Numerical Algorithms

Preliminaries Mathematics Review PRELIMINARIES: NUMBERS AND SETS VECTOR SPACES LINEARITY NONLINEARITY: DIFFERENTIAL CALCULUS Numerics and Error Analysis STORING NUMBERS WITH FRACTIONAL PARTS UNDERSTANDING ERROR PRACTICAL ASPECTS Linear Algebra Linear Systems and the LU Decomposition SOLVABILITY OF LINEAR SYSTEMS ADHOC SOLUTION STRATEGIES ENCODING ROW OPERATIONS GAUSSIAN ELIMINATION LU FACTORIZATION Designing and Analyzing Linear Systems SOLUTION OF SQUARE SYSTEMS SPECIAL PROPERTIES OF LINEAR SYSTEMS SENSITIVITY ANALYSIS Column Spaces and QR THE STRUCTURE OF THE NORMAL EQUATIONS ORTHOGONALITY STRATEGY FOR NONORTHOGONAL MATRICES GRAMSCHMIDT ORTHOGONALIZATION HOUSEHOLDER TRANSFORMATIONS REDUCED QR FACTORIZATION Eigenvectors MOTIVATION PROPERTIES OF EIGENVECTORS COMPUTING A SINGLE EIGENVALUE FINDING MULTIPLE EIGENVALUES SENSITIVITY AND CONDITIONING Singular Value Decomposition DERIVING THE SVD APPLICATIONS OF THE SVD Nonlinear Techniques Nonlinear Systems ROOTFINDING IN A SINGLE VARIABLE MULTIVARIABLE PROBLEMS CONDITIONING Unconstrained Optimization UNCONSTRAINED OPTIMIZATION: MOTIVATION OPTIMALITY ONE-DIMENSIONAL STRATEGIES MULTIVARIABLE STRATEGIES Constrained Optimization MOTIVATION THEORY OF CONSTRAINED OPTIMIZATION OPTIMIZATION ALGORITHMS CONVEX PROGRAMMING Iterative Linear Solvers GRADIENT DESCENT CONJUGATE GRADIENTS PRECONDITIONING OTHER ITERATIVE ALGORITHMS Specialized Optimization Methods NONLINEAR LEAST SQUARES ITERATIVELYREWEIGHTED LEAST SQUARES COORDINATE DESCENT AND ALTERNATION GLOBAL OPTIMIZATION ONLINE OPTIMIZATION Functions, Derivatives, and Integrals Interpolation INTERPOLATION IN A SINGLE VARIABLE MULTIVARIABLE INTERPOLATION THEORY OF INTERPOLATION Integration and Differentiation MOTIVATION QUADRATURE DIFFERENTIATION Ordinary Differential Equations MOTIVATION THEORY OF ODES TIMESTEPPING SCHEMES MULTIVALUE METHODS COMPARISON OF INTEGRATORS Partial Differential Equations MOTIVATION STATEMENT AND STRUCTURE OF PDES REPRESENTING DERIVATIVE OPERATORS SOLVING PARABOLIC AND HYPERBOLIC EQUATIONS NUMERICAL CONSIDERATIONS Exercises appear at the end of each chapter.