Browsing by Author "Guo, Chun-Hua"

Now showing 1 - 20 of 27

Open Access
Analysis and Numerical Methods for Algebraic Riccati Equations Associated with Regular M-Matrices
(Faculty of Graduate Studies and Research, University of Regina, 2015-12) Lu, Di; Guo, Chun-Hua; Argerami, Martin; Szechtman, Fernando
The thesis is a further study about algebraic Riccati equations for which the four coe cient matrices form a regular M-matrix K. We prove a property about minimal nonnegative solutions of such an algebraic Riccati equation and its dual equation. And we show that Newton's method, SDA, ADDA are well-de ned and quadratically convergent in non-critical case. Then we prove that ADDA is linearly convergent with rate 1=2 in critical case. As compared to earlier work on solving regular MARE by ADDA, the results we present here are more general. This thesis extends the knowledge of doubling algorithms.
Open Access
Analytical Modeling of Transient Heat Transfer Coupled With Fluid Flow in Heavy Oil Reservoirs During Thermal Recovery Processes
(Faculty of Graduate Studies and Research, University of Regina, 2014-11) Yu, Kuizheng; Zhao, Gang; Gu, Yongan (Peter); Guo, Chun-Hua; Jin, Yee-Chung
Tremendous resources of heavy oil are located in western Canada. Among the many heavy oil recovery methods proposed, thermal recovery methods which employing hot fluid injection or in-situ combustion, have been conventionally utilized to enhance heavy oil recovery. Temperature profiles in heavy oil reservoirs are important factors for making operation and production plans in thermal recovery processes. A significant factor in these processes is to study the heat energy transfer in the oil formations. In previous studies, heat transfer by conduction has been commonly considered as the main mechanism of heat transfer in porous media. However, this assumption is not reasonable for thermal recovery processes with fluid flow such as steam flooding, hot water flooding, steam-assisted gravity drainage (SAGD) and cyclic steam stimulation (CSS). Heat transfer and fluid flow occur simultaneously during thermal recovery processes, and heat transfer by convection cannot be neglected. Fluid flow motivates convective heat transfer, and increase the rate of energy transfer significantly. Heat conduction is dominated by temperature gradient, while heat convection is dominated by pressure gradient. By integrating conduction and convection, temperature and pressure domain can be coupled systematically. In this study, novel heat transfer models, integrating both conduction and convection, have been developed to describe the one-dimensional transient heat transfer coupled with fluid flow. In the models, the properties of the reservoir and the fluid are integrated into two important parameters, i.e., the thermal diffusivity of a reservoir/fluid system and the thermal convection velocity of the fluid. To derive the analytical solutions of these mathematical models, dimensionless variables are defined to reduce the models to the dimensionless form. After that, variable transformation and Laplace transformation are performed to derive the analytical solution in Laplace domain. By using the table of Laplace transformations, the solution in Laplace domain can be converted to dimensionless analytical solution in real time domain. Numerical simulations by COMSOL Multiphysics are conducted to validate the analytical solutions. Subsequently, case studies under both steady and unsteady flow conditions have been conducted. Satisfactory agreements of the results are achieved between analytical solutions and numerical simulation results. To prove the mathematical models could have practical application in the oil and gas industry, results comparison between the analytical solution and CMG simulation is conducted. A numerical simulation model for transient heat transfer in heavy oil reservoirs during the SAGD process was used for comparison. It is found that the shapes of temperature distributions and propagations of the analytical solution and the CMG simulation have the similar trends. The studies showed good agreement between the test results and those from the CMG simulation. The newly developed analytical solutions provide theoretical guidance for temperature transient analysis (TTA) and fluid injection strategies. These analytical solutions can be used to predict temperature profiles in heavy oil reservoirs during thermal recovery processes and improve the accuracy and efficiency of temperature transient analysis.
