The Best Approximation Method in Computational Mechanics [Paperback]

With the overwhelming use of computers in engineering, science and physics, the approximate solution of complex mathematical systems of equations is almost commonplace. The Best Approximation Method unifies many of the numerical methods used in computational mechanics. Nevertheless, despite the vast quantities of synthetic data there is still some doubt concerning the validity and accuracy of these approximations. This publication assists the computer modeller in his search for the best approximation by presenting functional analysis concepts. Computer programs are provided which can be used by readers with FORTRAN capability. The classes of problems examined include engineering applications, applied mathematics, numerical analysis and computational mechanics. The Best Approximation Method in Computational Mechanics serves as an introduction to functional analysis and mathematical analysis of computer modelling algorithms. It makes computer modellers aware of already established principles and results assembled in functional analysis.1 Topics in Functional Analysis.- 1.0 Introduction.- 1.1 Set Theory.- 1.2 Functions.- 1.3 Matrices.- 1.4 Solving Matrix Systems.- 1.5 Metric Spaces.- 1.6 Linear Spaces.- 1.7 Normed Linear Spaces.- 1.8 Approximations.- 2 Integration Theory.- 2.0 Introduction.- 2.1 Reimann and Lebesgue Integrals: Step and Simple Functions.- 2.2 Lebesgue Measure.- 2.3 Measurable Functions.- 2.4 The Lebesgue Integral.- 2.4.1 Bounded Functions.- 2.4.2 Unbounded Functions.- 2.5 Key Theorems in Integration Theory.- 2.6 Lp Spaces.- 2.6.1 m-Equivalent Functions.- 2.6.2 The Space Lp.- 2.7 The Metric Space, Lp.- 2.8 Convergence of Sequences.- 2.8.1 Common Modes of Convergence.- 2.8.2 Convergence in Lp.- 2.8.3 Convergence in Measure (M).- 2.8.4 Almost Uniform Convergence (AU).- 2.8.5 Is the Approximation Converging?.- 2.8.6 Counterexamples.- 2.9 Capsulation.- 3 Hilbert Space and Generalized Fourier Series.- 3.0 Introduction.- 3lC.