This is a draft cheat sheet. It is a work in progress and is not finished yet.
Numerical Integration
Area under the curve |
Single Integral |
Trapezoidal Rule |
fit linear function |
Single Segment A=(f(a)+f(b))*(b-a)/2 |
Multiple Segments A=(f(xo)+2sum(f(xi))+f(xn))*(b-a)/2n |
n is number of intervals- same width |
increase accuracy.....increase intervals |
Exception: -linear function -fluctuating function |
as dx decreases, come closer to function |
* inefficient but no limit on # of intervals |
if non equal intervals, calculate separately and add |
Simpson's 1/3 Rule |
fit quadratic function |
Single A=(f(x0)+4f(x1)+f(x2))*(b-a)/6 |
equidistant x1 |
Multiple A=(f(x0)+4sum_odd(f(xi))+2sum_even(f(xi))+f(xn))*(b-a)/3n |
# of intervals is even |
* even # of intervals |
most popular bec accuracy is not that significant from 3/8 with less computation |
Simpson's 3/8 Rule |
fit cubic function |
A= (f(x0)+3f(x1)+3f(x2)+f(x3))*(b-a)/8 |
1/3 rule is most widely used as computational efficiency it provides outweighs the accuracy provided by 3/8 rule |
Trapezoidal rule can reach same accuracy of 3/8 rule by increasing number of intervals |
* multiple of 3 # of intervals |
Multiple Integral |
Step 1 at y=0 find A repeat |
Step 2 find A of A(y) |
Tavg= A(A(y))/area or T=A(A(y)) |
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Numerical Differentiation
Taylor series |
f(xi+1)= f(xi)+f'(xi)h+f''(xi)h2/2!+............+fn(xi)hn/n! |
Exponential, you need infinite order because fn(xi)=ex which is never 0 |
First Forward difference |
f'(xi)= (f(xi+1)-f(xi))/h |
to increase accuracy, decrease h that will decrease the rest of Taylor series |
First Backward difference |
f'(xi)= (f(xi)-f(xi-1))/h |
First Centered difference |
f'(xi)= (f(xi+1)-f(xi-1))/2h |
Higher Order |
First Forward difference |
f'(xi)= -f(xi+2)+4f(xi+1) -3f(xi) /(2h) |
First Backward difference |
f'(xi) = 3f(xi) -4f(xi-1) +f(xi-2) /(2h) |
First Centered difference |
f'(xi)= -f(x+2) +8f(xi+1)-8f(xi-1)+f(xi-2) /(12h) |
Second Forward difference |
f''(xi)= (f(xi+2)-2f(xi+1)+f(xi))/h2 |
Second Backward difference |
f''(xi)= (f(xi)-2f(xi-1)+f(xi-2))/h2 |
Second Centered difference |
f''(xi)= (f(xi+1)-2f(xi)+f(xi-1))/h2 |
* more accurate |
Lagrange |
fit interpolated polynomial then differentiate- function that passes by all points |
general method |
f'(x)= pt1+pt2+pt3 |
pt1: 2x-xi-(xi+1)/ ((xi-1)-xi) ((xi-1)-(xi+1)) * f(xi-1) |
pt2: 2x-(xi-1)-(xi+1)/ (xi-(xi-1)) (xi-(xi+1)) * f(xi) |
pt3: 2x-(xi-1)-xi/ ((xi+1)-(xi-1)) ((xi+1)-xi) * f(xi+1) |
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Ordinary Differential Equations
Why solve numerically? |
efficiency or cannot solve analytically |
butterfly effect --sensitive to minor changes |
chaotic systems --system sensitive to initial conditions but with predictable behavior |
ODE with respect to 1 independent variable |
PDE with respect to more than 1 independent variable |
both can have many dependent variables |
Euler's Method |
f(xi+1)=f(xi)+ k1h |
to decrase error, either decrease timestep (h) or take more slopes |
* to decrease timestep (h), take into account round off error propagates and computational efficiency |
assume slope constant in interval |
Heun's Method |
second order |
yi+1=yi +kavg h |
implicit method (yi+1 predictor) ---iterative |
Midpoint Method |
yi+1= yi + k2 h |
Explicit Method |
ymid= yi+ (k1 h)/2 |
Runge Kutte |
generalized formula for methods |
yi+1= yi + phi h |
phi is weighted average of slopes |
number of ks reflects order |
there are infinite methods |
fourth order Runge Kutte |
yi+1 = yi + h/6 (k1+ 2k2+2k3+k4) |
Ralston's Method |
phi= 1/3 K1 + 2/3 k2 |
k2 calculated at 3/4th of interval |
third order Runge Kutte |
phi = (k1 + 4k2+ k3)/6 |
k2 is midway and k3 is at the end |
systems of odes |
need to look at each independently but solve simultanously |
#intial conditions = # dependent variables |
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