Solomon Numerical Algorithms

Methods for Computer Vision, Machine Learning, and Graphics
1. Auflage 2015
ISBN: 978-1-4822-5189-0
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 400 Seiten

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.

Zielgruppe

Advanced undergraduate/graduate computer science students, computer graphics professionals

Autoren/Hrsg.

Solomon, Justin

Fachgebiete

• Mathematik | Informatik EDV | Informatik Professionelle Anwendung
• Technische Wissenschaften Elektronik | Nachrichtentechnik Elektronik Robotik
• Mathematik | Informatik Mathematik Mathematik Allgemein Zahlensysteme

Inhaltsverzeichnis

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.

Über Autor(innen)

Justin Solomon is an NSF Mathematical Sciences Postdoctoral Fellow at Princeton’s Program in Applied and Computational Mathematics, where he is studying 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. Before his graduate studies, he was a member of Pixar’s Tools Research group.