Open Access
Complex symmetric stabilizing solution of the matrix equation $X+A^{T}X^{-1}A=Q$
(Elsevier, 2011) Guo, Chun-Hua; Kuo, Yueh-Cheng; Lin, Wen-Wei
We study the matrix equation $X+A^{T}X^{-1}A=Q$, where $A$ is a complex square matrix and $Q$ is complex symmetric. Special cases of this equation appear in Green's function calculation in nano research and also in the vibration analysis of fast trains. In those applications, the existence of a unique complex symmetric stabilizing solution has been proved using advanced results on linear operators. The stabilizing solution is the solution of practical interest. In this paper we provide an elementary proof of the existence for the general matrix equation, under an assumption that is satisfied for the two special applications. Moreover, our new approach here reveals that the unique complex symmetric stabilizing solution has a positive definite imaginary part. The unique stabilizing solution can be computed efficiently by the doubling algorithm.
Open Access
Convergence Analysis of the Doubling Algorithm for Several Nonlinear Matrix Equations in the Critical Case
(SIAM, 2009) Chiang, Chun-Yueh; Chu, Eric King-Wah; Guo, Chun-Hua; Huang, Tsung-Ming; Lin, Wen-Wei; Xu, Shu-Fang
In this paper, we review two types of doubling algorithm and some techniques for analyzing them. We then use the techniques to study the doubling algorithm for three different nonlinear matrix equations in the critical case. We show that the convergence of the doubling algorithm is at least linear with rate 1/2. As compared to earlier work on this topic, the results we present here are more general, and the analysis here is much simpler.
Open Access
Convergence and Comparison Theorems for Various Splittings of Matrices Based on Generalized Inverses
(Faculty of Graduate Studies and Research, University of Regina, 2015-03) Agasthian, Vijayaparvathy; Guo, Chun-Hua; Argerami, Martin; Fallat, Shaun; Yang, Boting; Wei, Yi-Min
The convergence of iterative methods for numerically solving the linear systems of equations associated with different types of splittings has been well studied in the literature. In this dissertation, we define new types of splittings based on generalized inverses (Moore-Penrose inverse, Drazin inverse, Group inverse) and present convergence results based on those splittings, namely, Weak Nonnegative Proper Splitting, Proper Regular Splitting, Index Proper Regular Splitting, Weak Nonnegative Index Proper Splitting, Group Proper Regular Splitting, and Weak Nonnegative Group Proper Splitting. We also present some new comparison theorems between the spectral radii of matrices, which are useful in the analysis of the rate of convergence of iterative methods for the different types of splittings of matrices.
Open Access
Convergence rates of some iterative methods for nonsymmetric algebraic Riccati equations arising in transport theory
(Elsevier, 2010) Guo, Chun-Hua; Lin, Wen-Wei
We determine and compare the convergence rates of various fixed-point iterations for finding the minimal positive solution of a class of nonsymmetric algebraic Riccati equations arising in transport theory.
Open Access
A convergence result for matrix Riccati differential equations associated with M-matrices
(2014-04-22) Guo, Chun-Hua; Yu, Bo
The initial value problem for a matrix Riccati differential equation associated with an $M$-matrix is known to have a global solution $X(t)$ on $[0, \infty)$ when $X(0)$ takes values from a suitable set of nonnegative matrices. It is also known, except for the critical case, that as $t$ goes to infinity $X(t)$ converges to the minimal nonnegative solution of the corresponding algebraic Riccati equation. In this paper we present a new approach for proving the convergence, which is based on the doubling procedure and is also valid for the critical case. The approach also provides a way for solving the initial value problem and a new doubling algorithm for computing the minimal nonnegative solution of the algebraic Riccati equation.
Open Access
Detecting and solving hyperbolic quadratic eigenvalue problems
(SIAM, 2009) Guo, Chun-Hua; Higham, Nicholas J.; Tisseur, Francoise
Hyperbolic quadratic matrix polynomials $Q(\lambda) = \lambda^2 A + \lambda B + C$ are an important class of Hermitian matrix polynomials with real eigenvalues, among which the overdamped quadratics are those with nonpositive eigenvalues. Neither the definition of overdamped nor any of the standard characterizations provides an efficient way to test if a given $Q$ has this property. We show that a quadratically convergent matrix iteration based on cyclic reduction, previously studied by Guo and Lancaster, provides necessary and sufficient conditions for $Q$ to be overdamped. For weakly overdamped $Q$ the iteration is shown to be generically linearly convergent with constant at worst 1/2, which implies that the convergence of the iteration is reasonably fast in almost all cases of practical interest. We show that the matrix iteration can be implemented in such a way that when overdamping is detected a scalar $\mu < 0$ is provided that lies in the gap between the $n$ largest and $n$ smallest eigenvalues of the $n \times n$ quadratic eigenvalue problem (QEP) $Q(\lambda)x = 0$. Once such a $\mu$ is known, the QEP can be solved by linearizing to a definite pencil that can be reduced, using already available Cholesky factorizations, to a standard Hermitian eigenproblem. By incorporating an initial preprocessing stage that shifts a hyperbolic $Q$ so that it is overdamped, we obtain an efficient algorithm that identifies and solves a hyperbolic or overdamped QEP maintaining symmetry throughout and guaranteeing real computed eigenvalues.
