Throughout the course examples of real world problems will be mathematically modeled and computationally solved.
Linear Equations Some real world applications and motivations, Systems of linear equations and LU factorization, Sensitivity and Conditioning, Overdetermined systems, Linear Least Squares, methods for QR factorisation, Linear Regression, Data Fitting and Interpolation, Eigenvalue Problems, methods for Eigen and SVD factorizations.
Nonlinear Equations and Optimization Basic 1D search methods with and without derivatives. Convergence rates and stopping criteria, Multidimensional root finding, Newton and Gauss-Newtons methods. Conditions for extrema, local and global min and max. Unconstrained optimization.
Numerical Quadrature (Integration) and Differentiation Numerical Quadrature, Numerical Differentiation, Richardson Extrapolation.
Numerical solution of Differential Equations Initial value problems, boundary value problems, partial differential equations.
Detailed topic listing: ----------------------- Scientific computing in Matlab Evaluating expressions Using built-in functions Writing scripts and functions Approximations in Scientific Computation Sources of Approximation Absolute Error and Relative Error Data Error and Computational Error Truncation Error and Rounding 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 Overdetermined Linear Systems Linear Least Squares Problems Existence and Uniqueness Normal Equations Normal Equations QR Factorization Sensitivity and Conditioning Interpolation and Approximation Existence, Uniqueness, and Conditioning Polynomial Interpolation Monomial Basis Interpolating Continuous Functions Intro to splines Eigenvalue Problems Eigenvalues and Eigenvectors Existence and Uniqueness Characteristic Polynomial Multiplicity and Diagonalizability Computing Eigenvalues and Eigenvectors Power Iteration Inverse Iteration SVD and Eigen values,vectors viewed as matrix factorizations. 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 Systems of Nonlinear Equations Newton's Method Optimization Optimization Problems Existence and Uniqueness Convexity Unconstrained Optimality Conditions Optimization in One Dimension Golden Section Search Newton's Method Multidimensional Unconstrained Optimization Direct Search Steepest Descent Newton's Method Nonlinear Least Squares Gauss-Newton Method Levenberg-Marquardt Method (Cursorly) Numerical Integration and Differentiation Integration Existence, Uniqueness, and Conditioning Numerical Quadrature Newton-Cotes Quadrature Other Integration Problems Tabular Data Numerical Differentiation Finite Difference Approximations Initial Value Problems (IVP) for Ordinary Differential Equations Ordinary Differential Equations (ODE) Existence, Uniqueness, and Conditioning Numerical Solution of ODEs Euler's Method Boundary Value Problems (BVP) for ODE Shooting Method Finite Difference Method