Numbers
- Babylonian base-60
- Egyptian fractional notation
- Roman numerals and operations
- conversions of fractions to floating-point representations
- Binary notation for integers
- 32-bit floating point number representations
Error
- absolute and relative
- common sources of error
- propagation of errors under addition and multiplication
Root finding in 1 variable
Fixed-point iteration
- difference equations
- fixed-points of difference equations
- stability of fixed-points of difference equations
- Square-roots by Greek method (Theon's ladder), convergence rates
- initial-condition dependence
- geometric square-root approximation formula (Heron's method)
- Newton--Raphson
* method
* local convergence criteria
* cube-root method
* order-of-convergence (linear, quadratic)
* requirements, pro's and con's
- Halley's method
* requirements, pro's and con's
- Secant Method
* requirements, pro's and con's
Bracketing
- Bisection (Bolzano's method)
* convergence rate
* requirements, pro's and con's
- Regula Falsi
* requirements, pro's and con's
- Muller's method
* requirements, pro's and con's
Matlab and fzero
Operation counts
Horner's method
Linear systems
- matrices, vectors, norms
- Forward problems, Matrix multiplication
* application to random-walk model
- Special matrices
- backward problems A x = b
- elimination and back-substitution.
* operations counts
* numerical stability of elimination
- pivoting, types, reasons
- stochiometry applications
- LU factorization
- pro's and con's
- rank, number of solutions of Ax = b.
- iterative methods: Jacobi
- operation counts for various algorithms
Non Linear Systems (nonlinear)
- function approximation
- local
* Calculate Taylor series
* Taylor series of common functions
* error approximation
- asymptotic, newton hulls