This thesis research has attempted to address two challenging problems in winter road maintenance, namely road surface condition (RSC) estimation and forecasting. For RSC estimation, the goal of the research was to develop models to discriminate RSC classes based on continuous friction measurements (CFM) and other available data such as temperature and precipitation history. A systematic exploratory study was conducted on an extensive field data set to identify the categorical relationship between RSC and various aggregate CFM measures, such as those related to probability distribution and spatial correlation. A new multi-level model structure was designed, under which binary logistic regression models were calibrated and validated utilizing several carefully chosen aggregate measures to classify major RSC types. This model structure was found to be effective in capturing the overlapping nature of CFM ranges over different RSC types -- a problem which has not been addressed adequately in the past studies. An alternative model with support vector machine (SVM) was also developed for benchmarking the performance of the proposed logit model. It was found that the two types of models are comparative in performance, confirming the high performance of the proposed multi-level model. For road surface condition forecasting, a novel conceptual framework for short-term road surface condition forecasting is proposed, under which the short-term changing process of surface temperature, friction level and contaminant layer depths, is comprehensively explored and analyzed. This study framework is designed to consider all important conditional factors, including weather, traffic and maintenance operations. The maintenance operations, especially salting, are handled by loosening the strict Markovian assumption, i.e., a history instead of one single time interval of salting operations is considered. In this way, the variation of snow/ice melting speed caused by both residual salt amounts and salt/contaminant mixing states is incorporated in the forecasting model, which enables accurate short-term forecasting for contaminant layers. This approach practically circumvents a major limitation of previous studies, making the post-salting RSC forecasting more reliable and accurate. Under the proposed model framework, several advanced time series modelling methodologies are introduced into the analysis, which can capture the highly complex interactions between RSC measures and conditional factors simultaneously. Those methodologies, especially the univariate and multivariate ARIMA methods, are for the first time applied to the winter RSC evolution process. The forecasting errors of surface temperature, friction level and contaminant layer depths are all found to be small, implying that both the proposed study framework and the resulting solutions closely match the real-world observations. The proposed forecasting models are simple in structure, easy to interpret and mostly consistent with physical knowledge. Compared to the existing models, the proposed models provide extra flexibility for refactory, tuning and deployment. Furthermore, all the modelled RSC measures are numerical and the forecast errors are relatively small, suggesting empirical models could be an efficient alternative to physical models. With the well-designed modelling methods, the resulting empirical models as calibrated in our study can be implemented into a decision support and simulation tool with high temporal resolution and accuracy.

Two efficient methods for solving generalized Lyapunov equations and their implementations in FORTRAN 77 are presented. The first one is a generalization of the Bartels--Stewart method and the second is an extension of Hammarling's method to generalized Lyapunov equations. Our LAPACK based subroutines are implemented in a quite flexible way. They can handle the transposed equations and provide scaling to avoid overflow in the solution. Moreover, the Bartels--Stewart subroutine offers the optional estimation of the separation and the reciprocal condition number. A brief description of both algorithms is given. The performance of the software is demonstrated by numerical experiments.

This Thesis contains contributions in two different but closely related subfields of Scientific and Parallel Computing which arise in the context of various eigenvalue problems: periodic and parallel eigenvalue reordering and parallel algorithms for Sylvestertype matrix equations with applications in condition estimation. Many real world phenomena behave periodically, e.g., helicopter rotors, revolving satellites and dynamic systems corresponding to natural processes, like the water flow in a system of connected lakes, and can be described in terms of periodic eigenvalue problems. Typically, eigenvalues and invariant subspaces (or, specifically, eigenvectors) to certain periodic matrix products are of interest and have direct physical interpretations. The eigenvalues of a matrix product can be computed without forming the product explicitly via variants of the periodic Schur decomposition. In the first part of the Thesis, we propose direct methods for eigenvalue reordering in the periodic standard and generalized real Schur forms which extend earlier work on the standard and generalized eigenvalue problems. The core step of the methods consists of solving periodic Sylvester-type equations to high accuracy. Periodic eigenvalue reordering is vital in the computation of periodic eigenspaces corresponding to specified spectra. The proposed direct reordering methods rely on orthogonal transformations and can be generalized to more general periodic matrix products where the factors have varying dimensions and ±1 exponents of arbitrary order. In the second part, we consider Sylvester-type matrix equations, like the continuoustime Sylvester equation AX −XB =C, where A of size m×m, B of size n×n, and C of size m×n are general matrices with real entries, which have applications in many areas. Examples include eigenvalue problems and condition estimation, and several problems in control system design and analysis. The parallel algorithms presented are based on the well-known Bartels–Stewart’s method and extend earlier work on triangular Sylvester-type matrix equations resulting in a novel software library SCASY. The parallel library provides robust and scalable software for solving 44 sign and transpose variants of eight common Sylvester-type matrix equations. SCASY also includes a parallel condition estimator associated with each matrix equation. In the last part of the Thesis, we propose parallel variants of the direct eigenvalue reordering method for the standard and generalized real Schur forms. Together with the existing and future parallel implementations of the non-symmetric QR/QZ algorithms and the parallel Sylvester solvers presented in the Thesis, the developed software can be used for parallel computation of invariant and deflating subspaces corresponding to specified spectra and associated reciprocal condition number estimates.

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Two efficient methods for solving generalized Lyapunov equations and their implementations in FORTRAN 77 are presented. The first one is a generalization of the Bartels--Stewart method and the second is an extension of Hammarling's method to generalized Lyapunov equations. Our LAPACK based subroutines are implemented in a quite flexible way. They can handle the transposed equations and provide scaling to avoid overflow in the solution. Moreover, the Bartels--Stewart subroutine offers the optional estimation of the separation and the reciprocal condition number. A brief description of both algorithms is given. The performance of the software is demonstrated by numerical experiments.