def wilk(n):
"""
wilk Various specific matrices devised/discussed by Wilkinson.
a, b = wilk(n) is the matrix or system of order N.
N = 3: upper triangular system Ux=b illustrating inaccurate solution.
N = 4: lower triangular system Lx=b, ill-conditioned.
N = 5: HILB(6)(1:5,2:6)*1.8144. Symmetric positive definite.
N = 21: W21+, tridiagonal. Eigenvalue problem.
References:
J.H. Wilkinson, Error analysis of direct methods of matrix inversion,
J. Assoc. Comput. Mach., 8 (1961), pp. 281-330.
J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied
Science No. 32, Her Majesty's Stationery Office, London, 1963.
J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
Press, 1965.
"""
b = []
if n == 3:
# Wilkinson (1961) p.323.
a = [ [1e-10, .9, -.4], \
[0, .9, -.4], \
[0, 0, 1e-10]]
b = [ 0, 0, 1]
elif n == 4:
# Wilkinson (1963) p.105.
a = [[0.9143e-4, 0, 0, 0], \
[0.8762, 0.7156e-4, 0, 0], \
[0.7943, 0.8143, 0.9504e-4, 0], \
[0.8017, 0.6123, 0.7165, 0.7123e-4]]
b = [0.6524, 0.3127, 0.4186, 0.7853]
elif n == 5:
# Wilkinson (1965), p.234.
a = hilb(6, 6)
# drop off the last row and the first column
a = a[0:5, 1:6] * 1.8144
# return zero array for b
b = np.zeros(5)
elif n == 21:
# Taken from gallery.m. Wilkinson (1965), p.308.
E = np.diag(np.ones(n - 1), 1)
m = (n - 1) / 2
a = np.diag(np.abs(np.arange(-m, m + 1))) + E + E.T
# return zero array for b
b = np.zeros(21)
else:
raise ValueError("Sorry, that value of N is not available.")
return np.array(a), np.array(b)
def wilk(n):
"""
wilk Various specific matrices devised/discussed by Wilkinson.
a, b = wilk(n) is the matrix or system of order N.
N = 3: upper triangular system Ux=b illustrating inaccurate solution.
N = 4: lower triangular system Lx=b, ill-conditioned.
N = 5: HILB(6)(1:5,2:6)*1.8144. Symmetric positive definite.
N = 21: W21+, tridiagonal. Eigenvalue problem.
References:
J.H. Wilkinson, Error analysis of direct methods of matrix inversion,
J. Assoc. Comput. Mach., 8 (1961), pp. 281-330.
J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied
Science No. 32, Her Majesty's Stationery Office, London, 1963.
J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University
Press, 1965.
"""
b = []
if n == 3:
# Wilkinson (1961) p.323.
a = [ [1e-10, .9, -.4], \
[0, .9, -.4], \
[0, 0, 1e-10]]
b = [0, 0, 1]
elif n == 4:
# Wilkinson (1963) p.105.
a = [[0.9143e-4, 0, 0, 0], \
[0.8762, 0.7156e-4, 0, 0], \
[0.7943, 0.8143, 0.9504e-4, 0], \
[0.8017, 0.6123, 0.7165, 0.7123e-4]]
b = [0.6524, 0.3127, 0.4186, 0.7853]
elif n == 5:
# Wilkinson (1965), p.234.
a = hilb(6, 6)
# drop off the last row and the first column
a = a[0:5, 1:6] * 1.8144
# return zero array for b
b = np.zeros(5)
elif n == 21:
# Taken from gallery.m. Wilkinson (1965), p.308.
E = np.diag(np.ones(n - 1), 1)
m = (n - 1) / 2
a = np.diag(np.abs(np.arange(-m, m + 1))) + E + E.T
# return zero array for b
b = np.zeros(21)
else:
raise ValueError("Sorry, that value of N is not available.")
return np.array(a), np.array(b)
def lotkin(n):
"""
lotkin lotkin matrix.
a = lotkin(n) is the Hilbert matrix with its first row altered to
all ones. A is unsymmetric, ill-conditioned, and has many negative
eigenvalues of small magnitude.
The inverse has integer entries and is known explicitly.
Reference:
M. Lotkin, A set of test matrices, MTAC, 9 (1955), pp. 153-161.
"""
a = hilb(n)
a[0, :] = np.ones(n)
return a
def invol(n):
"""
invol(n) an involutory matrix of order n.
a = invol(n) is an n-by-n involutory (a*a = eye(n)) and
ill-conditioned matrix.
It is a diagonally scaled version of hilb(n).
nb: b = (eye(n)-a)/2 and b = (eye(n)+a)/2 are idempotent (b*b = b).
Reference:
A.S. Householder and J.A. Carpenter, The singular values
of involutory and of idempotent matrices, Numer. Math. 5 (1963),
pp. 234-237.
"""
a = hilb(n)
d = -n
a[:, 0] = d * a[:, 0]
for i in range(n - 1):
d = -(n + i + 1) * (n - i - 1) * d / float((i + 1) * (i + 1))
a[i + 1, :] = d * a[i + 1, :]
return a