def chow(n, alpha=1, delta=0):
"""
chow(n, alpha, delta): chow matrix - a singular toeplitz
lower hessenberg matrix.
a = chow(n, alpha, delta) is a toeplitz lower hessenberg matrix
a = h(alpha) + delta*eye, where h(i,j) = alpha^(i-j+1).
h(alpha) has p = floor(n/2) zero eigenvalues, the rest being
4*alpha*cos( k*pi/(n+2) )^2, k=1:n-p.
defaults: alpha = 1, delta = 0.
References:
T.S. Chow, A class of Hessenberg matrices with known
eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395.
G. Fairweather, On the eigenvalues and eigenvectors of a class of
Hessenberg matrices, SIAM Review, 13 (1971), pp. 220-221.
"""
a = toeplitz(alpha ** np.arange(1, n + 1),
np.hstack((alpha, 1, np.zeros(n - 2)))) + delta * np.eye(n)
return a
def chow(n, alpha=1, delta=0):
"""
chow(n, alpha, delta): chow matrix - a singular toeplitz
lower hessenberg matrix.
a = chow(n, alpha, delta) is a toeplitz lower hessenberg matrix
a = h(alpha) + delta*eye, where h(i,j) = alpha^(i-j+1).
h(alpha) has p = floor(n/2) zero eigenvalues, the rest being
4*alpha*cos( k*pi/(n+2) )^2, k=1:n-p.
defaults: alpha = 1, delta = 0.
References:
T.S. Chow, A class of Hessenberg matrices with known
eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395.
G. Fairweather, On the eigenvalues and eigenvectors of a class of
Hessenberg matrices, SIAM Review, 13 (1971), pp. 220-221.
"""
a = toeplitz(alpha ** np.arange(1, n + 1), \
np.r_[alpha, 1, np.zeros(n - 2)]) + delta * np.eye(n)
return a
def prolate(n, w=0.25):
"""
PROLATE Prolate matrix - symmetric, ill-conditioned Toeplitz matrix.
A = PROLATE(N, W) is the N-by-N prolate matrix with parameter W.
It is a symmetric Toeplitz matrix.
If 0 < W < 0.5 then
- A is positive definite
- the eigenvalues of A are distinct, lie in (0, 1), and
tend to cluster around 0 and 1.
W defaults to 0.25.
Reference:
J.M. Varah. The Prolate matrix. Linear Algebra and Appl.,
187:269--278, 1993.
"""
a = np.zeros(n)
a[0] = 2 * w
a[1:n] = np.sin(2 * np.pi * w * np.arange(1, n)) / (np.pi *
np.arange(1, n))
t = toeplitz(a)
return t
def prolate(n, w=0.25):
"""
PROLATE Prolate matrix - symmetric, ill-conditioned Toeplitz matrix.
A = PROLATE(N, W) is the N-by-N prolate matrix with parameter W.
It is a symmetric Toeplitz matrix.
If 0 < W < 0.5 then
- A is positive definite
- the eigenvalues of A are distinct, lie in (0, 1), and
tend to cluster around 0 and 1.
W defaults to 0.25.
Reference:
J.M. Varah. The Prolate matrix. Linear Algebra and Appl.,
187:269--278, 1993.
"""
a = np.zeros(n)
a[0] = 2 * w
a[1:n] = np.sin(2 * np.pi * w * np.arange(1, n)) / \
(np.pi * np.arange(1, n))
t = toeplitz(a)
return t