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PYTHON PHYS > PART III: Computational Mathematics For Physics > Flashcards

PART III: Computational Mathematics For Physics Flashcards

1
Q

What’s the idea of numerical integration?

A

An approximation method to determine the area underneath the curve
There’s error since its an approximation

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2
Q

Riemann sum

A

I = sum(n-1,i=0)hf((x(i+1)+xi)/2)

From using midpoint rule

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3
Q

Derivation of the trapezoid rule

+ equation for the trapezoid rule

A 
A = 1/2*bh
I = sum(n-1,i=0)h[f(x(i+1))+f(xi)]/2
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4
Q

What is a Taylor series?

A

An expansion of a function into an infinite sum of terms, with increasing orders of a variable like x, x^2, x^3, ,,,
When it converges, it expresses the exact value of a function

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5
Q

What is a Taylor series expansion?

A

Formula given lol

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6
Q

What’s a Taylor polynomial?

A

Tn(x) from a convergent series allows us to approximate the value of a function up to a certain order

  • any orders we don’t care about it O(x^n+1)
  • Epsilon = f(x) - T(x), so epsilon proportional to O(x^n+1)
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7
Q

What’s special about Taylor expansions?

A
  • they’re smooth
  • allows us to rewrite a smooth function as an infinite sum of polynomial terms
  • can guess what a function will look like around a point
  • add infinite number of derivatives to get a single infinite sum
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8
Q

Calculating error in integration methods

A 
Taylor expand f around a=xi-1
Integrate it all
Sub u = x-a
Change integration bounds
Make it O(h^whatever it is)
Expand around second point xi
Average them together 
Add up all sub intervals from a to b
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9
Q

Error in trapezoid rule

A

1/4*h^2[f’(a)-f’(b)]

  • accurate to and including terms proportional to h (first order)
  • error O(h^2)
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10
Q

Accuracy of a first order rule

A

O(h) and error O(h^2)

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11
Q

Simpson’s rule: formula

A

I(a,b) = 1/3h[f(a) + f(b) +4sum(odd)f(a+kh) + 2sum(even)f(a+kh)]

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12
Q

What’s special about Simpson’s rule?

A
  • convenient to code
  • more precise with relatively little effort
  • moderately accurate with small number of steps
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13
Q

Error in simpson’s rule

A
  • 1/90*h^4[f”‘(a)-f”‘(b)]
  • 3rd order
  • 4th approximation error
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14
Q

Is Simpson’s rule always better than trapezoid rule?

A

No
When h>1, trapezoid is better
Both depend on derivative so if your 3rd derivative (what Simpson depends on) is large, the error is larger for Simpson’s rule

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15
Q

Adaptive integration

A

How many step sizes is enough?
Double N each time
Error is 1/3(Ii - I(i-1))

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16
Q

Romberg integration

A

Adjusting trapezoid rule (accuracy of Simpson’s method)
Calculate I1=R(1,1) and I2=R(2,1) with trapezoid rule
Use R(i,m+1) = R(i,m)+(R(i,m) - R(i-1,m))/((4^m)-1) to get more accurate estimate
I3 = R(3,1), use it to get R(3,2) and R(3,3)
At each successive stage, compute 1 more trapezoid rule estimate
For each estimate, we calculate error and stop when we get target

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17
Q

Error in Romberg integration

A

N levels of this process, the last estimate is R(n,n) which is accurate to order h^2n

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18
Q

Why is calculating numerical derivatives less common than integrals?

A
  • basic techniques for it are pretty simple
  • derivatives can usually be easily calculated analytically
  • noisy data poses a lot of problems
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19
Q

Forward difference method

A

f’ = [f(x+h)-f(x)]/h

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20
Q

What’s the types of error associated with numerical differentiation?

A 
Rounding error (due to use of computer/program/software)
Approximation error (due to technique)
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21
Q

Derivation of approximation error for forward difference method

A

Taylor expand f around x
Rearrange to solve for f’
You get f’ = forward diff + error
Epsilon = h/2*|f”(x)|

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22
Q

Derivation of round off error for forward difference method

A 
C = Python error constant
Apply to both f(x+h) and f(x)
f(x) becomes Cf(x)
Since f(x) is close to f(x+h), combined round off error is 2C|f(x)|
Take h into account
Epsilon = 2C|f(x)|/h
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23
Q

Total error in forward difference method

A

Epsilon = sum of approx + roundoff
Epsilon = 2C|f(x)|/h + h/2*|f”(x)|
Find h that minimizes it: derivative, set =0, solve for h, plug back in

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24
Q

Backward difference method

A

f’ = [f(x)-f(x-h)]/h

25
Q

Central difference method

A

Average of forward and backward difference

f’ = [f(x+0.5h)-f(x-0.5h)]/h

26
Q

Error on central difference method

A 
2 Taylor expansions (+ and -)
Subtract second from first, rearrange for f'
Same method as others from here
Epsilon is much smaller
Rounding error is the same
27
Q

Alternate central difference method: what is it and when do you use it?

A

Used if you have a discrete set of data points

f’ = [f(x+h) - f(x-h)]/2h

28
Q

When is central difference more accurate?

