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Edition: 96

Copyright: 1996

Publisher: Society for Industrial and Applied Mathematics

Published: 1996

International: No

Copyright: 1996

Publisher: Society for Industrial and Applied Mathematics

Published: 1996

International: No

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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.

**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.

**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

Convergence

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

Cancellation

The Quadratic Equation

That Fatal Bit of Rounding Error

Envoi

**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

Economics

Lecture 13

Inner Product Form of Cholesky Algorithm

Gaussian Elimination

Lecture 14

Pivoting

BLAS

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

Submission

**Polynomial Interpolation **

Lecture 18

Quadratic Interpolation

Shifting

Polynomial Interpolation

Lagrange Polynomials and Existence

Uniqueness

Lecture 19

Synthetic Division

The Newton Form of the Interpolant

Evaluation

Existence and Uniqueness

Divided Differences

Lecture 20

Error in Interpolation

Error Bounds

Convergence

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

Existence

Zeros of Orthogonal Polynomials

Gaussian Quadrature

Error and Convergence

Examples

**Numerical Differentiation **

Lecture 24

Numerical Differentiation and Integration

Formulas from Power Series

Limitations

Bibliography

Index

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Summary

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

**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

Convergence

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

Cancellation

The Quadratic Equation

That Fatal Bit of Rounding Error

Envoi

**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

Economics

Lecture 13

Inner Product Form of Cholesky Algorithm

Gaussian Elimination

Lecture 14

Pivoting

BLAS

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

Submission

**Polynomial Interpolation **

Lecture 18

Quadratic Interpolation

Shifting

Polynomial Interpolation

Lagrange Polynomials and Existence

Uniqueness

Lecture 19

Synthetic Division

The Newton Form of the Interpolant

Evaluation

Existence and Uniqueness

Divided Differences

Lecture 20

Error in Interpolation

Error Bounds

Convergence

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

Existence

Zeros of Orthogonal Polynomials

Gaussian Quadrature

Error and Convergence

Examples

**Numerical Differentiation **

Lecture 24

Numerical Differentiation and Integration

Formulas from Power Series

Limitations

Bibliography

Index

Publisher Info

Publisher: Society for Industrial and Applied Mathematics

Published: 1996

International: No

Published: 1996

International: No

*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.

**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.

**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

Convergence

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

Cancellation

The Quadratic Equation

That Fatal Bit of Rounding Error

Envoi

**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

Economics

Lecture 13

Inner Product Form of Cholesky Algorithm

Gaussian Elimination

Lecture 14

Pivoting

BLAS

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

Submission

**Polynomial Interpolation **

Lecture 18

Quadratic Interpolation

Shifting

Polynomial Interpolation

Lagrange Polynomials and Existence

Uniqueness

Lecture 19

Synthetic Division

The Newton Form of the Interpolant

Evaluation

Existence and Uniqueness

Divided Differences

Lecture 20

Error in Interpolation

Error Bounds

Convergence

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

Existence

Zeros of Orthogonal Polynomials

Gaussian Quadrature

Error and Convergence

Examples

**Numerical Differentiation **

Lecture 24

Numerical Differentiation and Integration

Formulas from Power Series

Limitations

Bibliography

Index