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New EIV Models for Solving Positive Definite and Positive Semidefinite Linear Systems with Application to Computing Positive Definite Solutions of Nonlinear Matrix Equations

Bagherpour, Negin | 2015

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 47769 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Mahdavi Amiri, Nezamodin
  7. Abstract:
  8. The need to estimate a positive definite or positive semi-definite solution to an overdetermined linear system of equations with multiple right hand side vectors arises in several process control contexts. A similar problem is encountered in certain contexts related to statistics and control theory with the unknown matrix being positive semi-definite. The coefficient and the right hand side matrices are respectively named data and target matrices. A number of optimization methods were proposed for solving such problems, in which the data matrix is unrealistically assumed to be error free. Here, considering error in measured data and target matrices, we present new approachs to solve two mathematical problems: (1) positive definite linear system of equations and (2) positive semi-definite linear system of equations. To solve the first problem, we define a new error function considering errors in both data and target matrices. To minimize the defined error function, we derive necessary and sufficient optimality conditions and outline a direct algorithm to compute the solution. We then outline another algorithm by using spectral decomposition instead of QR decomposition in the algorithm. Next, we study solving the second problem. First, we consider the problem assuming a specific rank for the unknown matrix. We define a new error function containing error in both data and target matrices. To minimize the error, a new technique for optimization problems on the Stiefel manifold is used. We prove the global convergence of our proposed approach. We also initiate the study of the asymptotic rate of convergence for optimization problems on Stiefel manifolds and establish the asymptotic superlinear convergence rate of our proposed approach. To extend our proposed method for solving positive semi-definite linear system of equations over all ranks, we solve the problem presuming all possible ranks to identify a solution with a minimal error. We further adapt the proposed approach to solve two well-known problems in control theory, the minimum rank problem and computing the correlation matrix. Moreover, we make use of our proposed method for solving the first problem to compute a symmetric and positive definite solution to some well-known nonlinear equations. Here, we propose new iterative algorithms for solving three well-known types of nonlinear matrix equations. Making use of an iterative process for inverse of a matrix, we convert the nonlinear matrix equation to an iterative linear one, and, in every iteration, we apply our algorithm for the first problem to solve the linear subproblem and update the newly defined variables and the matrix inverse terms using appropriate formulas. Our proposed algorithms have a number of useful features, one being that the computed unknown matrix remains symmetric and positive definite in all iterations. We finally implement all the proposed algorithms using MATLAB 2014b software environment and execute the programs on a PC with 2Gb RAM and a 3.2 GHz CPU. We report the numerical results for solving some well-known test problems as well as randomly generated test problems. To show the efficiency of our proposed methods, we also present the Dolan-Moré performance profiles. Numerical results confirm the efficiency of our proposed algorithms in computing more accurate solutions faster than the exisiting methods
  9. Keywords:
  10. Linear Systems ; Positive Definite Function ; Positive Semidefinite Solutions ; Error-In-Variable (EIV)Models ; Nonlinear Matrix Equations

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