Generate smooth functions in numpy

smooth.py

import numpy as np

def covar1(m, s):
    s = float(s)
    r = np.arange(m, dtype=np.float)
    d = np.reshape(r, (m, 1)) - np.reshape(r, (1, m))
    k = np.exp(-(0.5/s**2) * (d * d))
    return k

def smooth1(m, s):
    k = covar1(m, s)
    w, v = sqrt(k)
    z = np.random.randn(m)
    # A = V diag(w) such that E(x x') = E(A z z' A') = A A' = K
    x = np.dot(v, w * z)
    return x

def smooth2(m, s):
    m0, m1 = m
    s = float(s)
    k0 = covar1(m0, s)
    k1 = covar1(m1, s)
    w0, v0 = sqrt(k0)
    w1, v1 = sqrt(k1)
    z = np.random.randn(m0, m1)
    wz = np.reshape(w0, (m0, 1)) * np.reshape(w1, (1, m1)) * z
    x = np.dot(np.dot(v0, wz), v1.T)
    return x

def sqrt(s):
    w, v = np.linalg.eigh(s)
    w = np.sqrt(np.maximum(0, w))
    return w, v