A

h < |f”(x)/f”‘(x)|

29
Q

Define interpolation

A

When we want to approximate values in between known data points

30
Q

Define extrapolation

A

When we want to approximate values outside our range of data points

31
Q

Define interpolant

A

The model

32
Q

Define basis functions

A

Related to how much each datapoint is weighted

Multiply set of data values y by a set of basis functions W to get inter/extrapolate equation

33
Q

Higher order interpolation

A

Instead of 2 points, 3 points

More points = more weights

34
Q

Piecewise interpolation

A

For complex data

- break function into easier/manageable components

35
Q

Piece wise interpolation: how do we piece it together?

A

Splines!

36
Q

Define splines

A

Special basis set of cubic polynomials with two constraints

  • curve must pass through a point so two points are constrainted
  • tangent Fm curve must have a particular angle, so two derivatives are constrained
37
Q

How is a matrix different from an array?

A

Matrices are strictly 2D (arrays can be multi-dim)
Multiplication is different (mat does dot product)
Matrix class has methods of calculating inverse, transpose, etc
Syntax is different (mat(“[row1;row2]”)

38
Q

Solving linear equations

A
  • use matrices
  • convert system to matrix format
  • define Ax=v
  • use Gaussian elimination
  • back substitute
  • can use solve from numpy.linalg
39
Q

Relaxation method

A

For equations of form x=f(x)
Start with initial guess x = 1
Solve x’ = f(x) using x from previous step
Solve x” = f(x’) using x from step 2
Keep repeating until convergence or until accuracy reaches desired threshold

40
Q

When can we use the relaxation method?

A

The equation is invertible and not oscillating between different values (has convergence)

41
Q

Relaxation method: what happens if the function is oscillating or not invertible?

A

Use alternate method
Log both sides and solve for alternative expression for x
Apply relaxation to that

42
Q

Steps for binary search method

A 
Get initial pair of points x1,x2, check that f(x1) and f(x2) have opposite signs, pick target accuracy (check product=-)
Calculate midpoint on x'=0.5(x1+x2) and evaluate f(x')
If f(x') has the same sign as f(x1) set x1=x', or else set x2=x'
If |x1-x2| > target accuracy, repeat from calc midpoint
Else do it all again
43
Q

Binary search: how many steps are required?

A 
N = log2(del/e)
Del = initial interval
e = error (desired accuracy)
44
Q

Problems with binary search method

A

Multiple roots
Local extreme at 0 but doesn’t cross
End points don’t envelope a zero crossing

45
Q

Derivation for Newton’s method

A 
f' = f/del(x)
x' = x-del(x)
46
Q

What do we need to know to use Newton’s method?

A

Derivative

47
Q

Steps for Newton’s method

A

Start with initial guess of x
Solve for f(x), f’(x), plug into equation
Use x’ to plug back into equation for x”
Repeats until desired accuracy

48
Q

Error on Newton’s method + what does it mean?

A

Epsilon’ = [-f”(x)/2f’(x)]epsilon^2

Size of error is epsilon^2 on next estimate (quadratic convergence)

49
Q

Derivation for Euler’s method

A

With initial condition
x(t+h) = x(t) + hf(x,t) + O(h^2)
Assume O(h^2)->0
x(t+h) = x(t) + hf(x,t)

50
Q

Steps for Euler’s method

A 
Take x(t) that we know (initial condition)
Plug into x(t+h) = x(t)+hf(x,t) to solve for x(t+h)
Repeat that for another interval h after that
Can go as long as we want
51
Q

Error with Euler’s method

A

Summation of errors at each individual step
1/2h[f(x(b),b) - f(x(a),a)]
Linear so not great convergence

52
Q

Derivation for Runge-Kutta method

A

Instead of stopping Taylor expansion at O(h^2), change it to O(h^3) in Euler’s method
Use slope at time t+0.5h instead of at t exactly
x(t+h) = x(t) + h(dx/dt|(t+0.5h)) + O(h^3)
h^2 is gone, error O(h^3) so more accurate now

53
Q

Steps for Runge-Kutta method

A 
Approx x(t+0.5h) with Euler's method
Don't forget to sub 0.5 everywhere h is in the equation
K1 = hf(x,t)
K2 = hf(x+0.5k1, t+0.5h)
x(t+h) = x(t) + k2
This is SECOND ORDER
54
Q

Higher order runge-Kutta method

A

4th order exists (most common because 5th order error)

55
Q

Problems with Runge-Kutta method

A

Infinite ranges - have to change variables

Multiple variables - have to use vector equations and this gets complicated

56
Q

What is the Runge-Kutta method and why do we care about it?

A

Extension of Euler’s method with smaller error

Gets better estimates!

57
Q

Second order DEs

A 
Rewrite it as a first order vector DE
Solve for a condensed 1st order DE
Solve again when we replace the solution with original definitions
RK methods are better and faster here
This is called reducible
58
Q

Concerns with solving second order DEs?

A

Energy conservation and time reversal
Reducible method might not conserve energy
Meaning/interpretation of solutions could be lost

59
Q

Leapfrog method

A

Apply RK2 but conserve energy
Instead of calculating midpoint at each t+h, we use first midpoint and add h to base future midpoints off the first one
This is now time reversible! Aka energy is conserved

PYTHON PHYS (4 decks)

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