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Afternotes on Numerical Analysis

Afternotes on Numerical Analysis


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About the Book

This text presents the central ideas of modern numerical analysis. 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 landmarks of the author. Simple examples are used to introduce each topic, then the author moves on to the discussion of important methods and techniques. With its 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.

Table of Contents:

  • Part One: 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 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
  • Muller's Method
  • The Linear-Fractional Method
  • Lecture 5: A Hybrid Method
  • Errors, Accuracy, and Condition Numbers. Part Two: 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. Part Three: 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 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 a Linear System
  • Artificial Ill-Conditioning
  • Rounding Error and Gaussian Elimination
  • Comments on the 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. Part Four: 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. Part Five: 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
  • Lecture 24: Numerical Differentiation and Integration
  • Formulas From Power Series
  • Limitations
  • Bibliography: Introduction
  • References.


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    Product Details
    • ISBN-13: 9780898713626
    • Publisher: Society for Industrial & Applied Mathematics,U.S.
    • Publisher Imprint: Society for Industrial & Applied Mathematics,U.S.
    • Height: 229 mm
    • No of Pages: 210
    • Spine Width: 12 mm
    • Width: 152 mm
    • ISBN-10: 0898713625
    • Publisher Date: 31 Dec 1996
    • Binding: Paperback
    • Language: English
    • Returnable: Y
    • Weight: 394 gr


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