CompSciWeek11

Reading Assignment

• Numerical Recipes Pages 37-40
• Feynman Learning Technique
  • Try and identify the most challenging missing concept and explain it to someone else!
• Optional Additional reference: Basis and Singular Value lectures from codingthematrix.com

Class 1

• Tensor Manipulations - distance distributions in random point set exercise
  • pdist
  • histogram
• Rodrigues' Rotation Formula - the preferred method when you have to work with an angle.
• Matrix Multiplication
  • Basic linear algebra operations
  • Timing Strassen (see also Winograd)

Distribution Function Code

This illustrates dramatic simplifications that can be obtained by using tensors in numpy.

<source lang="python">

1. Radial distribution function calculator for un-scaled distributions.
2. (g(r) is get_rdf() * L**3/N and goes to 1).

from numpy import *

1. Get the unnormalized rdf in an isotropic cubic box of side length L
2. return unit is particles per L's length unit^3
3. Note that grid methods should be used for very large N.

def get_rdf(x, L, M):

   D2 = x - x[:,newaxis]
   D2 -= L*floor(D2/L+0.5) # wrap into box
   r2 = sqrt(sum(D2*D2, -1))
   s = arange(M+1)*0.5*L/M
   h, _ = histogram(r2, s)
   h[0] -= len(x) # remove self-distances
   # Divide by volume (times an extra (N-1), since we counted
   #                   N*(N-1) distances, but the density scales as N)
   h = h.astype(float)
   h /= (len(x)-1) * 4*pi/3.0*(s[1:]**3 - s[:-1]**3)
   return h

1. This should give a uniform RDF of 1000 pt / L**3 (= 1 here).

def test():

   N = 1000
   L = 10.0
   x = random.random((N,3))*L
   print get_rdf(x, L, 20)

</source>

Rodrigues' formula code. This is for completeness, so you can generate 3D rotations given in axis-angle notation. It is definitely not for memorization.

<source lang="python">

1. Build rotation matrix to rotate about an arbitrary vector using the right-hand rule.
2. Uses Rodrigues' rotation formula (in the quaternion representation).

def build_rotation(u, theta):

       n = u
       m = sum(n*n) # check that n is normalized
       if(m < 1.0-1.0e-10 or m > 1.0+1.0e-10):
               n /= sqrt(m)

       trans = zeros((3,3), float)

       s = sin(theta)
       c = cos(theta)
       trans[0,0] = c + n[0]*n[0]*(1.0-c)
       trans[0,1] = n[0]*n[1]*(1.0-c) - n[2]*s
       trans[0,2] = n[1]*s + n[0]*n[2]*(1.0-c)

       trans[1,0] = n[2]*s + n[0]*n[1]*(1.0-c)
       trans[1,1] = c + n[1]*n[1]*(1.0-c)
       trans[1,2] = -n[0]*s + n[1]*n[2]*(1.0-c)

       trans[2,0] = -n[1]*s + n[0]*n[2]*(1.0-c)
       trans[2,1] = n[0]*s + n[1]*n[2]*(1.0-c)
       trans[2,2] = c + n[2]*n[2]*(1.0-c)

       return trans

</source>

Class 2

• Implicit solutions to linear algebraic equations
  • Least-squares fitting
  • Repetition for point sets - project out and re-fit
  • Review projection and orthogonality (see also Gram-Schmidt)
  • Solution by factorization + forward / reverse substitution
• Minimization the hard way - nonlinear problems Optimize
  • Transcendental problem, 5th order polynomials, etc.
  • Geodesic paths via numerical optimization