def hadamard(n):
"""
HADAMARD Hadamard matrix.
HADAMARD(N) is a Hadamard matrix of order N, that is,
a matrix H with elements 1 or -1 such that H*H' = N*EYE(N).
An N-by-N Hadamard matrix with N>2 exists only if REM(N,4) = 0.
This function handles only the cases where N, N/12 or N/20
is a power of 2.
Reference:
S.W. Golomb and L.D. Baumert, The search for Hadamard matrices,
Amer. Math. Monthly, 70 (1963) pp. 12-17.
http://en.wikipedia.org/wiki/Hadamard_matrix
Weisstein, Eric W. "Hadamard Matrix." From MathWorld--
A Wolfram Web Resource. http://mathworld.wolfram.com/HadamardMatrix.html
"""
f, e = np.frexp(np.array([n, n / 12., n / 20.]))
try:
# If more than one condition is satified, this will always
# pick the first one.
k = [i for i in range(3) if (f == 0.5)[i] and (e > 0)[i]].pop()
except IndexError:
raise ValueError('N, N/12 or N/20 must be a power of 2.')
e = e[k] - 1
if k == 0: # N = 1 * 2^e;
h = np.array([1])
elif k == 1: # N = 12 * 2^e;
tp = rogues.toeplitz(np.array([-1, -1, 1, -1, -1, -1, 1, 1, 1, -1, 1]),
np.array([-1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1]))
h = np.vstack((np.ones((1, 12)), np.hstack((np.ones((11, 1)), tp))))
elif k == 2: # N = 20 * 2^e;
hk = rogues.hankel(
np.array([
-1, -1, 1, 1, -1, -1, -1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1, -1,
1
]),
np.array([
1, -1, -1, 1, 1, -1, -1, -1, -1, 1, -1, 1, -1, 1, 1, 1, 1, -1,
-1
]))
h = np.vstack((np.ones((1, 20)), np.hstack((np.ones((19, 1)), hk))))
# Kronecker product construction.
mh = -1 * h
for i in range(e):
ht = np.hstack((h, h))
hb = np.hstack((h, mh))
h = np.vstack((ht, hb))
mh = -1 * h
return h
def hadamard(n):
"""
HADAMARD Hadamard matrix.
HADAMARD(N) is a Hadamard matrix of order N, that is,
a matrix H with elements 1 or -1 such that H*H' = N*EYE(N).
An N-by-N Hadamard matrix with N>2 exists only if REM(N,4) = 0.
This function handles only the cases where N, N/12 or N/20
is a power of 2.
Reference:
S.W. Golomb and L.D. Baumert, The search for Hadamard matrices,
Amer. Math. Monthly, 70 (1963) pp. 12-17.
http://en.wikipedia.org/wiki/Hadamard_matrix
Weisstein, Eric W. "Hadamard Matrix." From MathWorld--
A Wolfram Web Resource:
http://mathworld.wolfram.com/HadamardMatrix.html
"""
f, e = np.frexp(np.array([n, n / 12., n / 20.]))
try:
# If more than one condition is satified, this will always
# pick the first one.
k = [i for i in range(3) if (f == 0.5)[i] and (e > 0)[i]].pop()
except IndexError:
raise ValueError('N, N/12 or N/20 must be a power of 2.')
e = e[k] - 1
if k == 0: # N = 1 * 2^e;
h = np.array([1])
elif k == 1: # N = 12 * 2^e;
tp = rogues.toeplitz(np.array([-1, -1, 1, -1, -1, -1, 1, 1, 1, -1, 1]),
np.array([-1, 1, -1, 1, 1, 1, -1, -1, -1, 1, -1]))
h = np.vstack((np.ones((1, 12)), np.hstack((np.ones((11, 1)), tp))))
elif k == 2: # N = 20 * 2^e;
hk = rogues.hankel(
np.array([-1, -1, 1, 1, -1, -1, -1, -1, 1,
-1, 1, -1, 1, 1, 1, 1, -1, -1, 1]),
np.array([1, -1, -1, 1, 1, -1, -1, -1, -1,
1, -1, 1, -1, 1, 1, 1, 1, -1, -1]))
h = np.vstack((np.ones((1, 20)), np.hstack((np.ones((19, 1)), hk))))
# Kronecker product construction.
mh = -1 * h
for i in range(e):
ht = np.hstack((h, h))
hb = np.hstack((h, mh))
h = np.vstack((ht, hb))
mh = -1 * h
return h
def test_hankel():
"""Simple test of Hankel matrix"""
a = np.arange(10)
b = np.arange(10, 20)
h = rogues.hankel(a, b)
t = np. array([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
[1, 2, 3, 4, 5, 6, 7, 8, 9, 11],
[2, 3, 4, 5, 6, 7, 8, 9, 11, 12],
[3, 4, 5, 6, 7, 8, 9, 11, 12, 13],
[4, 5, 6, 7, 8, 9, 11, 12, 13, 14],
[5, 6, 7, 8, 9, 11, 12, 13, 14, 15],
[6, 7, 8, 9, 11, 12, 13, 14, 15, 16],
[7, 8, 9, 11, 12, 13, 14, 15, 16, 17],
[8, 9, 11, 12, 13, 14, 15, 16, 17, 18],
[9, 11, 12, 13, 14, 15, 16, 17, 18, 19]])
npt.assert_array_equal(h, t)
def test_hankel():
"""Simple test of Hankel matrix"""
a = np.arange(10)
b = np.arange(10, 20)
h = rogues.hankel(a, b)
t = np.array([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9],
[1, 2, 3, 4, 5, 6, 7, 8, 9, 11],
[2, 3, 4, 5, 6, 7, 8, 9, 11, 12],
[3, 4, 5, 6, 7, 8, 9, 11, 12, 13],
[4, 5, 6, 7, 8, 9, 11, 12, 13, 14],
[5, 6, 7, 8, 9, 11, 12, 13, 14, 15],
[6, 7, 8, 9, 11, 12, 13, 14, 15, 16],
[7, 8, 9, 11, 12, 13, 14, 15, 16, 17],
[8, 9, 11, 12, 13, 14, 15, 16, 17, 18],
[9, 11, 12, 13, 14, 15, 16, 17, 18, 19]])
npt.assert_array_equal(h, t)
