- 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
- 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.
- 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 -= 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).
N = 1000 L = 10.0 x = random.random((N,3))*L print get_rdf(x, L, 20)
Rodrigues' formula code. This is for completeness, so you can generate 3D rotations given in axis-angle notation. It is definitely not for memorization.
- 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*n*(1.0-c) trans[0,1] = n*n*(1.0-c) - n*s trans[0,2] = n*s + n*n*(1.0-c)
trans[1,0] = n*s + n*n*(1.0-c) trans[1,1] = c + n*n*(1.0-c) trans[1,2] = -n*s + n*n*(1.0-c)
trans[2,0] = -n*s + n*n*(1.0-c) trans[2,1] = n*s + n*n*(1.0-c) trans[2,2] = c + n*n*(1.0-c)
- 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