James M. Ortega

Numerical analysis : a second course

Introduction 1. Linear Algebra Part I. Mathematical Stability and Ill Conditioning. 2. Systems of Linear Algebraic Equations 3. 4. Differential and Difference Equations Part II. Discretization Error 5. Discretization Error for Initial Problems 6. Discretization Error for Boundary Value Problems Part III. Convergence of Iterative Methods 7. Systems of Linear Equations 8. Systems of Nonlinear Equations Part IV. Rounding Error 9. Rounding Error for Gaussian Elimination Bibliography Index.

Monotone Iterations for Nonlinear Equations with Application to Gauss-Seidel Methods

Monotone iterations for nonlinear elliptic differential equations in boundary-value problems applied to Gauss-Seidel methods

The Curse of Dimensionality

This chapter discusses problems in more than one space dimension. As physical phenomena occur in a three-dimensional world, mathematical models in only one space dimension are usually considerable simplifications of the actual physical situation although in many cases they are sufficient for phenomena that exhibit various symmetries or in which events are happening in two of the three space dimensions at such a slow rate that those directions can be ignored. In the case of a single space variable, the use of an implicit method does not cause much computational difficulty as the solution of tridiagonal systems of equations can be accomplished so rapidly. In the case of a single space variable, the use of an implicit method does not cause much computational difficulty as the solution of tridiagonal systems of equations can be accomplished rapidly. However, each time step requires the solution of a two-dimensional Poisson-type equation, which is a much more difficult computational problem.

Stability of Difference Equations and Convergence of Iterative Processes

In this report, we review several connections between the theory of convergence of iterative processes and the theory of Lyapunov stability of ordinary difference equations. Among the topics discussed are local convergence and asymptotic stability, global convergence and global asymptotic stability, Lyapunov functions, domain of attraction, total stability and rounding errors, nonautonomous equations and variable operator iterations.

On Discretization and Differentiation of Operators with Application to Newton’s Method

In the numerical solution of operator equations

A General Convergence Result for Unconstrained Minimization Methods

Fx = 0

On a class of approximate iterative processes

, discretization of the equation and then application of Newton’s method results in the same linear algebraic system of equations as application of Newtons method followed by discretization. This leads to the general problem of determining when the two frequently used operations of discretization and (Frechet) differentiation applied to a nonlinear operator are commutative. A theory of discretization processes is developed here which proves that for a wide class of operators of interest in applications, discretization and differentiation indeed “commute”. The fundamental concept of the theory is a distinction between the discretization of the linear spaces involved and the replacement of the infinitesimal parts of the operator F, i.e., those parts involving, e.g., differentiation and integration, by a discrete analogue. Using this distinction in an abstract way, a “complete” discretization process is defined precisely and the cited commutativity res...

The World of Scientific Computing

This note extends an observation of J. Daniel and presents a general convergence theorem for iterative methods for unconstrained minimization problems. The key point is the concept of an essentially gradient-related sequence which includes the previously studied gradient-related sequences, as well as sequences which arise from univariate relaxation methods.

CHAPTER 1 – LINEAR ALGEBRA

Combination into unified setting of various results for approximate solution of fixed point equation, using iterative process

Conjugate Gradient-Type Methods

This chapter discusses the applications of scientific computing. Scientific computing is done on computers ranging from small PCs, which execute a few thousand floating point operations per second, to supercomputers capable of billions of such operations per second. Supercomputers that utilize hardware vector instructions are called vector computers, while those that incorporate multiple processors are called parallel computers. The scientific computing is viewed as the discipline that achieves a computer solution of mathematical models of problems from science and engineering. Hence, the first step in the overall solution process is the formulation of a suitable mathematical model of the problem at hand. The formulation of a mathematical model begins with a statement of the factors to be considered. In addition to the basic relations of the model, which in most situations in scientific computing take the form of differential equations, there usually will be a number of initial or boundary conditions. Once the model is deemed adequate from the validation and modification process, it is ready to be used for prediction.