Applied Linear Algebra and Numerical Analysis

Course syllabus (tentative)

• Introduction to MATLAB
  • m-files: scripts and functions
  • Plotting functions
• Measuring errors
• Review of Taylor series and Taylor polynomial
  • Error bounds
• Interpolation
  • Newton form and divided differences
  • Relation of Newton form to Taylor series
  • Error bounds
  • Hermite interpolation
  • Piecewise polynomial interpolation, splines
• Numerical integration or quadrature
  • Piecewise polynomial interpolation and composite rules
  • Adaptive quadrature
• Approximating derivatives
• Computer arithmetic and rounding errors
• Ordinary differential equations
  • The initial value problem
  • Euler's method
  • Taylor series methods
  • Runge Kutta methods
  • Explicit and implicit methods for systems of equations
• Root finding -- solving a single nonlinear equation in one variable
  • Bisection method
  • Newton's method and Secant method
  • Fixed point iteration
• Linear systems of equations
  • Matrix-vector notation
  • Backslash operator in MATLAB
• Nonlinear systems of equations
  • Multi-variate Taylor series and the Jacobian matrix
  • Newton's method
  • Application to implicit methods for ODEs
• Linear algebra
  • linear independence, column space, null space, rank of matrices
  • Gaussian elimination, pivoting, LU decomposition
  • vector and matrix norms, condition numbers
  • least squares problems and data fitting
  • eigenvalues and eigenvectors
  • function spaces