Open Access
Enhanced Solvent Vapour Extraction processes in Thin Heavy Oil Reservoirs
(Faculty of Graduate Studies and Research, University of Regina, 2014-01) Jia, Xinfeng; Zeng, Fanhua; Gu, Yongan; Jin, Yee-Chung; Guo, Chun-Hua; Shirif, Ezeddin; Torabi, Farshid; Chen, Zhangxing
Solvent-based techniques, such as solvent vapour extraction (VAPEX) and cyclic solvent injection (CSI), have emerged as promising processes to enhance heavy oil recovery. However, there are still a number of technical issues with these processes, such as the theoretical modeling and performance enhancement. This thesis aims at addressing the following major technical topics. Theoretical modeling of VAPEX. Heavy oil−solvent transition zone is where the VAPEX heavy oil recovery occurs. Existing analytical VAPEX models can neither fully characterize the transition zone nor accurately predict its growth. Numerical simulation models use grid sizes that are much larger than the transition-zone thickness (~1 cm) and thus cannot capture the characteristics of the transition zone. This study develops a new two-dimensional (2D) mathematical model for the VAPEX process on the basis of its major oil recovery mechanisms (i.e., solvent dissolution and gravity drainage) inside the transition zone. This VAPEX model is able not only to accurately describe the distributions of solvent concentration, oil drainage velocity, and diffusion coefficient across the transition zone, but also to predict the evolution of the solvent chamber. Theoretical modeling of the diffusionconvection mass transfer in CSI. CSI is a solvent huff-n-puff process. One of the differences between CSI and VAPEX is that the operating pressure is decreased and increased cyclically in CSI. Hence, in addition to molecular diffusion, CSI has another mass transfer mechanism, convection, which is attributed to the bulk motion of solvent caused by the pressure gradient between the solvent chamber and untouched heavy oil zone. This study develops a convection−diffusion mass-transfer model for the heavy oil−solvent mixing process of CSI. The diffusion coefficient and convection velocity are both considered as variables rather than constants. Results qualitatively show that pressure gradient can greatly enhance the mixing process. Enhancement of VAPEX and CSI. This study proposes a new process, namely foamy oil-assisted vapour extraction (F-VAPEX) to enhance the VAPEX performance. F-VAPEX combines merits of VAPEX (continuous production) and CSI (strong driving force) together. It is essentially a VAPEX process during which the operating pressure is cyclically reduced and restored. It is found that the foamy oil flow during the pressure reduction period can effectively move the partially diluted heavy oil toward the producer. Results show that F-VAPEX can increase both the average oil production rate and the ultimate oil recovery of VAPEX. In comparison with CSI, F-VAPEX has a higher oil production rate and a lower solvent−oil ratio. This thesis also proposes a new process to enhance the performance of CSI, namely gasflooding-assisted cyclic solvent injection (GA-CSI). GA-CSI uses dedicated solvent injector and oil producer to prevent the ‘back-and-forth movement’ of foamy oil inside the solvent chamber during the conventional CSI process. GA-CSI applies a gasflooding slug immediately after the pressure depletion process of CSI to produce the partially diluted foamy oil left in the solvent chamber. It is found that the motionless foamy oil due to pressure depletion and solvent liberation serves as a buffer zone, which effectively reduces the mobility ratio between the displacing solvent and the displaced oil and leads to a high sweeping efficiency. In comparison with the conventional CSI process, the GA-CSI process can increase the oil production rate by over 3 times and in the meantime decrease the solvent−oil ratio from ~4 to ~3 g solvent/g oil.
