Matrix algebra numerical example

In preparing the book, a special effort has been made to create self-contained chapters. Within each one, numerical examples and graphics have been provided to aid the reader in understanding the concepts and techniques presented. Notation, references, and material related to that covered in the text are included at the end of each chapter. It is assumed that the reader has a basic knowledge of matrix algebra and statistics however, an appendix covering pertinent statistical concepts is included at the end of the book. 

Using equation substitution, the fault tree based approximations can easily be solved. Using a spreadsheet tool, the Markov model solutions are obtained using linear algebra. For example, the numerical P matrix is ... 

Within esqjlicit schemes the computational effort to obtain the solution at the new time step is very small the main effort lies in a multiplication of the old solution vector with the coeflicient matrix. In contrast, implicit schemes require the solution of an algebraic system of equations to obtain the new solution vector. However, the major disadvantage of explicit schemes is their instability [84]. The term stability is defined via the behavior of the numerical solution for t —> . A numerical method is regarded as stable if the approximate solution remains bounded for t —> oo, given that the exact solution is also bounded. Explicit time-step schemes tend to become unstable when the time step size exceeds a certain value (an example of a stability limit for PDE solvers is the von-Neumann criterion [85]). In contrast, implicit methods are usually stable. 

Within the Matlab s numerical precision X is singular, i.e. the two rows (and columns) are identical, and this represents the simplest form of linear dependence. In this context, it is convenient to introduce the rank of a matrix as the number of linearly independent rows (and columns). If the rank of a square matrix is less than its dimensions then the matrix is call rank-deficient and singular. In the latter example, rank(X)=l, and less than the dimensions of X. Thus, matrix inversion is impossible due to singularity, while, in the former example, matrix X must have had full rank. Matlab provides the function rank in order to test for the rank of a matrix. For more information on this topic see Chapter 2.2, Solving Systems of Linear Equations, the Matlab manuals or any textbook on linear algebra. 

The shape types Tj are usually specified by various algebraic methods, for example, by a shape group or a shape matrix, or by some other algebraic or numerical means. The algebraic invariants or the elements of the matrices are numbers, and these numbers form a shape code. The (P,W)-shape similarity technique provides a nonvisual, algebraic, algorithmic shape description in terms of numerical shape codes, suitable for automatic, computer characterization and comparison of shapes and for the numerical evaluation of 3D shape similarity. 

Another important topic is the solution of nonlinear algebraic systems, which demands very robust algorithms. Chapter 7 illustrates the numerical methods for square systems in their sequential and parallel implementation. In addition to this, methods and techniques are proposed by separating the small and medium dimension problems, which are considered dense for large-scale systems, where the management of matrix sparsity is crucial. Many practical examples are provided. 

Here, we solve aboundary value problem from fluidmechanics numerically by converting it into a linear algebraic system. As this example makes clear, it is sometimes possible to reduce greatly the computational burden of elimination when the matrix is banded, i.e., aU nonzero elements are found near the principal diagonal. 

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