Numerical integration of gridded values

• The basics
• Newton-Cotes techniques
• Rectangular rule (also called the mid-point rule)
• 1) Trapezoid rule. 2) Summary of Trapezoid rule.
• 1) Simpson's rule  - note its only valid for even number of intervals (odd number of points)! 2) Summary of 1/3 Simpson's rule. 3) Summary of 3/8 Simpson's rule.
  
• How to use Simpson's rule for odd number of intervals - combine 1/3 and 3/8 Simpson's rules!

Numerical integration of analytical functions

• Introduction
• 1) Richards extrapolation. 2) Summary of Richard's extrapolation. 3) Definition of Richard's extrapolation.
• 1) Romberg integration. 2) Summary of Romberg integration. 3) Good explanation of Romberg integration. 4) This is an example of sequence (series) acceleration.
• Gaussian quadrature
• Improper integrals
• Advanced material - adaptive integration. Example with MPI FORTRAN code.

Random numbers 

• Random numbers
• Typical pseudo-random number generator equation - Linear congruential generator. What is a modulus operator? How to compute the modulus.
• A popular pseudo-random number generator -- the Mersenne Twister. The recurrence equation is complicated, but has a VERY LONG period.
• Random number generation in Excel. Summary of Excel function RAND
• Tests for true randomness: 1) DIEHARD (like the movie). 2) NIST Test suite.
• Online random number generator

Monte Carlo simulations

• Monte Carlo simulations
• The classic dice example
• Example: Collision-coalescnce process
• Example: Light scattering
• Example: Tornado risk assessment

Spreadsheet example, random number

• Random walk. Spreadsheet example. 
• Adding "noise" to a dataset. Spreadsheet example of a line with "noise" added.
• Create a randomized list. Create a random normal distrbution. Spreadsheet examples.

Spreadsheet examples, integration

• Area under curve. Spreadsheet examples.
• Monte Carlo integration. Spreadsheet examples.

Miscellaneous random number code from different computer languages

• DIEHARDER: contains both DIEHARD and NIST Test Suite
• R routine version of DIEHARDER
• Mersenne twister, FORTRAN routine
• Linear Congruent Method, FORTRAN routine
• Random number generation, Matlab
• Random number generator, Mathematica
• Random walk, FORTRAN routine

Miscellaneous numerical integration routines from different computer languages

• Matlab trapezoid rule function. Matlab routine.
• Matlab Simpson rule function. Matlab routine.
• Matlab Romberg function. Matlab routine.
• Matlab built-in functions for integration.
• Mathematica functions.
• R routine for integration.
• FORTRAN trapezoid rule subroutine. Driver program.
• FORTRAN Simpson rule subroutine .
• FORTRAN adaptive integration. Driver program.
• FORTRAN Monte Carlo integration. Driver program. 

Recorded lectures

• Numerical integration one - Audio only
• Numerical integration two - Audio only
• Random numbers - Audio only