Iterative methods to solve the constrained Sylvester equation

In this paper, the multiple constraint least squares solution of the Sylvester equation $AX+XB = C$ is discussed. The necessary and sufficient conditions for the existence of solutions to the considered matrix equation are given. Noting that the alternating direction method of multipliers (ADMM) is a one-step iterative method, a multi-step alternating direction method of multipliers (MSADMM) to solve the considered matrix equation is proposed and some convergence results of the proposed algorithm are proved. Problems that should be studied in the near future are listed. Numerical comparisons between MSADMM, ADMM and ADMM with Anderson acceleration (ACADMM) are included

Arbitrary generalized trapezoidal fully fuzzy sylvester matrix equation and its special and general cases

Many real problems in control systems are related to the solvability of the generalized Sylvester matrix equation either using analytical or numerical methods. However, in many applications, the classical generalized Sylvester matrix equation are not well equipped to handle uncertainty in real-life problems such as conflicting requirements during the system process, the distraction of any elements and noise. Thus, crisp number in this matrix equation is replaced by fuzzy numbers and called generalized fully fuzzy Sylvester matrix equation when all parameters are in fuzzy form. The existing fuzzy analytical methods have four main drawbacks, the avoidance of using near-zero fuzzy numbers, the lack of accurate solutions, the limitation of the size of the systems, and the positive sign restriction of the fuzzy matrix coefficients and fuzzy solutions. Meanwhile, the convergence, feasibility, existence and uniqueness of the fuzzy solution are not examined in many fuzzy numerical methods. In addition, many studies are limited to positive fuzzy systems only due to the limitation of fuzzy arithmetic operation, especially for multiplication between trapezoidal fuzzy numbers.Therefore, this study aims to construct new analytical and numerical methods, namely fuzzy matrix vectorization, fuzzy absolute value, fuzzy Bartle’s Stewart, fuzzy gradient iterative and fuzzy least-squares iterative for solving arbitrary generalized Sylvester matrix equation for special cases and couple Sylvester matrix equations. In constructing these methods, new fuzzy arithmetic multiplication operators for trapezoidal fuzzy numbers are developed. The constructed methods overcome the positive restriction by allowing the negative, near-zero fuzzy numbers as the coefficients and fuzzy solutions. The necessary and sufficient conditions for the existence, uniqueness, and convergence of the fuzzy solutions are discussed, and a complete analysis of the fuzzy solution is provided. Some numerical examples and the verification of the solutions are presented to demonstrate the constructed methods. As a result, the constructed methods have successfully demonstrated the solutions for the arbitrary generalized Sylvester matrix equation for special and general cases based on the new fuzzy arithmetic operations, with minimum complexity fuzzy operations. The constructed methods are applicable to either square or non-square coefficient matrices up to 100 × 100. In conclusion, the constructed methods have significant contribution to the application of control system theory without any restriction on the system

PREDICTION OF BREEDING VALUES FOR UNMEASURED TRAITS FROM MEASURED TRAITS

Henderson (1977, 1984) described a method for prediction of breeding values for traits not in the model for records. This method may be practical for animal or sire models for the case when no measurements can be obtained on any animals for some traits to be evaluated. The least squares equations are augmented with A-1⊗GN-1 rather than with A-1⊗G0-1 where A is the numerator relationship and G0and GN are the genetic covariance matrices for measured and for all traits. This method can be used for each unmeasured trait or simultaneously for measured and all unmeasured traits. An option in the MTDFRUN module of the Multiple Trait Derivative Free REML (MTDFREML) program of Boldman et al. (1993) is to obtain solutions for breeding values and their prediction error variances. However, the preparation program (MTDFPREP) must be tricked to set-up equation numbers for breeding values of the unmeasured traits. Adding dummy records for the unmeasured traits but with missing records for the measured traits for dummy animals to the data file of animals with measured traits but with missing unmeasurable traits will result in the needed equations. At least two dummy records are needed to avoid a divide by zero error in calculating the sample standard deviation. The dummy records need to be associated with a level of at least one fixed factor. The dummy animals also must be added to the pedigree file with unknown sires and dams before running the program to obtain the inverse of the numerator relationship matrix (MTDFNRM). In the program to obtain solutions to the multiple trait mixed model equations (MTDFRUN), the full genetic (co)vaiiance matrix, GN, for both measured and unmeasured traits is needed. The residual (co)variance matrix must have zero covariances between pairs of measured and unmeasured traits but the variance of the unmeasured trait must not be zero. This procedure provides direct solutions for breeding values of unmeasured traits based on mixed model predictions of breeding values of the measured traits and also allows calculation of standard errors of prediction for the solutions directly from elements of the inverse of the ugmented coefficient matrix. For example, this procedure can be used to predict breeding values of bulls (which have tenderness measurements) for the correlated trait of tenderness as a steer or heifer which cannot be measured on the bull