Week 5 Flashcards
Equate quantity and approximation for Richardson
Richardson extrapolation formula
Apply Richardson extrapolation formula to ‘right difference approximation of derivative’
Local approximation through trapezoidal rule
Global approximation through trapezoidal rule
Taylor expansion of f(x) on interval [x_i, x_i + h]
Taylor expansion of integral of f(x) on interval [x_i , x_i + h]
Midpoint rule local approximation
Midpoint rule local Taylor series
Simpson’s rule for [xi, xi+1]
General idea behind simpson rule
Trapezoidal rule on interval Si ? And error?
Estimate of error for trapezoidal rule across an entire interval
When is trapezoidal approximation exact
When second deriv is zero everywhere
And therefore function is linear
Approximation error of midpoint rule (analytically)
Simpson’s rule across the not small interval [a,b]
Derive Simpson’s rule
Using Richardson extrapolation on trapezoidal method
Setting h1 = b-a and h2 = (b-a)/2 = h1/2 (half interval)
p = 2
Derive approximation error for Simpson’s rule
Expand f(x) and xi in both Si and integral(f(x))
