def dramadah(n, k=1):
"""
dramadah a (0,1) matrix whose inverse has large integer entries.
An anti-hadamard matrix a is a matrix with elements 0 or 1 for
which mu(a) := norm(inv(a),'fro') is maximal.
a = dramadah(n, k) is an n-by-n (0,1) matrix for which mu(a) is
relatively large, although not necessarily maximal.
Available types (the default is k = 1):
k = 1: a is toeplitz, with abs(det(a)) = 1, and mu(a) > c(1.75)^n,
where c is a constant.
k = 2: a is upper triangular and toeplitz.
the inverses of both types have integer entries.
Another interesting (0,1) matrix:
K = 3: A has maximal determinant among (0,1) lower Hessenberg
matrices: det(A) = the n'th Fibonacci number. A is Toeplitz.
The eigenvalues have an interesting distribution in the complex
plane.
References:
R.L. Graham and N.J.A. Sloane, Anti-Hadamard matrices,
Linear Algebra and Appl., 62 (1984), pp. 113-137.
L. Ching, The maximum determinant of an nxn lower Hessenberg
(0,1) matrix, Linear Algebra and Appl., 183 (1993), pp. 147-153.
"""
if k == 1:
# Toeplitz
c = np.ones(n)
for i in range(1, n, 4):
m = min(1, n - i)
c[i:i + m + 1] = 0
r = np.zeros(n)
r[0:4] = np.array([1, 1, 0, 1])
if n < 4:
r = r[0:n]
a = rogues.toeplitz(c, r)
elif k == 2:
# Upper triangular and Toeplitz
c = np.zeros(n)
c[0] = 1
r = np.ones(n)
r[2::2] = 0
a = rogues.toeplitz(c, r)
elif k == 3:
# Lower Hessenberg.
c = np.ones(n)
c[1::2] = 0
a = rogues.toeplitz(c, np.r_['1', np.array([1, 1]), np.zeros(n - 2)])
return a
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_toeplitz():
"""
Simple test of toeplitz. This just checks that the
known values of determinant for a simple form at a
succession of matrix sizes
"""
dets = []
for i in range(3, 30):
a = np.arange(1, i)
b = rogues.toeplitz(a)
dets.append(nl.det(b))
dets = np.array(dets)
# Here are the known values of the determinants
ans = np.array([(-1) ** (i - 1) * (i + 1) * (2 ** (i - 2)) for i in \
range(2, 29)])
# pretty bad round off for some values
npt.assert_array_almost_equal(dets, ans, 5)
