def problem5():
logging.basicConfig(filename='problem5.log', level=logging.DEBUG)
logging.info('Starting problem 5')
lmin = 13
lmax = 13
matrix = [my_toeplitz(2**l) for l in range(lmin, lmax + 1)]
b = [ones(2**l) for l in range(lmin, lmax + 1)]
x0 = [zeros(2**l) for l in range(lmin, lmax + 1)]
preconditioner = [my_preconditioner(2**l) for l in range(lmin, lmax + 1)]
max_iterations = 1000
chotime = process_time()
L = cho_factor(matrix[0], lower=True)
x = cho_solve(L, b[0])
print('Dimension {}: Cholesky method in {}s'.format(
2**13,
process_time() - chotime))
pcg_time = process_time()
result_preconditioned_conjugate_gradient = (
preconditioned_conjugate_gradient(matrix[0], x0[0], b[0],
preconditioner[0], max_iterations))
pcg_time = process_time() - pcg_time
print('Dimension {}: PCG {} iterations in {}s'.format(
2**13, result_preconditioned_conjugate_gradient[1], pcg_time))
logging.info('Ending problem 5')
def problem4():
logging.basicConfig(filename='problem4.log', level=logging.DEBUG)
logging.info('Starting problem 4')
lmin = 10
lmax = 13
matrix = [my_toeplitz(2**l) for l in range(lmin, lmax + 1)]
preconditioner = [my_preconditioner(2**l) for l in range(lmin, lmax + 1)]
precond_matrix = [
dot(inv(preconditioner[i]), matrix[i]) for i in range(len(matrix))
]
matrix_eig = [eigvals(m) for m in matrix]
precond_matrix_eig = [eigvals(m) for m in precond_matrix]
matrix_cond = [
max(matrix_eig[i]) / min(matrix_eig[i]) for i in range(len(matrix_eig))
]
precond_matrix_cond = [
max(precond_matrix_eig[i]) / min(precond_matrix_eig[i])
for i in range(len(precond_matrix_eig))
]
logging.info('Ending problem 4')
mateig = plt.plot([2**i for i in range(lmin, lmax + 1)],
matrix_cond,
label='CondT')
pmateig = plt.plot([2**i for i in range(lmin, lmax + 1)],
precond_matrix_cond,
label='CondPT')
plt.xlabel('Matrix size (n)')
plt.ylabel('Condition')
mateig_line = mpatches.Patch(color='blue', label='$T_n$')
pmateig_line = mpatches.Patch(color='green', label='$P_n^{-1}T_n$')
plt.legend(handles=[mateig_line, pmateig_line], loc=0)
plt.savefig('4condition')
plt.close()
mateig = plt.plot([2**i for i in range(lmin, lmax + 1)],
log10(matrix_cond),
label='CondT')
pmateig = plt.plot([2**i for i in range(lmin, lmax + 1)],
log10(precond_matrix_cond),
label='CondPT')
plt.xlabel('Matrix size (n)')
plt.ylabel('$Log_{10}(Condition)$')
mateig_line = mpatches.Patch(color='blue', label='$T_n$')
pmateig_line = mpatches.Patch(color='green', label='$P_n^{-1}T_n$')
plt.legend(handles=[mateig_line, pmateig_line], loc=0)
plt.savefig('4logcondition')
def problem2():
logging.basicConfig(filename='problem2.log', level=logging.DEBUG)
logging.info('Starting problem 2')
lmin = 9
lmax = 13
matrix = [my_toeplitz(2**l) for l in range(lmin, lmax + 1)]
b = [ones(2**l) for l in range(lmin, lmax + 1)]
x0 = [zeros(2**l) for l in range(lmin, lmax + 1)]
max_iterations = 1000
gd_time = []
cg_time = []
result_conjugate_gradient = []
result_gradient_descent = []
for i in range(len(matrix)):
logging.info('Starting caclulation for matrix of size {}'.format(i))
cg_time.append(process_time())
result_conjugate_gradient.append(
conjugate_gradient(matrix[i], x0[i], b[i], max_iterations))
cg_time[-1] = process_time() - cg_time[-1]
gd_time.append(process_time())
result_gradient_descent.append(
gradient_descent(matrix[i], x0[i], b[i], max_iterations))
gd_time[-1] = process_time() - gd_time[-1]
logging.info('Ending calculation for matrix of size {}'.format(i))
logging.info('Ending problem 2')
for i in range(lmin, lmax + 1):
print('Dimension {}: GD {} iterations in {}s, CG {} iterations in {}s'.
