5 G. However it requires large-scale matrix computation with tremendous complexity, especially for data detection and precoding. Recently, many detection and precoding methods were proposed using approximate iteration methods, which meet the demand of precision with low complexity. In this paper, we compare these approximate iteration methods in precision and complexity, and then improve these methods with iteration refinement at the cost of little complexity and no extra hardware resource. By derivation, our proposal is a combination of three approximate iteration methods in essence and provides remarkable precision improvement on desired vectors. The results show that our proposal provides 27%-83% normalized mean-squared error improvement of the detection symbol vector and precoding symbol vector. Moreover, we find the bit-error rate is mainly controlled by soft-input soft-output Viterbi decoding when using approximate iteration methods. Further, only considering the effect on soft-input soft-output Viterbi decoding, the simulation results show that using a rough estimation for the filter matrix of minimum mean square error detection to calculating log-likelihood ratio could provideenough good bit-error rate performance, especially when the ratio of base station antennas number and the users number is not too large.

Weighted Gauss-Seidel Precoder for Downlink Massive MIMO Systems

In this paper,a novel precoding scheme based on the Gauss-Seidel(GS)method is proposed for downlink massive multiple-input multiple-output(MIMO)systems.The GS method iteratively approximates the matrix inversion and reduces the overall complexity of the precoding process.In addition,the GS method shows a fast convergence rate to the Zero-forcing(ZF)method that requires an exact invertible matrix.However,to satisfy demanded error performance and converge to the error performance of the ZF method in the practical condition such as spatially correlated channels,more iterations are necessary for the GS method and increase the overall complexity.For efficient approximation with fewer iterations,this paper proposes a weighted GS(WGS)method to improve the approximation accuracy of the GS method.The optimal weights that accelerate the convergence rate by improved accuracy are computed by the least square(LS)method.After the computation of weights,the different weights are applied for each iteration of the GS method.In addition,an efficient method of weight computation is proposed to reduce the complexity of the LS method.The simulation results show that bit error rate(BER)performance for the proposed scheme with fewer iterations is better than the GS method in spatially correlated channels.

The observation and interpretation of geomagnetic variations in a small area of Hainan Province,China

In late December of 1991, observation of three component geomagnetic short period variations was carried out in Chengmai county of Hainan province, along a short measurement profile with small spatial intervals of detection points. Within the period range of 20 s to 500 s, we have discovered that the vertical variations are basically correlated with the horizontal variations in north direction in all measurement points, the real parts of complex transfer functions demonstrate that an asymmetric spatial variation of short wavelength superimposed on regional monotonic tendencious spatial variation. Inversion of the observed data along the small profile was carried out by using the invention method of generalized inverse matrix of finite element forward calculation for the electromagnetic induction of the 2 D conductivity model to investigate the conductivity structure of the profile. We conclude that the anomaly of geomagnetic short periodic variation in the small area originated from the Wangwu Wenjiao fault in the northern part of Hainan island .

THE ANALYTICAL SOLUTION WITH RESPECT TO CHARACTERISTICS OF ELEMENTS' CROSS-SECTION AS VARIABLES OF THE PLANE FRAME

Taking the sectional area and the bending moment of inertia as variables for each beam element, the plane frame will possess a stiffness matrix containing parameters.In terms of the symbolic computation software, the inverse matrix is solved to obtain the new analytical solution with respect ic characteristics of elements cross-section.The general program is coded in the microcomputer and corresponding exmpales are computed.

Efficient Gauss-Seidel Precoding with Parallel Calculation in Massive MIMO Systems

A number of requirements for 5G mobile communication are satisfied by adopting multiple input multiple output(MIMO)systems.The inter user interference(IUI)which is an inevitable problem in MIMO systems becomes controllable when the precoding scheme is used.In this paper,the horizontal Gauss-Seidel(HGS)method is proposed as precoding scheme in massive MIMO systems.In massive MIMO systems,the exact inversion of channel matrix is impractical due to the severe computational complexity.Therefore,the conventionalGauss-Seidel(GS)method is used to approximate the inversion of channel matrix.The GS has good performance by using previous calculation results as feedback.However,the required time for obtaining the precoding symbols is too long due to the sequential process of GS.Therefore,the HGS with parallel calculation is proposed in this paper to reduce the required time.The rows of channel matrix are eliminated for parallel calculation inHGSmethod.In addition,HGSuses the ordered channelmatrix to prevent performance degradation which is occurred by parallel calculation.The HGS with proper number of parallelly computed symbols has better performance and reduced required time compared to the traditional GS.

The Ground and Excitation Levels of the Trimer

The ground and excited levels of the trimer (4He)3 are obtained theoretically by solving the 2-D Faddeev integral equation in the momentum space. In solving the equation, the direct matrix inversion method (I. M.) is used instead of the iteration and Pade approximate method. The new HFD-B Aziz potential ν is used in the calculation.