Open Access
Further Study of Some Iterative Methods for the Matrix Pth Root
(Faculty of Graduate Studies and Research, University of Regina, 2022-01) Lu, Di; Guo, Chun-Hua; Argerami, Martin; Szechtman, Fernando; Yao, JingTao; Poloni, Federico
In this thesis, we solve some problems arising in the study of some iterative methods for finding the principal pth root of a matrix. Our main interest is in Newton's method, Halley's method and Chebyshev's method. We also study Schroder's method (which includes Newton's method and Chebyshev's method as special cases). The study of these methods for the matrix case can be reduced to the study in the scalar case. Some theoretical properties of the iterative methods can be obtained by examining the Taylor series expansion of the iterates. It has been observed by Guo that the Taylor coefficients have a favorable sign pattern for Newton's method and Halley's method, and that some nice theoretical results can be proved after the conjectured sign pattern is confirmed. In this thesis, we are going to prove this conjecture raised by Guo. The key idea of the proof is to establish a monotonicity property for the coefficients as the iteration progresses. Using this idea, we prove the sign pattern of Newton's method and Halley's method without too much difficulty. We then move on to prove a similar sign pattern for Schroder's method. The proof is much more complicated, but the basic strategy remains the same. After we have confirmed the sign pattern of Taylor coefficients for Newton's method, Halley's method and Schroder's method, we present some related results for the matrix case. In this thesis, we also establish a convergence region for Newton's method that is much larger than previously available ones.
Open Access
An Improved Arc Algorithm for Detecting Definite Hermitian Pairs
(SIAM, 2009) Guo, Chun-Hua; Higham, Nicholas J.; Tisseur, Francoise
A 25-year old and somewhat neglected algorithm of Crawford and Moon attempts to determine whether a given Hermitian matrix pair $(A,B)$ is definite by exploring the range of the function $f(x) = x^*(A + iB)x/|x^*(A + iB)x|$, which is a subset of the unit circle. We revisit the algorithm and show that with suitable modifications and careful attention to implementation details it provides a reliable and efficient means of testing definiteness. A clearer derivation of the basic algorithm is given that emphasizes an arc expansion viewpoint and makes no assumptions about the definiteness of the pair. Convergence of the algorithm is proved for all $(A,B)$, definite or not. It is shown that proper handling of three details of the algorithm is crucial to the efficiency and reliability: how the midpoint of an arc is computed, whether shrinkage of an arc is permitted, and how directions of negative curvature are computed. For the latter, several variants of Cholesky factorization with complete pivoting are explored and the benefits of pivoting demonstrated. The overall cost of our improved algorithm is typically just a few Cholesky factorizations. Applications of the algorithm are described to testing the hyperbolicity of a Hermitian quadratic matrix polynomial, constructing conjugate gradient methods for sparse linear systems in saddle point form, and computing the Crawford number of the pair $(A,B)$ via a quasiconvex univariate minimization problem.
Open Access
Integrating PVT Properties for the Description of Well Responses in Gas Condensate Reservoirs
(Faculty of Graduate Studies and Research, University of Regina, 2015-06) Li, Jiawei; Zhao, Gang; Yang, Daoyong; Jin, Yee-Chung; Guo, Chun-Hua
A gas condensate reservoir exhibits complex behaviors when the bottomhole pressure falls below the dew point pressure at a given reservoir temperature. When the condensate oil begins to drop out from the gas, a two-phase fluid system develops and a bank of condensate oil builds up, inducing severe productivity losses. While the production rate is constant, different mobility zones are formed around the wellbore corresponding respectively to the original-gas-in-place (OGIP) away from the well, the condensate bank with only gas flow, and two-phase gas and oil flow near the wellbore. Thus, the behaviors of gas condensate systems are complex and difficult to interpret. In this thesis, a single well model is built to evaluate the dynamic performance of an infinite and homogeneous gas condensate reservoir. Firstly, apparent compressibility is defined by integrating PVT properties. The application of modified pseudo-pressure and pseudo-time linearizes the partial differential equations with the non-linearity caused by gas properties. Secondly, a three-region method accounts for the composition changes in the reservoir. Fluid flow towards the well during depletion can be divided into three concentric main flow regions, from the wellbore to the reservoir. An analytical model could have been built directly from the three-region method. Thirdly, on the basis of the three-region method, the semi-analytical model is developed by dividing the whole reservoir into multiple sub-radial regions. In the modeling process, the discretized subradial regions are hydraulically coupled with nearby sub-radial regions so that an ultimate linearized system is generated to obtain bottomhole pressure responses. Finally, a moving boundary is also taken into consideration to investigate the difference between a consistent boundary model and a moving boundary model. All models have been validated and can be successfully used to analyze pressure and production data of gas condensate production wells. This thesis has contributed to production from gas condensate reservoirs with detailed studies on the inherent PVT properties, condensate banks, and the interference of adjacent regions. The modeling results provide reliable perspectives of transient pressure analysis in gas condensate reservoirs and help characterize and estimate the drainage areas of the three regions mentioned above, which is critical in gas condensate reservoir development. Furthermore, this model builds a consolidated foundation for further investigation of reservoir heterogeneity in the development of unconventional reservoirs.
