Study points for the final exam

Mathematical preliminaries

• Taylor's theorem, integral mean value theorem, change of variable in integrals, matrix algebra

Floating point computation

• floating point number systems
• IEEE arithmetic
• propagation of errors by arithmetic operations and function evaluation, catastrophic cancellation

Numerical solution of nonlinear equations

• bisection method and its convergence theory
• Newton's method and its convergence theory
• secant method and its convergence theory
• adaptive hybrid methods
• fixed points and successive substitution, contractions, rates of convergence
• Muller's method
• Aitken extrapolation
• Newton's method for systems

Interpolation

• Lagrange interpolation: definition, existence and uniqueness, ...
• Lagrange's form
• divided differences and Newton's form, divided difference tables
• error formula
• Hermite cubic interpolation, Lagrange's form and Newton's form, error formula
• piecewise polynomial interpolation: piecewise Lagrange interpolation, piecewise Hermite cubic interpolation
• cubic spline interpolation: definition, defining equations, end conditions
• adaptive interpolation routines

Quadrature

• standard form of a quadrature rule (nodes and weights)
• derivation of quadrature rules from Lagrange interpolation
• determining the error by integrating the error formula for Lagrange interpolation
• scaling
• simple rules and composite rules, simplification of composite rules for equal spacing
• derivation of error estimates for the composite rule from those for the simple rule (for both equal and unequal spacing)
• endpoint corrections for composite trapezoidal rule with equal spacing
• Newton-Cotes rules: trapezoidal rule, Simpson's rule, Simpson's 3/8 rule, Boole's rule,...
• Peano kernel error representation for the simple trapezoidal rule; how to apply it
• adaptive quadrature schemes
• combining two equal order rules to get an error estimate and a better approximation, Richardson extrapolation
• Gaussian quadrature

Numerical solution of linear systems of equations

• review of matrix algebra
• solution of triangular systems, back substitution
• Gaussian elimination and reduction to triangular form
• the LU decomposition
• pivoting and scaling
• Cholesky decomposition
• elimination for tridiagonal matrices
• counting operations, operation counts for all the algorithms
• matrix and vector norms
• conditioning of linear systems, condition numbers