Topics for MA3457/CS4033

• RECURRING FUNDAMENTALS
• Taylor series and Taylor's theorem
• Big Oh notation
• BASIC ERROR ISSUES
• Roundoff/representation error
• Absolute and Relative error
• Effect of arithmetic operations on roundoff error
• Condition number of a function
• INTERPOLATION
• Interpolation vs. Approximation
• Reasons for interpolation
• Lagrange interpolation problem vs. other interpolation problems
• Uniqueness of the Lagrange interpolant
• Lagrange solution to the Lagrange interpolation problem
• Newton's formula for solution of the Lagrange interpolation problem
• Uniqueness and existence of Newton coefficients -- Newton's divided differences
• Error between a function and its interpolating polynomial
• Runge phenomena
• Spline interpolants -- linear, quadratic, cubic splines
• Why splines?
• Matching conditions
• Cubic splines -- boundary conditions, matrix solution
• Bezier curves -- quadratic, cubic
• Guide points for cubic splines
• APPROXIMATION
• What is ``the best fit'' to discrete data?
• Least squares minimization problem
• Linear least squares problem -- Normal equations
• Models and Curve Fitting -- Design matrix - connection to Normal matrix
• Approximation of given functions
• Norm of a function
• Hilbert matrix
• Orthogonal functions, e.g. Chebyshev, Legendre polynomials
• Approximation of periodic phenomena
• Fourier series and Fourier polynomials
• Orthogonality of Fourier polynomials
• NUMERICAL INTEGRATION
• Order of a numerical quadrature method
• The midpoint and trapezoid method
• Error analysis of the composite trapezoid method
• Derivation of Simpson's rule
• NUMERICAL DIFFERENTIATION
• First order derivatives -- forward, backward, and central difference schemes
• Higher order derivatives
• Partial derivatives
• NUMERICAL METHODS FOR INITIAL VALUE PROBLEMS FOR ODEs
• Single Step or Runge Kutta methods
• Euler's method, Midpoint Method, Modified Euler Method - derivation from quadrature rules
• Euler's method -- geometric derivation, Taylor series derivation
• Local truncation error -- derivation, implication
• 4th order Runge-Kutta method
• Multistep methods -- derivation from polynomial interpolation, truncation error
• Higher order ODEs and Systems of ODEs
• Stiffness in ODEs