Open Access
Iterative methods for a linearly perturbed algebraic matrix Riccati equation arising in stochastic control
(Taylor & Francis, 2013) Guo, Chun-Hua
We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problem for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods, and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.
Open Access
Lower Bounds and Algorithms for Searching Networks
(Faculty of Graduate Studies and Research, University of Regina, 2019-10) Xue, Yuan; Yang, Boting; Zilles, Sandra; Yang, Xue-Dong; Mouhoub, Malek; Guo, Chun-Hua; MacGillivray, Gary
Research on graph searching has recently gained interest in computer science, mathematics, and physics. This thesis provides new results on two graph search models, namely fast searching and the zero-visibility cops and robber model. Given a graph that contains an invisible fugitive, the fast searching problem is to find the fast search number, i.e., the minimum number of searchers to capture the fugitive in the fast search model. This model was first introduced by Dyer, Yang and Ya¸sar in 2008. Although the literature provides a number of results on fast searching, many properties of the fast search number have not yet been revealed. In this thesis, we give new lower bounds on the fast search number. Using the new lower bounds, we prove an explicit formula for the fast search number of the cartesian product of an Eulerian graph and a path. We also give formulas for the fast search number of variants of the cartesian product. We present an upper bound on the fast search number of hypercubes, and extend the results to a broader class of graphs including toroidal grids. In addition, we examine the complete k-partite graphs and provide lower bounds and upper bounds on their fast search number. We also investigate some special classes of complete k-partite graphs, such as complete bipartite graphs and complete split graphs. We solve the open problem of determining the fast search number of complete bipartite graphs, and present upper and lower bounds on the fast search number of complete split graphs. We also introduce the notion of k-combinable graphs, and propose an efficient method for computing the fast search number of such graphs. The zero-visibility cops and robber game is a variant of Cops and Robbers subject to the constraint that the cops have no information at any time about the location of the robber. We first study a partition problem in which for a given graph and an integer k, we want to find a partition of the vertex set such that the size of the boundary of the smaller subset in the partition is at most k while the size of this subset is as large as possible under some conditions. Then we apply such partitions to prove lower bounds on the zero-visibility cop number of graph products. We also investigate the monotonic zero-visibility cop number of graph products. In addition, we prove lower bounds on the zerovisibility cop number for various classes of graphs. In particular, we give lower bounds on the zero-visibility cop number for graph joins and lexicographic products of graphs.
Open Access
The matrix equation $X+A^TX^{-1}A=Q$ and its application in nano research
(SIAM, 2010) Guo, Chun-Hua; Lin, Wen-Wei
The matrix equation $X+A^TX^{-1}A=Q$ has been studied extensively when $A$ and $Q$ are real square matrices and $Q$ is symmetric positive definite. The equation has positive definite solutions under suitable conditions, and in that case the solution of interest is the maximal positive definite solution. The same matrix equation plays an important role in Green's function calculations in nano research, but the matrix $Q$ there is usually indefinite (so the matrix equation has no positive definite solutions) and one is interested in the case where the matrix equation has no positive definite solutions even when $Q$ is positive definite. The solution of interest in this nano application is a special weakly stabilizing complex symmetric solution. In this paper we show how a doubling algorithm can be used to find good approximations to the desired solution efficiently and reliably.
