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Afternotes on Numerical Analysis

Afternotes on Numerical Analysis - 96 edition

Afternotes on Numerical Analysis - 96 edition

ISBN13: 9780898713626

ISBN10: 0898713625

Afternotes on Numerical Analysis by G. W. Stewart - ISBN 9780898713626
Edition: 96
Copyright: 1996
Publisher: Society for Industrial and Applied Mathematics
International: No
Afternotes on Numerical Analysis by G. W. Stewart - ISBN 9780898713626

ISBN13: 9780898713626

ISBN10: 0898713625

Edition: 96

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There are many textbooks to choose from when teaching an introductory numerical analysis course, but there is only one Afternotes on Numerical Analysis. This book presents the central ideas of modern numerical analysis in a vivid and straightforward fashion with a minimum of fuss and formality. Stewart designed this volume while teaching an upper- division course in introductory numerical analysis. To clarify what he was teaching, he wrote down each lecture immediately after it was given. The result reflects the wit, insight, and verbal craftmanship which are hallmarks of the author.

Simple examples are used to introduce each topic, then the author quickly moves on to the discussion of important methods and techniques. With its rich mixture of graphs and code segments, the book provides insights and advice that help the reader avoid the many pitfalls in numerical computation that can easily trap an unwary beginner.

Written by a leading expert in numerical analysis, this book is certain to be the one you need to guide you through your favorite textbook.

Author Bio

Stewart, G. W. : University of Maryland College Park

G. W. Stewart is a Professor in the Computer Science Department and the Institute for Advanced Computer Studies at the University of Maryland at College Park.

Table of Contents

Table of Contents

Nonlinear Equations

Lecture 1

By the Dawn's Early Light
Interval Bisection
Relative Error

Lecture 2

Newton's Method
Reciprocals and Square Roots
Local Convergence
Slow Death

Lecture 3

A Quasi-Newton Method
Rates of Convergence
Iterating for a Fixed Point
Multiple Zeros
Ending with a Preposition

Lecture 4

The Secant Method
Rate of Convergence
Multipoint Methods
Muller's Method
The Linear-Fractional Method

Lecture 5

A Hybrid Method
Errors, Accuracy, and Condition Numbers

Floating-Point Arithmetic

Lecture 6

Floating-Point Numbers
Overflow and Underflow
Rounding Error

Floating-Point Arithmetic

Lecture 7

Computing Sums
Backward Error Analysis
Perturbation Analysis
Cheap and Chippy Chopping

Lecture 8

The Quadratic Equation
That Fatal Bit of Rounding Error

Linear Equations

Lecture 9

Matrices, Vectors, and Scalars
Operations with Matrices
Rank-One Matrices
Partitioned Matrices

Lecture 10

The Theory of Linear Systems
Computational Generalities
Triangular Systems
Operation Counts

Lecture 11

Memory Considerations
Row-Oriented Algorithms
A Column-Oriented Algorithm
General Observations on Row and Column Orientation
Basic Linear Algebra Subprograms

Lecture 12

Positive-Definite Matrices
The Cholesky Decomposition

Lecture 13

Inner Product Form of Cholesky Algorithm
Gaussian Elimination

Lecture 14

Upper Hessenberg and Tridiagonal Systems

Lecture 15

Vector Norms
Matrix Norms
Relative Error
Sensitivity of Linear Systems

Lecture 16

The Condition of a Linear System
Artificial Ill-Conditioning
Rounding Error and Gaussian Elimination
Comments on Error Analysis

Lecture 17

Introduction to a Project
More on Norms
The Wonderful Residual
Matrices with Known Condition Numbers
Invert and Multiply
Cramer's Rule

Polynomial Interpolation

Lecture 18

Quadratic Interpolation
Polynomial Interpolation
Lagrange Polynomials and Existence

Lecture 19

Synthetic Division
The Newton Form of the Interpolant
Existence and Uniqueness
Divided Differences

Lecture 20

Error in Interpolation
Error Bounds
Chebyshev Points

Numerical Integration

Lecture 21

Numerical Integration
Change of Intervals
The Trapezoidal Rule
The Composite Trapezoidal Rule
Newton-Cotes Formulas
Undetermined Coefficients and Simpson's Rule

Lecture 22

The Composite Simpson Rule
Errors in Simpson's Rule
Treatment of Singularities
Gaussian Quadrature: The Idea

Lecture 23

Gaussian Quadrature: The Setting
Orthogonal Polynomials
Zeros of Orthogonal Polynomials
Gaussian Quadrature
Error and Convergence

Numerical Differentiation

Lecture 24

Numerical Differentiation and Integration
Formulas from Power Series