Afternotes on Numerical Analysis - G W Stewart

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Résumé :
This book presents the central ideas of modern numerical analysis in a vivid and straightforward fashion.

Sommaire:
Part I. 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 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; Cramer's 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 Simpson's Rule; Lecture 22. The Composite Simpson's Rule; Errors in Simpson's 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.