Dense Cholesky update in Python

Question

Could anyone point me to a library/code allowing me to perform low-rank updates on a Cholesky decomposition in python (numpy)? Matlab offers this functionality as a function called 'cholupdate'. LINPACK also has this functionality, but it has (to my knowledge) not yet been ported to LAPACK and hence isn't available in e.g. scipy.

I found out that scikits.sparse offers a similar function based on CHOLMOD, but my matrices are dense.

Is there any code available for python with 'cholupdate''s functionality that's compatible with numpy?

Thanks!

Answer 1

Here is a Python package that does rank 1 updates and downdates on Cholesky factors using Cython: https://github.com/jcrudy/choldate

Example:

from choldate import cholupdate, choldowndate
import numpy

#Create a random positive definite matrix, V
numpy.random.seed(1)
X = numpy.random.normal(size=(100,10))
V = numpy.dot(X.transpose(),X)

#Calculate the upper Cholesky factor, R
R = numpy.linalg.cholesky(V).transpose()

#Create a random update vector, u
u = numpy.random.normal(size=R.shape[0])

#Calculate the updated positive definite matrix, V1, and its Cholesky factor, R1
V1 = V + numpy.outer(u,u)
R1 = numpy.linalg.cholesky(V1).transpose()

#The following is equivalent to the above
R1_ = R.copy()
cholupdate(R1_,u.copy())
assert(numpy.all((R1 - R1_)**2 < 1e-16))

#And downdating is the inverse of updating
R_ = R1.copy()
choldowndate(R_,u.copy())
assert(numpy.all((R - R_)**2 < 1e-16))

Answer 2

This should do a rank-1 update or downdate on numpy arrays R and x with sign '+' or '-' corresponding to update or downdate. (Ported from MATLAB cholupdate at the Wikipedia page: http://en.wikipedia.org/wiki/Cholesky_decomposition):

def cholupdate(R,x,sign):
    import numpy as np
      p = np.size(x)
      x = x.T
      for k in range(p):
        if sign == '+':
          r = np.sqrt(R[k,k]**2 + x[k]**2)
        elif sign == '-':
          r = np.sqrt(R[k,k]**2 - x[k]**2)
        c = r/R[k,k]
        s = x[k]/R[k,k]
        R[k,k] = r
        if sign == '+':
          R[k,k+1:p] = (R[k,k+1:p] + s*x[k+1:p])/c
        elif sign == '-':
          R[k,k+1:p] = (R[k,k+1:p] - s*x[k+1:p])/c
        x[k+1:p]= c*x[k+1:p] - s*R[k, k+1:p]
      return R

Answer 3

This guy is doing something similar using scikits and numpy/scipy.