import numpy as np
import numpy.random as random
[docs]def get_A100():
"""
Return A100 matrix for investigating QR factoration.
:return A: The 100x100 numpy array
"""
m = 100
random.seed(1111*m)
A = random.randn(m, m) + 1j*random.randn(m, m)
return A
[docs]def get_B100():
"""
Return B100 matrix for investigating QR factoration.
:return A: The 100x100 numpy array
"""
m = 100
random.seed(1111*m)
A = random.randn(m, m) + 1j*random.randn(m, m)
A[np.tril_indices(m, -2)] = 0
return A
[docs]def get_C100():
"""
Return C100 matrix for investigating QR factoration.
:return A: The 100x100 numpy array
"""
m = 100
random.seed(1111*m)
A = random.randn(m, m) + 1j*random.randn(m, m)
A = 0.5*(A + np.conj(A).T)
return A
[docs]def get_D100():
"""
Return D100 matrix for investigating QR factoration.
:return A: The 100x100 numpy array
"""
m = 100
random.seed(1111*m)
A = random.randn(m, m) + 1j*random.randn(m, m)
A = 0.5*(A + np.conj(A).T)
A[np.tril_indices(m, -2)] = 0
A[np.triu_indices(m, 2)] = 0
return A
[docs]def get_A3():
"""
Return A3 matrix for investigating power iteration.
:return A3: a 3x3 numpy array.
"""
return np.array([[ 0.76505141, -0.03865876, 0.42107996],
[-0.03865876, 0.20264378, -0.02824925],
[ 0.42107996, -0.02824925, 0.23330481]])
[docs]def get_B3():
"""
Return B3 matrix for investigating power iteration.
:return B3: a 3x3 numpy array.
"""
return np.array([[ 0.76861909, 0.01464606, 0.42118629],
[ 0.01464606, 0.99907192, -0.02666057],
[ 0.42118629, -0.02666057, 0.23330798]])
[docs]def pow_it(A, x0, tol, maxit, store_iterations = False):
"""
For a matrix A, apply the power iteration algorithm with initial
guess x0, until either
||r|| < tol where
r = Ax - lambda*x,
or the number of iterations exceeds maxit.
:param A: an mxm numpy array
:param x0: the starting vector for the power iteration
:param tol: a positive float, the tolerance
:param maxit: integer, max number of iterations
:param store_iterations: if True, then return the entire sequence \
of power iterates, instead of just the final iteration. Default is \
False.
:return x: an m dimensional numpy array containing the final iterate, or \
if store_iterations, an mxmaxit dimensional numpy array containing all \
the iterates.
:return lambda0: the final eigenvalue.
"""
raise NotImplementedError
return x, lambda0
[docs]def inverse_it(A, x0, mu, tol, maxit, store_iterations = False):
"""
For a Hermitian matrix A, apply the inverse iteration algorithm
with initial guess x0, using the same termination criteria as
for pow_it.
:param A: an mxm numpy array
:param mu: a floating point number, the shift parameter
:param x0: the starting vector for the power iteration
:param tol: a positive float, the tolerance
:param maxit: integer, max number of iterations
:param store_iterations: if True, then return the entire sequence \
of inverse iterates, instead of just the final iteration. Default is \
False.
:return x: an m dimensional numpy array containing the final iterate, or \
if store_iterations, an mxmaxit dimensional numpy array containing \
all the iterates.
:return l: a floating point number containing the final eigenvalue \
estimate, or if store_iterations, a maxit dimensional numpy array containing \
all the iterates.
"""
raise NotImplementedError
[docs]def rq_it(A, x0, tol, maxit, store_iterations = False):
"""
For a Hermitian matrix A, apply the Rayleigh quotient algorithm
with initial guess x0, using the same termination criteria as
for pow_it.
:param A: an mxm numpy array
:param x0: the starting vector for the power iteration
:param tol: a positive float, the tolerance
:param maxit: integer, max number of iterations
:param store_iterations: if True, then return the entire sequence \
of inverse iterates, instead of just the final iteration. Default is \
False.
:return x: an m dimensional numpy array containing the final iterate, or \
if store_iterations, an mxmaxit dimensional numpy array containing \
all the iterates.
:return l: a floating point number containing the final eigenvalue \
estimate, or if store_iterations, an m dimensional numpy array containing \
all the iterates.
"""
raise NotImplementedError
[docs]def pure_QR(A, maxit, tol):
"""
For matrix A, apply the QR algorithm and return the result.
:param A: an mxm numpy array
:param maxit: the maximum number of iterations
:param tol: termination tolerance
:return Ak: the result
"""
raise NotImplementedError
