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Full Description
This third book in a suite of four practical guides is an engineer's companion to using numerical methods for the solution of complex mathematical problems. The required software is provided by way of the freeware mathematical library BzzMath that is developed and maintained by the authors. The present volume focuses on optimization and nonlinear systems solution. The book describes numerical methods, innovative techniques and strategies that are all implemented in a well-established, freeware library. Each of these handy guides enables the reader to use and implement standard numerical tools for their work, explaining the theory behind the various functions and problem solvers, and showcasing applications in diverse scientific and engineering fields. Numerous examples, sample codes, programs and applications are proposed and discussed. The book teaches engineers and scientists how to use the latest and most powerful numerical methods for their daily work.
Contents
Preface FUNCTION ROOT-FINDING Introduction Substitution Algorithms Bolzano's Algorithm Function Approximation Use of a Multiprocessor Machine with a Known Interval of Uncertainty Search for an Interval of Uncertainty Stop Criteria Classes for Function Root-Finding Case Studies Tests for BzzFunctionRoot and BzzFunctionRootMP Classes Some Caveats ONE-DIMENSIONAL OPTIMIZATION Introduction Measuring the Efficiency of the Search for the Minimum Comparison Methods Parabolic Interpolation Cubic Interpolation Gradient-Based Methods Combination of Algorithms in a General Program Parallel Computations Search for the Interval of Uncertainty Stop Criteria Classes for One-Dimensional Minimization Case Studies Tests UNCONSTRAINED OPTIMIZATION Introduction Heuristic Methods Gradient-Based Methods Conjugate Direction Methods Newton's Method Modified Newton Methods Quasi-Newton Methods Narrow Valley Effect Stop Criteria BzzMath Classes for Unconstrained Multidimensional Minimization Case Study Tests LARGE-SCALE UNCONSTRAINED OPTIMIZATION Introduction Collecting a Sparse Symmetric Matrix Ordering the Hessian Rows and Columns Quadratic Functions Hessian Evaluation Newton's Method Inexact Newton Methods Practical Preconditioners openMP Parallelization Class for Large-Scale Unconstrained Minimization ROBUST UNCONSTRAINED MINIMIZATION Introduction One-Dimensional Minimization Classes for One-Dimensional Robust Minimization Examples in One-Dimensional Space Examples in Multidimensional Space Two-Dimensional Space Classes for Robust Two-Dimensional Minimization Examples for BzzMinimizationTwoVeryRobust Class Multidimensional Robust Minimization Class for Robust Multidimensional Minimization ROBUST FUNCTION ROOT-FINDING Introduction Class and Examples NONLINEAR SYSTEMS Introduction Comparing Nonlinear Systems to Other Iterative Problems Convergence Test Substitution Methods Minimization Methods Jacobian Evaluation Newton's Method Gauss-Newton Method Modified Newton Methods Newton's Method and Parallel Computations Quasi-Newton Methods Quasi-Newton Methods and Parallel Computing Stop Criteria Classes for Nonlinear System Solution with Dense Matrices Tests for the BzzNonLinearSystem Class Sparse and Large-Scale Systems Large Linear System Solution with Iterative Methods Classes for Nonlinear System Solution with Sparse Matrices Continuation Methods Solution of Certain Equations with Respect to Certain Variables Case Studies Special Cases Some Caveats UNDERDIMENSIONED NONLINEAR SYSTEMS Introduction Underdimensioned Linear Systems Class for Underdimensioned Nonlinear System Solution CONSTRAINED MINIMIZATION Introduction Equality Constraints Equality and Inequality Constraints Lagrangian Dual Problem LINEAR PROGRAMMING Introduction Basic Attic Method Concepts Attic Method Differences between the Attic Method and Traditional Approaches Explosion in the Number of Iterations Degeneracy Duality General Considerations QUADRATIC PROGRAMMING Introduction KKT Conditions for a QP Problem Equality-Constrained QP Equality- and Inequality-Constrained Problems Class for QP Projection or Reduced Direction Search Methods for Bound-Constrained Problems Equality, Inequality, and Bound Constraints Tests CONSTRAINED MINIMIZATION: PENALTY AND BARRIER FUNCTIONS Introduction Penalty Function Methods Barrier Function Methods Mixed Penalty-Barrier Function Methods CONSTRAINED MINIMIZATION: ACTIVE SET METHODS Introduction Class for Constrained Minimization Successive Linear Programming Projection Methods Reduced Direction Search Methods Projection or Reduced Direction Search Methods for Bound-Constrained Problems Successive Quadratic Programming or Projected Lagrangian Method Narrow Valley Effect The Nonlinear Constraints Effect Tests PARAMETRIC CONTINUATION IN OPTIMIZATION AND PROCESS CONTROL Introduction Algebraic Constraints
