Afternotes on Numerical Analysis

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.