# CompSciWeek11

From Predictive Chemistry

# 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
- 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">

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

from numpy import *

- Get the unnormalized rdf in an isotropic cubic box of side length L
- return unit is particles per L's length unit^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

- 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">

- Build rotation matrix to rotate about an arbitrary vector using the right-hand rule.
- 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