Afternotes on numerical analysis: a series of lectures on elementary numerical analysis presented at the University of Maryland at College Park and recorded after the fact

Abstract

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.