format(2**i, result_gradient_descent[i - lmin][1],
gd_time[i - lmin], result_conjugate_gradient[i - lmin][1],
cg_time[i - lmin]))
gd_plot = plt.plot([2**i for i in range(lmin, lmax + 1)],
log10(gd_time),
label='Gradient Descent')
cg_plot = plt.plot([2**i for i in range(lmin, lmax + 1)],
log10(cg_time),
label='Conjugate Gradient')
plt.xlabel('Matrix size (n)')
plt.ylabel('$Log_{10}(time (s))$')
gd_line = mpatches.Patch(color='blue', label='Gradient Descent')
cg_line = mpatches.Patch(color='green', label='Conjugate Gradient')
plt.legend(handles=[gd_line, cg_line], loc=0)
plt.savefig('2logtime')
plt.close()
gd_plot = plt.plot([2**i for i in range(lmin, lmax + 1)],
gd_time,
label='Gradient Descent')
cg_plot = plt.plot([2**i for i in range(lmin, lmax + 1)],
cg_time,
label='Conjugate Gradient')
plt.xlabel('Matrix size (n)')
plt.ylabel('Time (s)')
gd_line = mpatches.Patch(color='blue', label='Gradient Descent')
cg_line = mpatches.Patch(color='green', label='Conjugate Gradient')
plt.legend(handles=[gd_line, cg_line], loc=0)
plt.savefig('2time')
plt.close()
gd_plot = plt.plot([2**i for i in range(lmin, lmax + 1)],
log10([
result_gradient_descent[i][1]
for i in range(len(result_gradient_descent))
]),
label='Gradient Descent')
cg_plot = plt.plot([2**i for i in range(lmin, lmax + 1)],
log10([
result_conjugate_gradient[i][1]
for i in range(len(result_conjugate_gradient))
]),
label='Conjugate Gradient')
plt.xlabel('Matrix size (n)')
plt.ylabel('$Log_{10}(iterations)$')
gd_line = mpatches.Patch(color='blue', label='Gradient Descent')
cg_line = mpatches.Patch(color='green', label='Conjugate Gradient')
plt.legend(handles=[gd_line, cg_line], loc=0)
plt.savefig('2logiterations')
plt.close()
gd_plot = plt.plot([2**i for i in range(lmin, lmax + 1)], [
result_gradient_descent[i][1]
for i in range(len(result_gradient_descent))
],
label='Gradient Descent')
cg_plot = plt.plot([2**i for i in range(lmin, lmax + 1)], [
result_conjugate_gradient[i][1]
for i in range(len(result_conjugate_gradient))
],
label='Conjugate Gradient')
plt.xlabel('Matrix size (n)')
plt.ylabel('Iterations')
gd_line = mpatches.Patch(color='blue', label='Gradient Descent')
cg_line = mpatches.Patch(color='green', label='Conjugate Gradient')
plt.legend(handles=[gd_line, cg_line], loc=0)
plt.savefig('2iterations')
def problem3():
logging.basicConfig(filename='problem3.log', level=logging.DEBUG)
logging.info('Starting problem 3')
lmin = 6
lmax = 10
matrix = [my_toeplitz(2**l) for l in range(lmin, lmax + 1)]
b = [ones(2**l) for l in range(lmin, lmax + 1)]
x0 = [zeros(2**l) for l in range(lmin, lmax + 1)]
preconditioner = [my_preconditioner(2**l) for l in range(lmin, lmax + 1)]
max_iterations = 1000
pcg_time = []
result_preconditioned_conjugate_gradient = []
for i in range(len(matrix)):
logging.info('Starting caclulation for matrix of size {}'.format(i))
pcg_time.append(process_time())
result_preconditioned_conjugate_gradient.append(
preconditioned_conjugate_gradient(matrix[i], x0[i], b[i],
preconditioner[i],
max_iterations))
pcg_time[-1] = process_time() - pcg_time[-1]
logging.info('Ending calculation for matrix of size {}'.format(i))
logging.info('Ending problem 3')
for i in range(lmin, lmax + 1):
print('Dimension {}: PCG {} iterations in {}s'.format(
2**i, result_preconditioned_conjugate_gradient[i - lmin][1],
pcg_time[i - lmin]))
pcg_plot = plt.plot([2**i for i in range(lmin, lmax + 1)],
log10(pcg_time),
label='Preconditioned Conjugate Gradient',
color='red')
plt.xlabel('Matrix size (n)')
plt.ylabel('$Log_{10}(time (s))$')
pcg_line = mpatches.Patch(color='red',
label='Preconditioned Conjugate Gradient')
plt.legend(handles=[pcg_line], loc=0)
plt.savefig('3logtime')
plt.close()
pcg_plot = plt.plot([2**i for i in range(lmin, lmax + 1)],
pcg_time,
label='Preconditioned Conjugate Gradient',
color='red')
plt.xlabel('Matrix size (n)')
plt.ylabel('Time (s)')
pcg_line = mpatches.Patch(color='red',
label='Preconditioned Conjugate Gradient')
plt.legend(handles=[pcg_line], loc=0)
plt.savefig('3time')
plt.close()
pcg_plot = plt.plot(
[2**i for i in range(lmin, lmax + 1)],
log10([
result_preconditioned_conjugate_gradient[i][1]
for i in range(len(result_preconditioned_conjugate_gradient))
]),
label='Preconditioned Conjugate Gradient',
color='red')
plt.xlabel('Matrix size (n)')
plt.ylabel('$Log_{10}(iterations)$')
pcg_line = mpatches.Patch(color='red',
label='Preconditioned Conjugate Gradient')
plt.legend(handles=[pcg_line], loc=0)
plt.savefig('3logiterations')
plt.close()
pcg_plot = plt.plot([2**i for i in range(lmin, lmax + 1)], [
result_preconditioned_conjugate_gradient[i][1]
for i in range(len(result_preconditioned_conjugate_gradient))
],
label='Conjugate Gradient',
color='red')
plt.xlabel('Matrix size (n)')
plt.ylabel('Iterations')
pcg_line = mpatches.Patch(color='red',
label='Preconditioned Conjugate Gradient')
plt.legend(handles=[pcg_line], loc=0)
plt.savefig('3iterations')
from numpy import flipud, insert, append, zeros, multiply, split
from numpy.fft import rfft, irfft
def precompute_g(T):
return rfft(append(T[:, 0], insert(flipud(T[0, 1:]), 0, 0)))
def fftmul(g, x):
h = multiply(g, rfft(append(x, zeros(len(x)))))
return split(irfft(h), 2)[0]
if __name__ == '__main__':
from numpy import ones
from project3.my_toeplitz import my_toeplitz
fftmul(my_toeplitz(5), ones(5))