Numerical algorithms : methods for computer vision, machine learning, and graphics / Justin Solomon.

"An A K Peters Book."

Justin Solomon is an assistant professor in the Department of Electrical Engineering and Computer Science at MIT, where he studies problems in shape analysis, machine learning, and graphics from a geometric perspective. He received a PhD in computer science from Stanford University, where he was also a lecturer for courses in graphics, differential geometry, and numerical methods. Subsequently he served as an NSF Mathematical Sciences Postdoctoral Fellow at Princeton?s Program in Applied and Computational Mathematics. Before his graduate studies, he was a member of Pixar?s Tools Research group.

Includes bibliographical references (pages 361-368) and index.

Table of Contents

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.

Features

Contains classroom-tested material for a one- to two-semester course in numerical algorithms, with a focus on modeling and applications
Introduces themes common to nearly all classes of numerical algorithms
Covers algorithms for solving linear and nonlinear problems, including popular techniques recently introduced in the research community
Includes comprehensive end-of-chapter exercises that push students at all levels to derive, extend, and analyze numerical algorithms

Solutions manual and figure slides available upon qualifying course adoption
Summary

Numerical Algorithms: Methods for Computer Vision, Machine Learning, and Graphics presents a new approach to numerical analysis for modern computer scientists. Using examples from a broad base of computational tasks, including data processing, computational photography, and animation, the textbook introduces numerical modeling and algorithmic design from a practical standpoint and provides insight into the theoretical tools needed to support these skills.

The book covers a wide range of topics?from numerical linear algebra to optimization and differential equations?focusing on real-world motivation and unifying themes. It incorporates cases from computer science research and practice, accompanied by highlights from in-depth literature on each subtopic. Comprehensive end-of-chapter exercises encourage critical thinking and build students? intuition while introducing extensions of the basic material.

The text is designed for advanced undergraduate and beginning graduate students in computer science and related fields with experience in calculus and linear algebra. For students with a background in discrete mathematics, the book includes some reminders of relevant continuous mathematical background.