Staffed labs Tuesday 8-10:50 CSC 1-59 (Workstations), Tue 5-7:50 (Your laptop)
Heath provides a mathematically careful treatment of core topics.
A hands on tutorial on numerical computing in Matlab
Linear Equations Some real world applications and motivations, Gauss and Gauss-Jordan Elimination, LU factorization, Sensitivity and Conditioning, Overdetermined systems and Linear Least Squares, Linear Regression, Data Fitting and Interpolation, Eigenvalue Problems.
Nonlinear Equations and Optimization Basic 1D search methods with and without derivatives. Convergence rates and stopping criteria, Multidimensional root finding. Conditions for extrema, local and global min and max. Unconstrained optimization. Constrained optimization, Comparison of root finding and optimization, Trust region methods, Homotopy methods and Embeddings.
Numerical Integration and Differentiation Numerical Quadrature, Numerical Differentiation, Richardson Extrapolation.
Selected CS topics and Applications: Fast Fourier Transform and Dynamic Programming, Strassens algorithm, SwEng for numerical computing, Writing Vectorized code, Revisit of problem and algorithm sensitivity and stability.
Approximations in Scientific Computation Sources of Approximation Absolute Error and Relative Error Data Error and Computational Error Truncation Error and Rounding Error Forward and Backward Error Sensitivity and Conditioning Stability and Accuracy Systems of Linear Equations Solving Linear Systems Problem Transformations Triangular Linear Systems Elementary Elimination Matrices LU Factorization Implementation of LU Factorization Complexity of Solving Linear Systems Existence and Uniqueness Sensitivity and Conditioning Vector Norms Matrix Norms Matrix Condition Number Error Bounds Residual Solving Modified Problems Improving Accuracy Special Types of Linear Systems Symmetric Positive Definite Systems Banded Systems Iterative Methods for Linear Systems Overdetermined Linear Systems Linear Least Squares Problems Existence and Uniqueness Normal Equations Orthogonality and Orthogonal Projectors Sensitivity and Conditioning Problem Transformations Normal Equations Augmented System Method Orthogonal Transformations Triangular Least Squares Problems QR Factorization Orthogonalization Methods (Overview coverage only) Householder Transformations Givens Rotations Gram-Schmidt Orthogonalization Singular Value Decomposition Other Applications of SVD Comparison of Methods Interpolation and Approximation Existence, Uniqueness, and Conditioning Polynomial Interpolation Monomial Basis Lagrange Interpolation Newton Interpolation Orthogonal Polynomials (Cursorly) Interpolating Continuous Functions Piecewise Polynomial Interpolation Hermite Cubic Interpolation Cubic Spline Interpolation B-splines Eigenvalue Problems Eigenvalues and Eigenvectors Existence and Uniqueness Characteristic Polynomial Multiplicity and Diagonalizability Eigenspaces and Invariant Subspaces Properties of Matrices and Eigenvalue Problems Localizing Eigenvalues Sensitivity and Conditioning Problem Transformations Diagonal, Triangular, and Block Triangular Forms Computing Eigenvalues and Eigenvectors Power Iteration Inverse Iteration Deflation Simultaneous Iteration QR Iteration Jacobi Method Comparison of Methods Generalized Eigenvalue Problems Computing the Singular Value Decomposition Nonlinear Equations Existence and Uniqueness Sensitivity and Conditioning Convergence Rates and Stopping Criteria Nonlinear Equations in One Dimension Interval Bisection Fixed-Point Iteration Newton's Method Secant Method Safeguarded Methods Systems of Nonlinear Equations Fixed-Point Iteration Newton's Method Secant Updating Methods Robust Newton-Like Methods Optimization Optimization Problems Existence and Uniqueness Convexity Unconstrained Optimality Conditions Constrained Optimality Conditions Sensitivity and Conditioning Optimization in One Dimension Golden Section Search Newton's Method Safeguarded Methods Multidimensional Unconstrained Optimization Direct Search Steepest Descent Newton's Method Quasi-Newton Methods Secant Updating Methods Conjugate Gradient Method Truncated or Inexact Newton Methods Nonlinear Least Squares Gauss-Newton Method Levenberg-Marquardt Method Numerical Integration and Differentiation Integration Existence, Uniqueness, and Conditioning Numerical Quadrature Newton-Cotes Quadrature Gaussian Quadrature Composite Quadrature Adaptive Quadrature Other Integration Problems Tabular Data Romberg Integration Double Integrals Multiple Integrals Transforming Integrals Improper Integrals Integral Equations Numerical Differentiation Finite Difference Approximations Automatic Differentiation Richardson Extrapolation