Open Access
Modeling The Fluid Flow in Low-Permeability Unconventional Reservoirs Across Scales
(Faculty of Graduate Studies and Research, University of Regina, 2019-05) Yao, Shanshan; Zeng, Fanhua; Torabi, Farshid; Gu, Yongan (Peter); Guo, Chun-Hua; Zeng, Hongbo
In the last decade low-permeability unconventional reservoirs (i.e., shale and tight formations) become an increasingly important source of oil and especially gas supply in the world. Low permeability reservoirs are characterized with small grain sizes (m), low permeability (< 0.1mD), small porosity (<10%) and high total organic carbon (TOC) (0.8-20wt%). The productivity of shale and tight reservoirs heavily depends on the interaction between reservoir rock matrix and multi-stage fractured horizontal wells (MFHWs). To predict and optimize unconventional reservoirs’ production behavior, this study tries to model the fluid flow across scales-from the pore scale to the reservoir- scale. Shale matrix permeability is important in interpreting permeability measurement experiments as well as modeling the reservoir-scale flow in shale reservoirs. This study utilizes 2D SEM images and the process-based modeling approach to reconstruct 3D multi-scale shale pore networks. When compared with pore models in the literature, the pore network model is advantaged in describing a realistic, wide range of pore size distribution from micrometer (m) to several nanometers (nm) in a sub-millimeter-sized rock volume. The porescale no-slip flow modeling on pore networks provides intrinsic matrix permeabilities under the effect of multi-scale pore structures and different geological-forming processes. The intrinsic matrix permeability cannot fully represent the gas transport capability of an unconventional reservoir rock when the gas flow velocity at pore surfaces is no longer zero. Unified models are developed for the rarefied gas flow in single conduits of various cross-sections at elevated pressure. Apparent permeabilities are calculated with running unified models on all throats of pore networks. The relationship among pore space structures, gas pressure and apparent permeability reveals the limitation of Klinkenberg equation in describing the high-pressure rarefied gas flow in shale matrix. This study further develops a new equation of apparent permeability vs. pore pressure. Hydrocarbon flows out of rock matrix and then flows into hydraulic fractures then to the horizontal wellbore. Models of coupled flow in matrix and hydraulic fractures can be applied to interpret and/or predict the flow rates/pressure at wellbore vs. time. Distinguished from most models in the literature, this study develops a semi-analytical model with considering the dynamic declining rates of hydraulic fracture conductivity vs. increasing effective stress. This study validates that ignoring such fracture stress-sensitivity can underestimate MFHWs’ productivity at late-time stage. Many low-permeability unconventional reservoirs have a mixture of various conditions, such as rarefied flow, fracture and matrix stress-sensitivity, reservoir heterogeneity and gas adsorption/desorption. In order to easily model multiple flow phenomena, this work develops a composite methodology that combines simple linear flow, radial flow and/or source/sink flow equations. One of this composite method’s applications is validated by the fast and accurate composite modeling of the fluid flow in heterogeneous unconventional reservoirs. In the future work, the composite methodology will be applied in the modeling of gas adsorption/desorption and the rarefied flow in stress-sensitive reservoirs.
Open Access
Monotone convergence of Newton-like methods for M-matrix algebraic Riccati equations
(Springer, 2013) Guo, Chun-Hua
For the algebraic Riccati equation whose four coefficient matrices form a nonsingular $M$-matrix or an irreducible singular $M$-matrix $K$, the minimal nonnegative solution can be found by Newton's method and the doubling algorithm. When the two diagonal blocks of the matrix $K$ have both large and small diagonal entries, the doubling algorithm often requires many more iterations than Newton's method. In those cases, Newton's method may be more efficient than the doubling algorithm. This has motivated us to study Newton-like methods that have higher-order convergence and are not much more expensive each iteration. We find that the Chebyshev method of order three and a two-step modified Chebyshev method of order four can be more efficient than Newton's method. For the Riccati equation, these two Newton-like methods are actually special cases of the Newton-Shamanskii method. We show that, starting with zero initial guess or some other suitable initial guess, the sequence generated by the Newton--Shamanskii method converges monotonically to the minimal nonnegative solution. We also explain that the Newton-like methods can be used to great advantage when solving some Riccati equations involving a parameter.
