Week 8 Flashcards
Binomial coefficient
Probability distribution of Monte Carlo method
For p1 = s1/(s1 + s2) and p2 = s2/(s1+s2)
Convergence rate of Monte Carlo method
1/sqrt(N)
RV for mont Carlo method
α = n/N
What is expectation value for Monte Carlo
Error in Monte Carlo in terms of α
MGF for Monte carlo
Use MGF of Monte Carlo to show error
Standard formula for exponential limit
Limit of MGF of Monte Carlo
Stirling’s approximation of factorial
m! ~ m^m
For large m
Use stirlings approximation of factorial on Pn(N>
How to do shooting method
Where boundary conditions y(a) = α and y(b) = β are imposed on an IVP
Where β = y(b) = g(u)
1) use Euler/RK to solve with one boundary condition
2) use secant to find solution that connects with other boundary
Define a first order differential equation for a vector of functions of a single variable
Write an n order ODE as a vector
How to equate an n’th order ODE to a first order ODE of a vector of n functions
Theory of Euler method
Approximate deriv by:
y’(x) ~ (y(x+h)-y(x))/h
Then solve recursively for xi = a + ih
Where y(a+ih) =: y, i = 0,1,2,…
Then
Correct Euler method to include error
Fundamental difference between Euler and Runge Kutta
Express y(x+h) in terms of y(x) and its derivatives by Taylor series
Runge Kutta method and derivation
Using the Taylor series exp of y(x+h) up to 2nd deriv
4th order Runge Kutta
