Afternotes on Numerical Analysis

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.

Spis treści:
Part I. Nonlinear Equations, Lecture 1. By the Dawns Early Light
Interval Bisection
Relative Error
Lecture 2. Newtons Method
Reciprocals and Square Roots
Local Convergence Analysis
Slow Death
Lecture 3. A Quasi-Newton Method
Rates of Convergence
Iterating for a Fixed Point
Multiple Zeros
Ending with a Proposition
Lecture 4. The Secant Method
Convergence
Rate of Convergence
Multipoint Methods
Mullers Method
The Linear-Fractional Method
Lecture 5. A Hybrid Method
Errors, Accuracy, and Condition Numbers. Part II. Computer 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. Part III. Linear Equations, Lecture 9. Matrices, Vectors, and Scalars
Operations with Matrices
Rank-One Matrices
Partitioned Matrices
Lecture 10. 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 the 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 Linear Systems
Artificial Ill Conditioning
Rounding Error and Gaussian Elimination
Comments on the Analysis
Lecture 17. The Wonderful Residual, A Project
Introduction
More on Norms
The Wonderful Residual
Matrices with Known Condition
Invert and Multiply
Cramers Rule
Submission
Part IV. 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
Divided Differences
Lecture 20. Error in Interpolation
Error Bounds
Convergence
Chebyshev Points. Part V. Numerical Integration and Differentiation, Lecture 21. Numerical Integration
Change of Intervals
The Trapezoidal Rule
The Composite Trapezoidal Rule
Newton-Cotes Formulas
Undetermined Coefficients and Simpsons Rule
Lecture 22. The Composite Simpsons Rule
Errors in Simpsons Rule
Weighting Functions
Gaussian Quadrature
Lecture 23. The Setting
Orthogonal Polynomials
Existence
Zeros of Orthogonal Polynomials
Gaussian Quadrature
Error and Convergence
Examples
Lecture 24. Numerical Differentiation and Integration
Formulas From Power Series
Limitations
Bibliography.