Open Access
A note on the fixed-point iteration for the matrix equations $X\pm A^*X^{-1}A=I$
(Elsevier, 2008) Fital, Sandra; Guo, Chun-Hua
The fixed-point iteration is a simple method for finding the maximal Hermitian positive definite solutions of the matrix equations $X\pm A^*X^{-1}A=I$ (the plus/minus equations). The convergence of this method may be very slow if the initial matrix is not chosen carefully. A strategy for choosing better initial matrices has been recently proposed by Ivanov, Hasanov and Uhlig. They proved that this strategy can improve the convergence in general and observed from numerical experiments that dramatic improvement happens for the plus equation with some matrices $A$. It turns out that the matrices $A$ are normal for those examples. In this note we prove a result that explains the dramatic improvement in convergence for normal (and thus nearly normal) matrices for the plus equation. A similar result is also proved for the minus equation.
Open Access
Numerical solution of nonlinear matrix equations arising from Green's function calculations in nano research
(Elsevier, 2012) Guo, Chun-Hua; Kuo, Yueh-Cheng; Lin, Wen-Wei
The Green's function approach for treating quantum transport in nano devices requires the solution of nonlinear matrix equations of the form $X+(C^*+{\rm i} \eta D^*)X^{-1}(C+{\rm i} \eta D)=R+{\rm i}\eta P$, where $R$ and $P$ are Hermitian, $P+\lambda D^*+\lambda^{-1} D$ is positive definite for all $\lambda$ on the unit circle, and $\eta \to 0^+$. For each fixed $\eta>0$, we show that the required solution is the unique stabilizing solution $X_{\eta}$. Then $X_*=\lim_{\eta\to 0^+} X_{\eta}$ is a particular weakly stabilizing solution of the matrix equation $X+C^*X^{-1}C=R$. In nano applications, the matrices $R$ and $C$ are dependent on a parameter, which is the system energy $\mathcal E$. In practice one is mainly interested in those values of $\mathcal E$ for which the equation $X+C^*X^{-1}C=R$ has no stabilizing solutions or, equivalently, the quadratic matrix polynomial $P(\lambda)=\lambda^2 C^*-\lambda R+ C$ has eigenvalues on the unit circle. We point out that a doubling algorithm can be used to compute $X_{\eta}$ efficiently even for very small values of $\eta$, thus providing good approximations to $X_*$. We also explain how the solution $X_*$ can be computed directly using subspace methods such as the QZ algorithm by determining which unimodular eigenvalues of $P(\lambda)$ should be included in the computation. In some applications the matrices $C, D, R, P$ have very special sparsity structures. We show how these special structures can be expoited to drastically reduce the complexity of the doubling algorithm for computing $X_{\eta}$.
Open Access
On a nonlinear matrix equation arising in nano research
(SIAM, 2012) Guo, Chun-Hua; Kuo, Yueh-Cheng; Lin, Wen-Wei
The matrix equation $X+A^{T}X^{-1}A=Q$ arises in Green's function calculations in nano research, where $A$ is a real square matrix and $Q$ is a real symmetric matrix dependent on a parameter and is usually indefinite. In practice one is mainly interested in those values of the parameter for which the matrix equation has no stabilizing solutions. The solution of interest in this case is a special weakly stabilizing complex symmetric solution $X_*$, which is the limit of the unique stabilizing solution $X_{\eta}$ of the perturbed equation $X+A^{T}X^{-1}A=Q+i\eta I$, as $\eta\to 0^+$. It has been shown that a doubling algorithm can be used to compute $X_{\eta}$ efficiently even for very small values of $\eta$, thus providing good approximations to $X_*$. It has been observed by nano scientists that a modified fixed-point method can sometimes be quite useful, particularly for computing $X_{\eta}$ for many different values of the parameter. We provide a rigorous analysis of this modified fixed-point method and its variant, and of their generalizations. We also show that the imaginary part $X_I$ of the matrix $X_*$ is positive semi-definite and determine the rank of $X_I$ in terms of the number of unimodular eigenvalues of the quadratic pencil $\lambda^2 A^{T}-\lambda Q+A$. Finally we present a new structure-preserving algorithm that is applied directly on the equation $X+A^{T}X^{-1}A=Q$. In doing so, we work with real arithmetic most of the time.