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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.
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
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